Category Archives: Writing: Trying To Land A Jumbo Jet On A Postage Stamp

Epistemically Accessible Possible Worlds, Compatibility, Failure Of Bivalence, And Inconsistent Situations

Caravaggio’s painting the CARDSHARPS nicely encapsulates the structure of a situation, and illustrates the three-place relation Rstu. Situations and their structure I will be discussing extensively in this essay in order to arrive at a satisfactory statement of the truth conditions for negation. This statement will be needed to resolve the paradoxes of Classical Logic’s Material Implication, paradoxes which Strict Implication does not completely resolve. Rstu I will be discussing in a different essay

I start by attempting to define situations (as opposed to possible worlds) in terms of parts of possible worlds that are epistemically accessible to me from my situation in the actual world. This will let me show how A v ~A can be false for a situation. In other words, both A and ~ A are false for that situation, though not for a possible world. The situation therefore counts as non-bivalent. I will show this by way of a set of truth conditions for negation that rely on a concept of a compatibility relation between situations that relies on possible worlds that are epistemically accessible from the actual world for a sentient being S.

I will then add ‘ostensible objects’ to my account of situations. This will let me show how ‘A and not A‘ can be true for a situation, a situation that would therefore count as inconsistent. I will show this employing a concept of an incompatibility relation between situations that relies on (impossible?) worlds that are epistemically inaccessible from the actual world for a sentient being S.

Both non-bivalent and inconsistent situations are needed for Relevant Logic, a paraconsistent logic, in order to eliminate countless irrelevant implication propositions that would otherwise be true.

I am willing to ‘go there’, so to speak, because I will do practically anything (even becoming one of those squeegee car-window washers waiting to pounce on cars at intersections in the hopes of gaining a quarter) to avoid making the following true:

If Cliff lives in Houston, then the earth has at least one moon

If Paris, Texas is the capital of France, then Calypso music originated in Wisconsin

If I vote for Hillary Clinton, we will end up with a criminal president who is under investigation from day one

All of the above statements are true in Classical Logic but false in Relevant Logic.

Let me start, then.

Nota Bene: in what follows, I will be enclosing the names of words and phrases in double quotes, and the names of concepts, relations, and propositions within single quotes. I will use | to mean “alternatively”. The first occurrence of an important technical term will be in bolded italic.

Although the contents of this paragraph will probably seem a bit opaque to a reader who is not already familiar with these issues, I will list out anyway the points I will be making here. We can analyze ‘compatibility’/’incompatibility’ in terms of (parts of) epistemic possible worlds. We do not need to leave these concepts as primitives. Since, as it will turn out, situations have everything to do with knowledge/information, it should not surprise us that the concept ‘epistemic possible world’, as well as related epistemic concepts, should be useful for analyzing them. The method I employ in the following will itself also rely heavily on the concept of epistemic possible worlds. All of the above will be used to analyze a truth condition for negation that will allow for the existence of situations that fail to bivalent and that fail to be consistent. And ultimately, of course (though I will not be discussing this in detail here), the aim of doing that is to show how the above implications can be regarded as false.

Here is the crumb-trail for this disquisition. First I will be discussing possible worlds in general, then possible worlds that are epistemically accessible for an agent S from another possible world (especially the actual world). I will then discuss situations, which discussion will branch off into two streams. In one stream, I will discuss what in a situation (as opposed to a possible world) is truth-making for a sentence. In the other stream, I will discuss the binary compatible/incompatible relation among situations. These two streams will converge in a statement of the truth condition for negation, which is (spoiler alert) as follows:

A situation makes ~A true if and only if every situation compatible with it fails to make A true

RL p. 75

This truth condition for negation will then let me show how situations can fail to be bivalent and fail to be consistent. The discussion of inconsistent situations will include a discussion of ostensible objects.

But to get to this truth condition, I will need to discuss what it means to “make” a sentence true, and what it means to say one situation is compatible with another. And in turn, to get to those discussions, I will need to discuss possible worlds, and those parts of possible worlds called “situations”.

Here are the topics I will be discussing, in order: Possible Worlds Spawned-From-The-Known Possible Worlds Spawned-From-The-Unknown Possible Worlds Possible Worlds That Are Epistemically Accessible To Me From The Actual World Situations Epistemic Possibility Epistemic Truth Compatibility/Incompatibility Situations Drawn From Spawned-From-The-Unknown Possible Worlds Situations Drawn From Spawned-From-The-Known Possible Worlds Bivalent and Inconsistent Situations Making A True

Possible Worlds

At least for now — until I am forced to change — I will take a possible world to be defined by a set of natural-language descriptions, each one of which corresponds | fails to correspond to a state of affairs which, because it obtains | fails to obtain at a time t, is (to follow Chisholm) identical with a proposition. There is therefore a one-to-one relationship between a state of affairs and a proposition that obtains | fails to obtain at t. There is, of course, a many-to-one relationship between natural-language descriptions and a proposition aka state of affairs at t. “Umuulan” uttered in Manila as one holds an umbrella and “It is raining” also uttered in Manila while holding an umbrella are natural-language descriptions in Tagalog and English respectively which both express the proposition named by ‘it is raining’.

A sufficient condition for a state of affairs’ obtaining is its having a set of objects located in space-time and bearing the relevant relations to one another. For example, the state of affairs expressed by the sentence “Our Unix SA is standing to Genghis Khan’s right; and my brother Dan is standing to Leon Trotsky’s left who in turn is standing to Genghis Khan’s left” obtains if our Unix SA, Genghis Khan, Dan, and Leon Trotsky have a location in space-time at t and are spatially related to one another in the way just described.

The description that expresses the proposition are sentences in a natural language that are used to perform a speech act of describing. If my cat Tiger is now sitting on the mat in my apartment, that state of affairs obtains, and the description performed by uttering the English sentence “My cat Tiger is now sitting on the mat” expresses a proposition that is true because it corresponds to this state of affairs. In a derivative way, the description/sentence also corresponds to this state of affairs and is also true. If Tiger is not now sitting on this mat, the aforementioned state of affairs does not now obtain, and the aforementioned proposition does not correspond to anything in my apartment that would make it true. The proposition is false, as is the sentence expressing that proposition whose utterance performs the speech act of making a false description.

I will leave ‘state of affairs’ as an unanalyzed primitive, explicating it only by providing examples. ‘Corresponds’ I will also not say much about, except to say that it is a relation between a proposition and a state of affairs, and the relation might be, as Chisholm thinks, identity at a time t.

As one possible world among others (if something is actual, it is also possible), the actual world comprises that set of descriptions, each one of which expresses a proposition that corresponds to a state of affairs that obtains. In the actual world, the color of the walls in my apartment is a nice, subtle portobello mushroom. The description “my apartment has portobello-mushroom colored walls” corresponds to this particular state of affairs, this particular piece of the actual world. It is a member of the set of descriptions that defines the actual world. I will say that the description maps to the state of affairs. I conceive of states of affairs that obtain as akin to bricks building up the actual world.

In the actual world the color of the walls in my apartment is not a wild fuchsia. The description “fuchsia-colored walls” does not correspond to this particular piece of the actual world. That this state of affairs fails to obtain (i.e., my apartment walls being wild fuchsia) means that it is not part of the actual world. The actual world contains just the one portobello mushroom brick, so to speak, and not a fuchsia brick.

Spawned-from-the-known possible worlds. Through a kind of wave of the hand, I can specify a possible world distinct from the actual world by saying “everything is the same as in the actual world, except the color of the walls of my apartment is a wild fuchsia.” In doing this, I have just spawned, so to speak, or “accessed” a possible world (which, henceforth, I will call the “fuchsia-wall possible world”) by taking a description of a state of affairs I know to obtain in the actual world (“the walls of my apartment are a nice subtle portobello mushroom”), a state of affairs that is exposed to me, so to speak, and substituting for it a different description (“the walls of my apartment are a wild fuchsia”), a description which fails to map to any obtaining state of affairs in the actual world. I will call possible worlds accessed this way “spawned-from-the-known possible worlds.”

These possible worlds are defined by sets of descriptions some of which map to the states of affairs that “build up” so to speak the actual world and others which do not map to or correspond to any states of affairs obtaining in the actual world. When a description does correspond to a state of affairs obtaining at t in the actual world, I will say that the description is satisfied by the state of affairs.

A state of affairs is actual when the objects it comprises have a location in space time. For all I know, there may be more than one space-times. Here, however, I will assume there is just one, and that there is, therefore, just one actual world comprising all actual states of affairs. will assume that there is only one actual world. The actual world is one possible world among others, but it is the only one in which every description in the set of descriptions that defines it maps to the corresponding state of affairs which obtains.

If I may be allowed to indulge in wild metaphor for the moment, I will say that the fuchsia-wall possible world is composed of the same bricks as the actual world, with the exception that the portobello mushroom brick has been taken out and replaced with a written note saying “this is a fuchsia brick”.

I will call phrases such as “fuchsia walls” “key descriptions”, since these are the descriptions that, with the aid of the “everything else is the same” hand wave, are key to defining the fuchsia-wall possible world and opening up so to speak that world to me. It will be seen shortly that key descriptions are not limited to generating or “accessing” spawned-from-the-known possible worlds. When a key description is not satisfied by a state of affairs in the actual world, I will say that the content of that description is merely “posited“.

A positing may be actually realized by a person performing it, or it could be something potentially realized. An hour ago, for example, I had not posited a butterfly with black and white wings fluttering about in the Amazon at GPS location xyz, but I am now. An hour ago this particular possible world was just a potential one, contemplated perhaps by an infinite mind but by no one else. Compare this notion with that of a line in geometry that, potentially, is infinitely divisible, but each of whose segments is known by an infinite mind.

While I am still here, let me “access” (henceforth I will not be placing scare quotes around the word “access”) a couple more spawned-from-the-known possible worlds. I can say “everything is the same (even the color of my apartment walls) as in the actual world mutatis mutandis, except that water flows uphill.” Again, I have “generated” or accessed a possible world from a known fact in the actual world, namely, that water flows downhill. Or: “everything is the same as in the actual world, except that donkeys talk”. I have accessed a possible world by using a known fact in the actual world namely, that donkeys don’t talk. Naturally, the “mutatis mutandis” (would very much else really be the same if donkeys talked or water ran uphill?) makes me a bit uncomfortable, but I won’t try to deal here with the issues it may raise.

Spawned-from-the-unknown possible worlds. I can access a possible world, though not one necessarily distinct from the actual world, by using as my raw material a piece of the actual world about which I do not have knowledge. Suppose, for example, that, in the Pilsen neighborhood of Chicago, near Western Avenue, Elizarraraz (from whom I was renting living quarters and studio space for a while … “Elizarraraz” is Ladino, the Iberian Sephardic counterpart to the Ashkenazim Yiddish, for “poor king”; many people of Jewish descent managed to make their way to Mexico, from which country Elizarraraz’ paternal ancestors hail, in order to place as much distance as possible between themselves and the Spanish Inquisition … but I digress) . . . as I was saying, suppose that Elizarraraz has set up a shell game comprising three shells under one of which lies a peanut. I do not know, of course, under which shell the peanut lies hidden.

Might this be the infamous rogue peanut?

I can generate three possible worlds from these three unknowns by filling in the “holes”, so to speak, with key descriptions taking the form “the peanut lies under shell #x”. In one possible world, the peanut lies under shell #1 and everything else is the same as in the actual world; in the other it lies under shell #2 and everything else is the same as in the actual world; in the third it lies under shell #3 and everything else is the same as in the actual world.

Possible Worlds That Are Epistemically Accessible To Me From The Actual World: I do not know anything that would rule out the identity of any of these possible worlds with the actual world. Following the literature, I will say that these are possible worlds that are epistemically accessible for me from my location in the actual world.

We can talk about the relation of espistemic [sic] accessibility. A world w0 is epistemically accessible from w for an agent S (in w) iff S knows nothing that would rule out the hypothesis that w0 = w. Then, p is epistemically necessary (for S) iff p is true at all possible worlds that are epistemically accessible from w (for S).

http://fitelson.org/125/accessibility.pdf

I modify the above to the following:

A world w0 is epistemically accessible from w for an agent S (in w) iff nothing in the body of knowledge possessed by S would rule out the hypothesis that w0 = w.

I want to talk about ‘a body of knowledge’ to segue more easily into a discussion of probability which I will be doing shortly. According to Henry Kyburg’s account of probability, which I find useful, a probability is always asserted relative to a body of knowledge or evidence:

My own conception of probability is, like Keynes’s and Carnap’s, a logical one. Probability statements are logically true, if they are true at all. A probability is asserted always relative to a body of evidence; relative to different bodies of evidence different probability statements will be true. … I do not want to regard this body of knowledge as fixed in any absolute sense; over a period of time, however brief it changes. But in order to clarify the process of inference we introduce the logician’s standard idealization and regard the inference as taking place at a timeless moment. In this moment there are statements are open to question and statements that are not open to question — though in another moment, or from another point of view, they could of course be questioned.

Henry E. Kyburg Jr., EPISTEMOLOGY AND INFERENCE (Minneapolis, University of Minnesota Press, 1983), p. 137

Adding to this account the claim that one does not necessarily know or even believe they know something that the they actually do know will save me a fair amount of agony and grief. A skeptic, for example might persuade me that I do not know that I am awake and not dreaming when in fact I do know that I am awake and not dreaming. Someone may be told by someone who completely convinces them for a moment “Who are you going to believe: me or your lying eyes?” [[It is possible to hold conflicting beliefs at the same time.]]

The actual world is, of course, a possible world that is epistemically accessible for me and to everyone existing in it because I, along with everyone else, know nothing that would rule out the identity of the actual world with the actual world. Since the actual world is my primary concern here, I will assume that the possible world w that S is located in and from which they access other possible worlds, is the actual world. I will sometimes replace ‘S knows nothing that would rule out the hypothesis that’ with ‘so far as S knows’ or ‘to the extent of S‘s knowledge’.

Not every spawned-from-the-unknown possible world is a possible world epistemically accessible to me from my location in the actual world. I could, for example, also generate a possible world from the three unknowns just mentioned by positing a peanut that is always, of its own accord, switching between shells and sometimes even finding itself on the nose of the Mona Lisa, mimicking thereby the possible behavior of a rogue electron as described by a chemistry TA I once had as an undergraduate. But we know things (I think … I hope) that rule out the identity with the actual world of the possible world in which peanuts behave this way: namely, that peanuts do not behave this way in the actual world. Although electrons may behave this way (if I can trust my memory of what my undergraduate chemistry TA told me), peanuts do not jump from shell to shell and they never find themselves of their own accord on the nose of the Mona Lisa. This possible world is not epistemically accessible for me from my location in the actual world.

No peanut here — so far

Possible worlds spawned from unknown pieces of the actual world, then, fall into two categories: those that are epistemically accessible for a knower S from S‘s location in the actual world (every now and then I will be referring to these just as “epistemically accessible possible worlds”), and those that are not.

The possible world w0 in which Bigfoot is roaming the forests of western Washington state is epistemically accessible for me because I know nothing that would absolutely rule out the identity of this possible world with the actual world. Likewise, the possible world w1 in which Bigfoot is not roaming the forests of western Washington state is also epistemically accessible for me because I know nothing that would rule out the identity with the actual world of this possible world. I think the chances are greater than 50% that Bigfoot is not roaming these forests, but that is not the same as my knowing this.

These two possible worlds are incompatible; they cannot be the same possible world. The one world precludes the other. One makes ‘Bigfoot is roaming…’ (A) true; the other makes ‘Bigfoot is not roaming…’ (~A) true. A possible world answers every question: for any given possible world, either Bigfoot is roaming in the possible world or she is not. One must be true but not the other. Both cannot be true. So in all possible worlds A ^ ~A is false. A possible world cannot be inconsistent.

(At least for now, I won’t try to deal with possible counter-examples to this assertion. One such possible counterexample is Graham Priest’s contention that a person walking out of a room is, for one moment, both inside the room and outside the room when the center of their gravity aligns with the center of the door space. I am tempted to think that ‘inside the room’ and ‘outside the room’ are two different situations which have a common boundary, with the result that the description “outside the room” is not always the negation of “inside the room”. But I won’t try to expand on this now. At the moment I will simply assume that in all possible worlds A ^ ~A is false.)

Which possible world we are talking about hinges on how it answers the Bigfoot question (A or not A?). The identities of w0 and w1 depend upon this answer. The identity of possible world w0 depends upon that world’s not being possible world w1. If I exist in a possible world in which A is true at a time t then all of a sudden per impossibile I am existing in a possible world in which A is not true at this same time t, then I have been transported to a different possible world. (Maybe Scottie beamed me into it.)

And one of them (at any given time), A or not A, must hold in w0 and w1. In neither world can neither ‘Bigfoot is roaming the forests of Western Washington State’ nor ‘Bigfoot is not roaming the forests of Western Washington State’ be true. The worlds must be bivalent. One or the other — at least one and at most one — A or not A, must be true for these possible worlds to be the possible worlds — mutually exclusive worlds — they are. To remove A from w0 would be to strip that world of its identity; likewise, to remove not A from w1 would be to strip that world of its identity. Neither w0 nor w1 can be a non-bivalent world. In all possible worlds, ‘A v not A‘ is true. ‘A v not A‘ is necessarily true, which means that ‘If A then A‘ is also necessarily true.

So the falsity of ‘A ^ ~A‘ and the truth of ‘A v ~A‘ stems from the criteria for the identity of possible worlds. Possible worlds have to be complete, answering every question (Is there, or is there not, a butterfly with violet wings fluttering about in the Amazon at GPS point xyz?). They have to be consistent. They must be bivalent. They would undergo serious ‘criteria of identity’ crises were these conditions not fulfilled.

To sum up this discussion of possible worlds: possible worlds fall into two categories: those possible worlds that are spawned from the known and those that are spawned from the unknown. In turn, spawned-from-the unknown possible worlds fall into two categories: possible worlds that are epistemically accessible to a sentient being S and those that are not so accessible. Unlike situations, to which I am about to turn, possible worlds must be complete (must answer every question, including what is happening at GPS point xyz in the Amazon), must be consistent, and must be bivalent.

I now turn to a discussion of situations and their compatibility/incompatibility. I will be discussing almost exclusively what I will call informational or epistemic situations, which, naturally, will be analyzed in terms of information and as parts of epistemic possible worlds. Unlike possible worlds, situations can be incomplete, inconsistent, and non-bivalent.

Situations

I now relate the concept of epistemically accessible possible worlds to the concept of a situation. A situation is a part of a possible world and comprises one or more states of affairs. A situation is not necessarily a proper part of a possible world; for I count each possible world, including especially the actual world, as a situation. Nonetheless, when I talk about situations, I will have mainly in mind situations that are proper parts of possible worlds.

Among those situations that are proper parts of a possible world are those whose states of affairs they comprise correspond to the key descriptions defining possible worlds. As we have seen, the key description “the walls of my apartment are fuchsia” defines, along with the hand-wave “and every other state of affairs is the same as in the actual world” the possible world in which the walls of my apartment are fuchsia instead of portobello mushroom. This state of affairs, had it obtained, would have had the aforementioned key description corresponding to it. I will call the situation which comprises this state of affairs a “key-description situation”.

A situation that comprises just a single state of affairs I will call a “singleton situation”. There is a many-to-one relationship between single states of affairs and the set of objects existing (actually or possibly) in whatever relationships among one another and having whatever properties. The state of affairs named by 788UIOYOIYTYTIURRRBBBB4yt76876%^%$##DYCbHGFUIYTDTFJvDIYOBOvo ‘Tiger is now sitting on the keyboard of my laptop’ FI^T%$^%DXGDIGVHLOUFC:”}{PVTDXERhjkakb sdkdvsdwlebwhcjhsasssdff is a single state affairs obtaining at t; the state of affairs named by ‘Tiger’s color is silver’ also obtains at t and is a distinct single state of affairs. There are therefore at least two singleton states of affairs here mapping to the set comprising my cat Tiger at t.

Situations that are proper parts of possible worlds are limited in some way. They are finite in one way or another. A major cause of finitude of course, is being born. One’s (merely possible or actual — but from now on I will assume ‘actual’ unless otherwise noted) entry into a (merely possible or actual — but see the previous parenthesis) world results in their getting plunged into a world which has them as one center (among countless others, not totally unlike Nicholas of Cusa’s description of space as infinite and having infinite centers) and a horizon that extends only so far. One exists in the thick of a mileau with which they must constantly cope. A certain portion of the actual world gets exposed, or unconcealed (I try to define “exposure” below) to one, starting at birth and continuing through one’s lifetime. That is to say, one acquires and has available to them information. I will be leaving the term “information” undefined, except to accept Dretske’s claim that information can be either conceptual (information grasped intellectually or in terms of categories) or non-conceptual (information not grasped intellectually, for example information felt, as Nelson Goodman put it in his LANGUAGES OF ART, in the bones and in the muscles). I use the term “exposure” partly to avoid the implication that what I am talking about is necessarily conceptual in character. It is thoroughly obvious that any being that is not omniscient will have only a limited portion of the actual world exposed to them. Like being born, not being omniscient is a major cause of finitude.

Caravaggio’s painting THE CARDSHARPS very nicely illustrates, I think, four typical features of situations. First the boundaries of the information that is available to one form the boundaries of a situation. Second, to be in a situation is to be placed in the thick of things: one is grappling with the exigencies of a situation. Third, the information might be information grasped not by the mind but felt by the bones and muscles, emotionally. Fourth, there can be a conduit of information from one situation to another.

Caravaggio,, THE CARDSHARPS
 

Caravaggio’s painting depicts three situations. One situation comprises the situation of the callow young man, apparently of some wealth and status and vulnerable to being considered as a mark, who is pondering his cards. Exposed to him are one side of the cards, the table, and the front of the worldly-wise teenage cardsharp. Although he is directly looking at just the cards at the moment, I think it is safe to assume that in the very recent past he has acquired information about and has available to him information about the teenager and about the middle-age-ish man behind him. The young man, the mark, is a center of a situation within which a certain amount is exposed and from which much is hidden. The cards nicely serve as a visual symbol of the fact that his situation has boundaries shaped by a perspective and a point of view.

The teenage cardsharp of course has his own perspective, his own point of view, on the scene. What he is exposed to includes the back of the mark’s cards, the table, the cards he feels with his hands as he draws them from his back, the knife’s-edge tension he feels arising from the fact he has to accomplish his move perfectly, the feeling of danger arising from the fact the scene could very easily explode into violence — fists or swords — if he is not smooth enough. His own vulnerability, the fragility of his enterprise and its danger he cognizes in the bones and the muscles. Included in his situation is a kind of very broad present perfect. He has been shaped by a larger world which — if I remember my European economic history correctly — that has been suffering from harder times for a while now; and this card game is where he is now. All of this is his situation.

Oh, and yes — I almost forgot — part of the cardsharp’s situation includes a view of his confederate signalling to him the content of the mark’s cards. The confederate is in the thick of his own situation and of course he feels the knifes’s edge tension and danger, the fragility of their endeavor and the not at all improbable chance it could go wrong. These aspects of his situation are exposed to him, just as they are to his younger confederate. This situation serves as a conduit of information linking the mark’s situation with the teenager’s.

If I may be permitted to foreshadow what belongs to another post, we have here the 3-place accessibility relation between situations Rstu, where s is the signaler’s situation, t the teenager’s situation — a situation that includes a view of the signalling — and u is the mark’s situation. But more of that later, in a different post.

Any key or potential description that opens up a possible world maps to a situation. The key description ‘the walls of my apartment are fuchsia’ opens up, along with the phrase ‘and everything else is the same as in the actual world’ for me that possible world. The key description maps to the key situation that is part of this possible world, namely, the situation in which the walls of my apartment are fuchsia. I will say that situations hold in a possible world, including the actual world. That the walls of my apartment are portobello mushroom is a situation that holds in that possible world which happens to be identical with the actual world.

A situation comprises one or more state of affairs that obtains or could obtain. I use “comprise” for its suggestion that a situation surrounds, encloses, includes the states of affairs. The situation of my apartment in the actual world, for example, comprises the states of affairs ‘my cat Tiger is sleeping at my feet’ and ‘the walls of my apartment are portobello mushroom’, both of which obtain. Each of these states of affairs can be splintered off to comprise a distinct situation by itself.

“Known” aka “exposed to” aka “unconcealed” counts as a key description. That part of the actual world which I actually know, i.e., actually have information about, am cognitively exposed to, comprises my comprehensive actual situation. Everything that Smith has information about, everything that Morgenstern has information about, everything that Elizarraraz has information about, everything that an omniscient being has information about, comprises the comprehensive actual situation of Smith, Morgenstern, Elizarraraz and the omniscient being respectively. I will dub situations carved out this way from a possible world (including and primarily the actual world, of course) “informational” or “epistemic situations”; henceforth I will be using “situation” to refer to epistemic situations only, unless otherwise noted. A sentient being exists in an informational situation; this being is in the thick of things which press upon them and with which they must cope. That situation is defined by the information that is available to this being.

My own informational situation, the situation I am in, includes everything I am familiar in my apartment, as well as the view of the courtyard outside. It includes my knowledge that Houston is in Texas, Seattle is in Washington state, and St. George and Salt Lake City are in Utah. It includes all my old haunts in downtown Chicago and in that city’s Rogers Park, Near West, and Pilsen neighborhoods. Taken together, all this knowledge, all this information — both knowledge that and in-the-bones-and-muscles knowledge, comprises, along with the objects the information is about, what I will call my comprehensive actual situation.

My comprehensive actual situation includes numerous sub-situations — my cat Tiger’s sitting on a mat, the doorbell’s ringing inside my apartment. I will say that one is in a comprehensive actual situation or sub-situation; sometimes I will say that a situation — especially a sub-situation — holds for one. Sometimes, to emphasize the fact that the things within one’s situation are, typically, things with which they must cope, I will say one is grappling with and in the thick of a situation.

A bit later, when I discuss ostensible objects, I will be dropping the word “actual” and talk only about my “comprehensive situation.”

Now anything which blocks information from me … distance, shells, the practical difficulties (including getting the needed funding) of searching for probably-mythical creatures … will create a space for a possible world epistemically accessible for me from the actual world in which I exist. All that is required for the generation of a possible world that is epistemically accessible for me from my location in the actual world is that what I posit to “fill in” these spaces (a peanut in either shell #1 or shell #2 or shell #3; a Bigfoot | alternatively an absence of a Bigfoot roaming the forests of western Washington state; the sea-glass green or Venetian red or goldenrod yellow or periwinkle blue color of my neighbor’s walls) not be ruled out by what I know about the actual world.

The range of information available to me is obviously limited. It does not include enough information to absolutely rule out the identity of w0 (Bigfoot is roaming the forests of western Washington state) with the actual world, nor the identity of w1 (no Bigfoot is roaming the forests of western Washington state) with the actual world. It will not include enough information to rule out the identity of any other possible world that is epistemically accessible for me from the actual world — for example, the possible world in which the peanut lies under shell #1, the possible world in which it lies under shell #2, or the possible world in which it lies under shell #3. Nor does my comprehensive actual situation tell us whether, for any given GPS point in the Amazon, a butterfly is flapping its wings there, and if one is, whether its wings are purple or chartreuse. Nor does it tell us whether the walls of my neighbor’s apartment are colored sea-glass (viridian) green or Venetian red, a brick wall blocking this information from me, and my not having any other source of information that would tell me what the color is.

So unlike a possible world (including the actual world) which is “dense” in the sense that it answers every question, a situation has “holes” in it in the sense that it does not answer every question. It is littered with pockets of missing information. The situation comprises the actual world minus all the holes in which epistemically accessible possible worlds can be generated by any positing whose content does not rule out, given what I know about the actual world, its being a component of the that world. If a possible world is dense like a slab of most cheeses, a situation possesses many “holes” — pockets of missing information that can be filled in by positings consistent with what I know about the actual world — and is in that regard like a slab of Swiss cheese.

Or, to switch metaphors, my comprehensive actual situation is like a mesa (comprising all the knowledge/information I have) in southern Utah surrounded by a fog-covered plain. Lurking within within this fog are countless possible worlds that are epistemically accessible for me from my location on this mesa in the actual world.

Caspar David Friedrich, WANDERER. Okay — this isn’t Utah … but it’s close enough for government work

Since my comprehensive actual situation could be (in a sense of “could” that I will be clarifying shortly) identical with the Bigfoot possible world, or with the non-Bigfoot possible world, or with the ‘my neighbor’s walls are wine red’ possible world, and so on, my comprehensive actual situation could exist in more than one possible world. One is constantly finding out which possible world their comprehensive actual situation belongs to (and therefore which possible world the actual world belongs to). Is this the possible world the one in which the peanut is lying under shell #1 or the one in which shell #1 hides nothing except a piece of table top? Is this the world in which Bigfoot is roaming the forests of western Washington state, or the one in which no such creature is roaming those forests? Is this the world in which my neighbor’s walls are wine red, or the world in which they are some other color?

The key descriptions of these (epistemically accessible for me from my location in the actual world) possible worlds define circumscribed situations within those worlds. These descriptions might or might not be satisfied by the actual world. On the table in the Pilsen neighborhood of Chicago (that somber city) on which Elizarraraz has set up a shell game, the peanut may or may not be under shell #1. In the forests of western Washington state, Bigfoot might or might not be roaming. I will call these situations “derived from epistemically accessible possible worlds situations”, or “epistemic situations”. These are situations which the various epistemically accessible possible worlds that lurk in the holes in the swiss cheese, or in the fog surrounding the mesa, comprise.

Truth And Possibility

Epistemic Possibility. If I know nothing that rules out the identity with the actual world of the possible world which has as its key description ‘the peanut is under shell #1’, there is one possible world epistemically accessible for me from my location in the actual world in which the peanut is under shell #1. It is therefore possible — epistemically possible — that the peanut is under shell #1. That the peanut is under shell #1 is a possibility. As part of the possible world that has this as its key description, the peanut’s being under shell #1 could be a situation that is part of the actual world and therefore could become part of my comprehensive exposed actual situation. It’s being so, if I may be permitted to risk wearing out the word, is a definite possibility. Possibility defined in terms of what I do not know — my ignorance. This is the sense of “could” that is operative when, before the shell is turned over, I can truthfully and non-misleadingly say that the peanut could be under the shell, but then can no longer truthfully say this once the shell is turned over and turns out to be hiding nothing but empty air and a patch of table surface. This is a sense of “possible” that requires ignorance, not knowing. [See P.Z. Myers.] I will call possibility understood this way “epistemic possibility”.

Given my knowledge that the peanut lurks underneath one of the shells and that Elizarraraz would not do anything funny to manipulate the odds, there is a probability of 1 in 3 that the shell I select hides the peanut. Say I select shell #1. The credence I can assign to there being a peanut there is 1 in 3. This is the measure of the credence I can give to ‘this shell’ (say shell #1) ‘hides the peanut’. Once I overturn the shell to expose the peanut | a small expanse of tabletop the credence that there is a peanut there | nothing there becomes 1. Before that, when I observed just a shell, the credence to assign to there being a shell there is 1. Likewise, when I first observed the color of my apartment walls, the credence to assign to that color’s being portobello mushroom is also 1. This credence, or “probability”, is assigned relative to a body of knowledge that includes such propositions as those expressed by “I am not dreaming at the moment” and “I am not hallucinating at the moment”. I might be wrong about what propositions are to be included in this reference body of knowledge — for example, notoriously, I might believe that I am awake when in fact I am dreaming — but we should not automatically assume that having a false belief about what is included in my body of knowledge means that I do not know that thing. Also, I might not necessarily be able to express in a natural language (or in a formal language, for that matter) a proposition that is in fact part of my body of knowledge. [[[Present pluperfect]]]

Let me say, then, that a state of affairs is exposed, or unconcealed (I will use the two terms interchangeably; I will be using “unconcealed” when I am in a more Heideggerian mood), in a situation when, given the body of knowledge possessed by the sentient being whose situation it is, the credence to be assigned to that state of affairs is 1. When that credence is less than 1, that state of affairs counts as hidden or concealed.

Since my comprehensive actual situation is that part of the actual world that is exposed for me, every state of affairs in it is an exposed state of affairs. If a state of affairs is hidden, it is not part of my comprehensive actual situation.

Epistemic Truth: Correspondence is the truth-making factor, whether the state of affairs lies within a possible world or within a situation. But a sentence can correspond to a state of affairs with in a situation only if that state of affairs has been exposed within that situation: a situation comprises exposed states of affairs only. So when I utter a sentence that is true because it corresponds to a state of affairs within my situation, for example:

The walls of my apartment are portobello mushroom

it corresponds to an exposed state of affairs. All sentences that are true because they correspond to a state of affairs in my situation will be corresponding to an exposed, an unconcealed state of affairs.

Let me say that a sentence is true in a situation when it corresponds to a state of affairs within that situation. (For the best defense of the correspondence theory of truth that I have seen so far, see Joshua Rasmussen, Defending The Correspondence Theory Of Truth, Cambridge, Cambridge University Press, 2014). Let me also say that a situation makes the sentence true when a state of affairs within that situation corresponds to that sentence. Since a state of affairs is not comprised by a situation unless it is exposed within that situation, exposure, unconcealment, is a necessary condition for a situation’s making the sentence true. A situation cannot make true a sentence that corresponds to a hidden state of affairs, since that state of affairs cannot count as within or being a part of, that situation.

So if any of the following sentences is true, my comprehensive actual situation does not make them so. The states of affairs that would make them true are states of affairs in different situations.

1) The butterfly fluttering about in the Amazon at GPS point xyz has chartreuse wings

2) Shell #1 is hiding the peanut

3) The walls of my neighbor’s apartment are wine red

4) Bigfoot is roaming the forests of western Washington state

The first sentence, 1), might be generated by my laptop which I have programmed to spit out sentences of that form. Relative to my situation, it is an epistemic possibility: the situation that comprises that state of affairs could be drawn from a possible world whose identify with the actual world cannot be ruled out by anything in my body of knowledge. The number of variants of this situation is humongous, so the credence/probability to be assigned to 1) is extremely low. The probability to be assigned to 2) is 1/2. The credence to be assigned to 3 is 1 divided by the number of paint colors out there. My own Bayesian priors for 4) is about 1 in 100. (I know, I know — I have just lost any credibility any reasonable person might have been willing to grant me.)

Sentences 1 through 4 may or may not be made true by states of affairs exposed in other situations. For example, they may be made true by the comprehensive actual situation of an omniscient being to whom every state of affairs in the actual world is exposed, and for whom there are no brick walls, no funding difficulties blocking off the flow of information — a situation to which I will apply the label ‘worldly situation’. But none of them is made true by my comprehensive actual situation. Nor, as we will see, does my situation make true the sentences formed by prefixing “It is not the case that” to each one. The worldly situation does make at least one of A or ~A true in each case, so one would be justified in exclaiming “but surely A v ~A is true for each of these!” Yes — but it isn’t my comprehensive actual situation that makes them true.

Were however a conduit of some sort to link the situations relevant to 1 through 4 to my situation, either each sentence or its negation would be made true by my situation and in my situation. If my neighbor is reliable and he tells me that the walls of his apartment are wine red, he would have provided a conduit linking his comprehensive actual situation to mine and exposing in my situation the wine-red color of his walls. This state of affairs would then become part of my situation, having a probability of 1. [[[Need to discuss how something can fail to be a logical truth and still have a probability of 1.]]] Likewise, I might employ something like Google Earth to zoom in on that particular spot in the Amazon and discover there is indeed a butterfly with chartreuse wings fluttering about there. A sonar peanut detector might reveal to me that shell #1 is hiding the peanut, a conduit that perhaps is not quite totally kosher, like the one depicted in Caravaggio’s CARDSHARPS.

And a cryptozoologist with impeccable credentials might inform me that she was able to ascertain beyond a shadow of a doubt that the Bigfoot-like creature gallivanting around the various campsites in western Washington state impishly waving hello to campers and clearly enjoying their reaction, really is Bigfoot and not, as I had at first assumed it would have to be, my DFH friend Shaggy.

In each of these cases, an epistemic possibility — the ‘could’ generated by my ignorance — gets converted to an exposed actuality. Something that has an location in space-time gets unconcealed. Something gets made true in my situation by the exposure of a previously hidden state of affairs in a different situation. A probability of less-than-one gets turned into a probability of 1. To get exposed or to have been exposed is to have a probability of 1, where that probability is relative to one’s body of knowledge.

Knowledge has as one necessary component truth. I do not know that p unless p is true. Epistemic truth would reverse this relationship; p is not true in a situation (made true in that situation) unless p is known, exposed unconcealed — unless information, conceptual or non-conceptual, is available about it.

Here then, is the main take-away from this section: a situation cannot make true a sentence that corresponds to a hidden state of affairs. As a complete possible world, the actual world has sufficient resources, so to speak, to make true any sentence that corresponds to any state of affairs, hidden or exposed, within that world. But a situation does not have these resources. It is unable to expose every truth, and no sentence is true in a situation unless it has been exposed. Just as one does not go to Judge Judy to expose the truth in the murder trial of the century, I do not use my (current) comprehensive actual situation to expose the truth regarding whether there is a butterfly fluttering about at GPS location xyz in the Amazon, and, if so, what color its wings are.

Judge Judy trying to expose the truth of the Amazonian Chartreuse Butterfly Sentence

So much for truth. Let me turn now to compatibility/incompatibility. I will then combine these two separate discussion streams into a discussion of the truth condition for negation that will leave room for relevant implication.

Compatibility/Incompatibility

Compatibility and Incompatibility are binary relations holding between situations. I discuss these relations as they hold between situations drawn from spawned-from-the-unknown possible worlds and spawned from the known possible worlds respectively. I give a nod to the concept of defeasibility.

Situations Drawn From Spawned-From-The-Unknown Possible Worlds: The possible world in which the walls of my neighbor’s apartment are Venetian red is epistemically accessible to me because nothing in my body of knowledge rules out the identity with the actual world of this possible world. This possible world has “my neighbor’s walls are Venetian red” as its key description. The hand-wave “and everything else is the same as in the actual world” accomplishes the rest of this definition of the possible world. This world could (epistemic “could”) be identical with the actual world, and, upon the exposure to me of my neighbor’s walls as in fact being Venetian red, that particular situation (a singleton comprising just one state of affairs) drawn from this possible world would become part of my comprehensive actual situation.

Epistemic Compatibility: When a key-description situation s1 is drawn this way from a possible world that is epistemically accessible for me from my location in the actual world (remember, I will sometimes refer to these simply as “epistemically possible worlds” to save breath) and would therefore become part of my comprehensive actual situation s0 upon exposure, I will say that s1 is epistemically compatible with s0. Every situation s1 defined this way will be epistemically compatible with s0; this manner of definition provides a sufficient condition for epistemic compatibility.

So the key descriptions of all the epistemically accessible possible worlds lurking within the fog surrounding the mesa, all those ghost-worlds haunting all those holes in the lump of swiss cheese … all of these key descriptions define situations that are epistemically compatible with my comprehensive actual situation. No epistemically possible world will have a situation-defining key description that is incompatible with my comprehensive actual situation.

If I may be allowed to go out on a limb for a moment, compatibility is to be analyzed in terms of possibility, which in turn is to be analyzed in terms of possible worlds. As I will show shortly, different accessibility relations will allow some pairs of compatible situations and disallow others.

[[[A little reflection will show that every situation mapping to a key description of a possible world that is epistemically accessible for me from my location in the actual world will be epistemically compatible with my comprehensive actual situation. For any given key description, were there anything in my body of information and knowledge that ruled out its being part of the actual world, it would not be a key description for a possible world that is epistemically accessible for me. It therefore would not map to — really, be — a situation within an epistemically accessible world. It is not the case that it could be a part of the actual world; a fortiori, it is not the case that it is could be part of the actual world known to me or about which I have information. That is to say, it is not the case that it could be part of my comprehensive actual situation. Therefore all epistemic situations [define these above] are epistemically compatible with my comprehensive actual situation.]]]

Two situations can be epistemically compatible but incompatible in other ways. It is not completely inconceivable that, were I completely omniscient, or at least close to it, I might discover that the following two situations are nomically incompatible: the walls of my apartment are portobello mushroom, and the walls of my neighbor’s apartment are primrose pink. For suppose the universe is deterministic, at least on the post-quantum level. An incredibly complicated chain of events starting from the big bang, knowable only by an omniscient being, has resulted in the color of my apartment being portobello mushroom. This situation holds in all possible worlds that are a) nomically accessible from this world — i.e., the same laws of physics hold as in this actual world; and b) have the same initial conditions. Conceivably, there is just one such world, the actual world.

Another incredibly complicated chain of events starting from the big bang with the same initial conditions, also knowable only by an omniscient being, has resulted in my neighbor’s walls being chicory-flower blue (a beautifully paled even somewhat chalky ultramarine with a hint perhaps of violet). This situation also holds in all possible worlds nomically accessible from this world which have the same initial conditions. There might be just one such world, but it would still be true that in all possible worlds accessible from the actual world in the way described my walls are portobello mushroom and my neighbor’s walls are chicory-flower blue.

Necessarily, then, (nomically-plus-initial conditions necessary), my neighbors’ walls cannot be primrose pink at the same time my walls are portobello mushroom. The two situations preclude one another. They are incompatible.

To talk about two situations existing in the binary relation ‘incompatible’ is to talk about their necessarily not holding at the same time in the same possible world. In one sense or another of “cannot”, they cannot hold at the same time in the same possible world. Here the sense of the modal “cannot” is nomic-and-initial-conditions necessity. There is no possible world accessed this way from the actual world in which my walls are portobello mushroom and my neighbor’s walls are primrose pink.

Which sense of “cannot” is in play, which situations get included in the incompatibility relation will depend upon which accessibility relation one chooses. ‘Portobello mushroom in my apartment’ precludes and is incompatible with ‘primrose pink next door’ if the accessibility relation is ‘nomic and same initial conditions’, but not if the accessibility relation is ‘nothing in my body of knowledge rules out the identity with the actual world of the possible world in which my walls are portobello mushroom and my neighbor’s walls are primrose pink’. Because I am definitely not omniscient, my knowledge being limited and my ignorance vast, the two situations remain epistemically compatible. The latter epistemic accessibility relation renders the two situations compatible — epistemically compatible.

The situation described above whose corresponding description forms the key description for the rogue peanut world is not epistemically compatible with my comprehensive actual situation or with any of its sub-situations. For that situation would have to be part of the actual world; yet in the actual world peanuts do not jump around from shell to shell and never end up on the nose of the Mona Lisa. This is true of any world for which this is the key description: for example, the world in which Elizarraraz has painted the exterior of the building he owns goldenrod yellow instead of its very bright viridian green but the peanuts in his shell games went rogue; or the possible world in which the peanuts have also gone rogue but the sunflower in front is ever so slightly to the left … and so on ad infinitum. If I may be permitted a slight leap, then, I will assert that in all possible worlds in which peanuts have gone rogue, the key situation is epistemically incompatible with my comprehensive actual situation or any of its sub-situations. My situation (and each of its sub-situations) precludes any of the key situations, which means that never in any world shall the twain hold together at the same time.

Looking at situations drawn from spawned-from-the-known possible worlds will illuminate two dimensions of the epistemic incompatibility relation: the idea of difference, and the defeasibility of the aforementioned “cannot”.

Situations Drawn From Spawned-From-The-Known Possible Worlds: All the sub-situations within my comprehensive actual situation are, of course compatible with one another. That they are all holding at the same time is proof of their compatibility. Clearly, they do not preclude one another.

It will be worth keeping in mind (if only for the sake of clarity) that among these compatible situations are singleton situations mapping to a set comprising the same objects. For example, the situation that maps to the state of affairs named by ‘Tiger is sitting on my keyboard’ is clearly compatible with the numerically distinct situation named by ‘Tiger’s color is silver and the color of the Chromebook keyboard he is sitting on is Payne’s grey with a slight greenish undertone’. Clearly, ‘Tiger is sitting on my keyboard’ is compatible with ‘Tiger’s color is silver and the color of the Chromebook keyboard he is sitting on is Payne’s grey with a slight greenish undertone’

One would normally expect key situations defining spawned-from-the known possible worlds to be incompatible with the situation serving as its “raw material” so to speak if those two situations are taken to hold within the same possible world. If I access a possible world by taking the description “portobello mushroom walls” (the description for what I will call the “raw material situation”) and substituting for it the description “fuchsia walls”, the portobello mushroom wall situation and the fuchsia wall situations surely must be in different possible worlds. (Warning — this “must” is about to be caveated to death.) I am not, after all, trying to land in the same possible world I started with! I am trying to imagine what the effect would be if the color of the walls were different from what they are now.

And my walls — at least not in the same expanses — cannot be fuchsia and portobello mushroom at the same time in the same possible world. This I am willing to bet my paint brushes on. Unable to hold at the same time in the actual world — or in any other possible world — the two situations are incompatible. The one precludes the other from holding. If one holds in a possible world, the other cannot hold in that same world.

In positing a different possible world, one needs a feature, after all, that is, well, different from the raw material feature one is using. Without a difference in the key descriptions, one does not access a possible world different from the actual world. This leaves open the possibility that while one thinks there is a difference in key descriptions, there is in fact no difference.

What does count as a difference in the key descriptions? If we take color as our guide, we can, I think, confidently say that the fact my apartment walls are portobello mushroom means that the same expanse of wall cannot be fuchsia, periwinkle yellow, cornflower blue, or any other color at the same time it is portobello mushroom. The sub-situation within my comprehensive actual situation that is defined by ‘my walls are portobello mushroom’ cannot be identical with any situation defined by ‘my walls are x‘, where for x one can substitute any color name except “portobello mushroom”. Again, I am willing to bet my paint brushes on this. I mean, I certainly have never seen any wall that is both (in the same expanse) portobello mushroom and some other color. Nor can I imagine this. I will go so far as to say I cannot conceive this — not in this actual world nor in any other possible world. Necessarily, a wall cannot be portobello mushroom and some other color at the same time and in the same expanse.

But why should one think this? Knowing that two situations hold at the same time in the actual world if dispositive for their being compatible. But what is dispositive for the incompatibility of two situations within the same possible world? Look as hard as I may in the actual world for a wall that is both fuchsia and portobello mushroom in the same expanses at the same time, I will surely never find one. But isn’t a search like this a bit like searching for an elephant-sized swan? I may never find one — but that is not an absolute guarantee I never will! As Tom Kyte, the database guru likes to remark, you can’t prove a negative such as “no swans the size of an elephant exist”.

Certainly I am unable to imagine a wall that is both of the aforementioned colors at the same time. It would seem, then that I cannot conceive of these two situations holding at the same time in the actual world, right?. (Spoiler alert: no.) And isn’t the inability even to conceive this co-holding evidence that the concept of their compatibility is incoherent?

It is also true, however, that I cannot imagine a mass being at the same time a wave. Yet on my layman’s Wikipedia-level understanding of physics, every mass is also a wave. The situation to which corresponds the description “lumbering mass furiously scribbling away on a Chromebook” exists at the same time as the situation corresponds to the description “tiny petite wave furiously scribbling away (more precisely, keyboarding away) on a Chromebook”. So the two situations must be compatible. How can that be?

The apparent incompatibility can be resolved, I think, if we heed Leibniz when he tells us that imagining is not necessarily conceiving. I cannot really imagine a icosahedron with its 20 equal faces. But I can conceive of one. I suggest, then, that while one may not be able to imagine the situations “lumbering mass” and “tiny petite wave” holding for the same entity at the same time, one may be able to conceive this.

Certainly I cannot conceive this — my conceptual resources are too limited. But I am willing to bet that the physicist, as they delve into the properties of this mass in an ever-more sophisticated way, will end up ascribing properties to this mass that are also properties of a wave. For all I know, I have wave-properties such that, were the wave (per impossibile) long enough, I would be able to pass through two separated doors at the same time, just as, if I am not mistaken, an electron is able to pass through two separated slits at the same time. Let us suppose that there is a long series of mathematical equations that describe this; and let’s call this description a “conception of a lumbering mass that is identical with a tiny petite wave”. This description would correspond to a single situation — a mass that is identical with a wave. Given enough knowledge, the description ‘mass’ merges into the description ‘wave’; and the situation named by ‘is a mass’ is identical with the situation named by ‘is a wave’. The identity of the “two” situations is, of course, a sufficient condition for their compatibility.

Let me pause for a moment and elucidate a couple of matters, just for the sake of clarity. First, identity is a two place relation, with one singleton situation bearing one name related to the same singleton situation bearing a different name. The “two” situations are numerically one, comprising the same proposition aka state of affairs at t. Other propositions/states of affairs might comprise the same objects, but be comprised by different situations. The same lumbering mass/petite tiny wave may be part of the state of affairs named by ‘lives in Houston’ and also part of a different state affairs named by ‘studied philosophy in Chicago at Loyola University’, with the result that it is involved in three numerically distinct situations each comprising a different state of affairs.

I continue. I have never encountered a wall that is both fuchsia and portobello mushroom in the same expanse and at the same time — just as I have never encountered a large mass that in any obvious way is also a wave. I am unable to picture imaginatively a wall that is both colors at hte same time — just as I am unable (at present) to picture imaginatively a large mass that is also a wave. I am lacking a non-pictorial conception of a wall that is both colors (henceforth “at the same time and in the same expanse” will be understood) — and until I have such a conception, I will continue to regard the two wall-color situations as incompatible. In no possible world in which walls (or anything for that matter) have colors are the walls of my apartment both fuchsia and portobello mushroom. As parts of states-of-affairs-comprised-by-situations-that-are-parts-of-possible-worlds these two situations must be parts of possible worlds. To put the matter a bit more simply, the walls cannot be these colors without being in different possible worlds. I am in the same position with regard to the wall colors as one would have been with regard to mass/wave situations before our knowledge of physics expanded enough to give us a conception of the fact that a lumbering mass is also a wave.

For all I know, we may eventually broaden our knowledge enough to allow us to conceive an identity in the same possible world of the fuchsia wall situation and the portobello mushroom situation. Or maybe an infinite or at least extremely large intelligence already has such a conception. I am not willing to bet my paint brushes, however, on anyone arriving at such a conception in my lifetime, or ever. Nor am I able to ground the assertion that this conception will never arrive, no more than I can ground the assertion that we will never discover an elephant-sized swan. Someone in 1801 might have made the same assertion regarding large masses and tiny waves. All I can say is “so far as I know, the same expanse of wall cannot be, in the same world, at the same time, and in the same expanse both fuchsia and portobello mushroom.

In other words, there is nothing in my body of knowledge that would rule out the hypothesis that, among all the possible worlds in which a wall can be some color and in which what it is to be a color (however that may be articulated) is the same as it is in the actual world, there is no world in which the same expanse of wall is both fuchsia and portobello mushroom at the same time. So far as I know there is no possible world in which ‘fuchsia-wall’ and ‘portobello-mushroom wall’ are two names for the identical situation. The necessity, and therefore the incompatibility, are epistemic, grounded in my lack of knowledge, my ignorance. The necessity and the incompatibility are defeasible.

Side note for the sake of clarity: I have moved, then, from discussing individual worlds that are epistemically accessible for a sentient being S (the possible world in which Bigfoot does indeed roam the forests of western Washington state; the possible world in which the walls of my neighbor on the other side of the brick wall I am facing as I write this are wine-red) to sets of possible worlds that ground epistemic necessity (there is nothing in my body of knowledge that would rule out the hypothesis that, in the set of possible worlds that are accessible in such and such a way from the actual world, none comprise situation s).

Strictly speaking, epistemic necessity is compatible with, does not preclude epistemic contingency (there is nothing in my body of knowledge that would rule out the hypothesis that, in the set of possible worlds that are accessible in such and such a way from the actual world, there is at least one that comprises situation s). This is perhaps a bit like saying “all unicorns (all zero of them) are yellow ocher” and “all unicorns (all zero of them) are rainbow-colored”, where the particular set of rainbow colors in question excludes yellow ocher. Unlike the usual uses of “necessary” and “possible” “epistemic necessity” cannot be defined as “not epistemically possible” and vice versa.

Nonetheless I do think there is a certain asymmetry between epistemic necessity and epistemic contingency. Reviewing possible worlds to try to find one in which the walls in it are (at the same time and in the same expanses) both fuchsia and portobello mushroom is rather pointless given that one does not know what it means for a wall to be both these colors at once. And one does not know what this means because one does not have access to the truth conditions for “this wall is fuchsia and portobello mushroom at the same time”. (Henceforth I will call this the the “fuchsia/portobello mushroom wall sentence”.) Currently, one lacks the means for getting these truth conditions. Imagination would have been one means. My imagining Tiger sitting on my Navajo mat suffices to give me the truth conditions for the English sentence “Tiger is sitting on my Navajo mat”. I know the meaning of that English sentence because, via the imagination, I know what would be the case if the sentence were true. But imagination certainly does not give me a way to get at the (quite possibly non-existent) truth conditions for the fuchsia/portobello mushroom wall sentence. My imagination can fail me in giving me the truth conditions for the English sentence “this humongous lumbering mass is also a tiny petite wave”, but someone (I am confident) has a conception stated in the mathematical language of physics of precisely this. (Maybe, in honor of Hilary Putnam’s division of semantic labor which relegates to certain experts the distinction between a beech and an elm, I will ask a physics undergraduate to write this down for me.) But no such mathematically-expressed conception seems to exist to give us the truth conditions for the fuchsia/portobello mushroom wall sentence. That is to say, so far as I know no such conception exists.

At the time of this writing, I can state confidently that I do not have a conception for the fuchsia/portobello mushroom wall situation that would give me access to the truth conditions for the fuchsia/portobello mushroom wall sentence. I also lack a conception for the humongous lumbering mass/petite tiny wave sentence (I will need to make sure above that I have named this sentence). But I am fairly confident that I can appeal to someone in my linguistic community (some physics undergraduate, for example) who does have a conception ghat gives them access to the relevant truth conditions, just as Hilary Putnam in his MEANING OF MEANING can appeal to a tree expert for the truth conditions of ‘is an elm’ versus ‘is a beech’ even though he lacks access to those truth conditions by himself. The access is distributed across members of a linguistic community unlike epistemic accessibility, which — at least given how the concept has been defined so far — is restricted to individual sentient beings. Let me define, then, what I will call semantic accessibility in the following way:

A world w0 is semantically accessible from w for an speaker S (in w) in linguistic community L to which S belongs iff S is able to consult with speakers in the linguistic community L to which S belongs who have a conception giving them access to the truth conditions of the key description of w0.

Then, p is semantically necessary (for S in L in w) iff p is true in all possible worlds that are semantically accessible from w (for S in L in w). And p is semantically possible (for S in L in w) iff p is true in at least one possible world that is semantically accessible from w (for S in L in w).

I am call this accessibility relation “semantic” of course because it is concerned with truth conditions. A “conception” that gives one access to a particular set of truth conditions is whatever resources one possesses that lets one know what it would be for the sentence “snow is white” to be true (namely, that snow is white). Since semantics is concerned with language — the meanings of words — and language is a communal enterprise, it is natural that the definition of semantic accessibility should include a reference to a linguistic community.

I started my discussion of possible worlds by asserting them to be identical with sets of descriptions, some of which are satisfied by states of affairs in the actual world (a “brick”, to speak metaphorically), and others which are left unsatisfied (a “written description” is in place where the brick should be). Without access to the truth conditions of a description, one has no access to what the description means. One therefore has no access to the possible world the key description is meant to open up. If, say, an omniscient mind knows the truth conditions, that mind has access to that possible world. For us more limited creatures, however, we may not necessarily know if there is a possible world out there we can ever hope to gain access to in the first place. As time goes on and — as in the case of humongous lumbering masses/tiny petite waves worlds — we may gain a conception which gives us access to the world and opens it up to us. I can no more definitively say this won’t happen for the fuchsia/portobello mushroom wall world any more than I can say that I will never encounter an elephant-sized swan. I am willing to bet that the communal body of knowledge — the resources available to members of the linguistic community to form the needed conception — will never expand so as enable anyone to access the truth conditions for the fuchsia/portobello world, but….

So we can divide worlds — the sets of descriptions, satisfied or unsatisfied, real or merely purported [[expand on this — difference between nonsense such as “trd4r3e3bbubijvbhjihyiurhtrg” and words that are intended to be used to describe but fail in that function because the truth conditions are lacking) into those worlds regarding which there is nothing in the communal body of knowledge that would rule out (preclude) their belonging to the set of possible worlds, i.e., the set of worlds, that is to say, the set of descriptions whose truth conditions one has access to. If a world is not a possible world, it is an impossible world, a status that it may (epistemic “may”) or may not have permanently.

[[[Let me note parenthetically that the entities I have defined with reference to a personal or communal body of knowledge — epistemically accessible possible worlds, epistemic probability, semantically accessible possible worlds rather paradoxically depend for their definition precisely on what is not in the body of knowledge. For probability part of the relevant body of knowledge consists in knowing what one doesn’t know — I don’t know under which shell the peanut lies, though I know that there is one peanut and that it lies under one of the shells.]]]

[[[Explain this a bit more. Compare with probability as a number relative to a body of knowledge.]]]

To continue: I propose, then that every key situation to which corresponds a key description generating a spawned-from-the-known possible world be regarded at first as (epistemically) incompatible with the situation in the actual world that formed the raw material of this key description. As far as one knows — that is to say, unless and until one’s body of knowledge has expanded sufficiently — the possible situation has to be seen as belonging to a possible world different from the actual world and as numerically distinct from the situation that forms its raw material.

Epistemic incompatibility, then, is the converse of epistemic accessibility. A possible world is epistemically accessible for me from my location in the actual world if there is nothing in my body of knowledge that would rule out its identity with the actual world; and a situation within that possible world could (epistemic “could”) be part of the actual world. For example, there is nothing in my body of knowledge that would rule out the identity with the actual world of the possible world in which my neighbor’s walls are wine red; and the situation named by ‘my neighbor’s walls are wine red’ could be a situation within the actual world. Correlatively, there is nothing in my body of knowledge that would give me the conceptual resources to conceive of my neighbor’s walls being both wine red and sea glass green in the same possible world; or of my own walls as being both fuchsia and portobello mushroom in the same actual world; or any wall in any possible world in which there are things called “walls” that have colors having two distinct colors at the same time and in the same expanses. There is nothing in my body of knowledge that would rule out the non-identity with the actual world of the possible world in which my apartment walls are of a different color. More generally:

A world w0 is epistemically distinct from w for an agent S (in w) iff there is nothing in S‘s body of knowledge that would rule out the hypothesis that w0 != w.

And:

A key description d0 is epistemically incompatible with a key description d for an agent S in world w for whose definition d is key iff every world w0 accessible for S by way of a key description d0 is epistemically distinct from w.

[[[[Two descriptions d0 and d1 are epistemically incompatible for an agent S (in w) iff, given the body of knowledge possessed by S, d0 and d1 together fail to provide a key description giving S access to a possible world.]]]]]

And:

Two situations s and s0 are epistemically incompatible iff the key descriptions to which they correspond are epistemically incompatible.

That two situations are incompatible is defeasible. [[Need a similar definition of compatibility.]]

To sum up this discussion of epistemic compatibility/incompatibility: Key description situations drawn from spawned-from-the-unknown possible worlds are epistemically compatible with the corresponding situation in the actual world when they are epistemically accessible from the actual world. Otherwise they are incompatible. Key description situations drawn from spawned-from-the-known possible worlds may at first be assumed to be epistemically incompatible with their “raw material” situations in the exposed part of the actual world, but then may become epistemically compatible as our knowledge grows. I dare say, however, that it is rarely — if ever — the case that one should bet the ranch — much less one’s paint brushes — on the incompatibility between two situations ever morphing into compatibility.

[[[[Lumberoom: Imagination, then is not an absolutely reliable guide to incompatibility. The mere inability to imagine two situations as being the same situation is not exactly an absolutely reliable guide to their being incompatible and therefore forever doomed to non-identity. My mere inability to imagine that the walls of my apartment, in the same expanses, are at once fuchsia and portobello mushroom does not suffice to guarantee that they cannot be. At the same time, I do not know what it would be for the walls (in the same expanses — henceforth this will be understood) of my apartment to be both portobello mushroom and fuchsia at the same time. I do not know — I haven’t the faintest idea — what would be the case if the sentence “the walls of my apartment are both fuchsia and portobello mushroom at the same time” were true. The truth conditions for that sentence are lacking for me — I don’t have access to these truth conditions. I am therefore unable to posit a possible world which has this sentence as its key description.]]]]

[[[[But I do have access to the truth condition for the possible world in which the walls of my apartment are fuchsia, not portobello mushroom. I know what would be the case if the sentence “the walls of my apartment are fuchsia” were true. I have the knowledge expressed by the sentence “this is fuchsia” and the knowledge of what it is to paint walls a certain color and whatever other knowledge is required to give me the conceptual resources to know this truth condition. I am therefore able to posit the fuchsia apartment wall possible world.

At one time, no one would have had the conceptual resources to know the truth conditions for the possible worlds (one of which which happens to be the actual world) in which “lumbering mass at work on its laptop scribbling away” is describes the same situation as does “petite tiny wave at work on its laptop scribbling away”. I am not sure I have the conceptual resources to know these truth conditions. I am confident, however, that a competent physicist would. Perhaps even they would not be able to imagine these truth conditions pictorially; but perhaps the ability to produce a series of equations describing the situation would count as knowing the truth conditions. At one point in time (t0) no sentient being S had the conceptual resources afforded by the needed body of knowledge; then at another point in time (t1) some agents do. At t0 the prudent thing to say would have been “so far as I know, ‘is a wave’ and ‘is a particle with mass’ are incompatible descriptions of the identical situation (namely, this person scribbling away)”; at t1 one can now say they are indeed compatible descriptions of the same situation.

Now, I submit, at the time of this writing, the prudent thing to say is that so far as I know, ‘the walls of my apartment are fuchsia” and “the walls of my apartment are portobello mushroom” are incompatible descriptions of the identical situation (namely, the walls of my apartment being such and such a color). But obviously I have no guarantee that my current body of knowledge and the resulting conceptual resources available to me will always be the same. Perhaps these can change just as they did in the case of the mass that is also a wave. The only entity that could provide such a guarantee would be an omniscient being. My being finite — i.e., my being situated — means that the ability to provide such a guarantee is likely to be forever beyond my reach. So I have to rest content with the “so far as I know”.

Let’s say, then, that incompatibility is relative to a body of knowledge, just as epistemic accessibility and probability (at least according to Kyburg) are. The incompatibility of descriptions d0 and d1 is relative to the body of knowledge possessed by a (possibly fictional) omniscient being who knows (if they do know this) that the two can never describe the same situation at the same time. The epistemic incompatibility of d0 and d1 is relative to a body of knowledge that fails to provide the resources — a failure that quite possibly will never be remediated — to give one access to the truth conditions for a single situation’s holding for which both d0 and d1 serve as key descriptions at the same time. [[[To have access to the truth conditions is to have access to a possible world.]]] This incompatibility means that d0 and d1 can never, together, provide a key description giving one access to a possible world. ]]]]]

[[[[[[

To gain access to a possible world is to have the ability, provided by one’s body of knowledge plus whatever other abilities, to conceive or represent the truth conditions of that world’s key description. If, given one’s body of knowledge and one’s cognitive abilities, one cannot cognize the truth conditions of the combination of d0 and d1, that combined description cannot open up or give one access to a possible world. The one description precludes the other. And there is no situation that they, in combination, correspond to. This absence of truth conditions is what provides the meaning of epistemic incompatibility.

Generally, one would expect that coming up with an ostensibly different key description (my walls are fuchsia, not portobello mushroom; Tiger is from planet XZDOIN#K deep in the Orion galaxy, not planet Earth) would result in pairs of incompatible descriptions and therefore incompatible situations which cannot hold at the same time in the same possible world (especially in the actual world). “Different” means “not identical”. Different description, different situation.

The one exception to different descriptions generating different situations would be when the descriptions are parts of a longer description whose individual sentences (perhaps in the forms of mathematical statements) cohere — maybe surprisingly –into a single description. The physicist starts with a mathematical description of a lumbering mass…and when they finish, end up with something that also describes a very small wave. They may still be unable to picture something’s being both a lumbering mass and a tiny petite wave at the same time — nonetheless, their long mathematical description constitutes a setting forth of the truth conditions. ]]]]]

[[[Need now to show how room is left for a situation not to be compatible with itself]]]

I now join these two streams: the discussion of truth, and the discussion of compatibility/incompatibility.

Non-Bivalent And Inconsistent Situations

Non-Bivalent Situations: The binary relation ‘compatibility’ holding between two situations gives us a truth condition for negation that will let us see how situations can be both bivalent and inconsistent. The case for bivalent situations is perhaps a bit less … venturesome … than the case for inconsistency. We will see where I end up. Let’s take a look at this truth condition for negation, but modified so as to be talking about epistemic compatibility, not compatibility per se.

A situation makes ~A true if and only if every situation [epistemically] compatible with it fails to make A true.

RL, p. 75

Making A True: Above, we have seen that a situation makes A true when that situation contains a state of affairs to which A corresponds. But the situations we are concerned with, epistemic situations, contain only states of affairs that are exposed, that is to say, unconcealed. So a situation makes A true only when A corresponds to an exposed state of affairs within the situation.

Epistemic Compatibility: As we have seen, every sub-situation within my comprehensive actual situation is epistemically compatible with every other such sub-situation and with my comprehensive situation itself. The fact that all these sub-situations hold at the same time in the actual world ices their all being compatible. [[[Be sure to reconcile this with the impossible cube. The impossible cube can never be actual.]]] Drawn from the actual world, these sub-situations are also epistemically compatible with any other situation drawn form the actual world by virtue of the fact that the actual world is epistemically accessible from the actual world. For any key description that is used to define the actual world, there is nothing in my body of knowledge that would rule out, i.e., preclude its being a part of the actual world.

All these actual situations are compatible with one another by virtue of the fact that that all situations drawn from possible worlds (one of which is the actual world) that are epistemically accessible for me from my location in the actual world are epistemically compatible with my comprehensive actual situation. And as we have seen, all possible situations — not just those possible situations that happen also to be actual — that are drawn from epistemically accessible possible worlds are epistemically compatible with my comprehensive actual situation. [[[Make sure I am not committing some part/whole fallacy.]]]

Negation: Negation works unproblematically for the dense parts of the lump of swiss cheese, for the mesa surrounded by the fog, for that part of the actual world which is exposed to S. Let’s take, for example, ~A to be the sentence “It is not the case that the walls of my apartment are fuchsia.” My comprehensive actual situation makes that sentence true.

Now we have seen that a sub-situation is compatible with the situation of which it is a part. So every sub-situation within my comprehensive actual situation s0 is epistemically compatible with s0. For no sub-situation within my comprehensive actual situation makes A (“It is the case that the walls of my apartment are fuchsia”) true. One of these sub-situations is, of course, the singleton situation comprising the state of affairs at t identical with the proposition expressed by “the walls of my apartment are portobello mushroom”, namely A. That situation especially fails to make A true. Tiger trying to get onto the keyboard as I write this; the doorbell ringing, the chalky Venetian red color of the cliffs in Southern Utah — list them all — each is epistemically compatible with s0 and each fails to make A true.

And surely every situation outside my comprehensive actual situation will also fail to make A true. [[[Discuss situations outside s0.]]] For example, whatever the color of my neighbor’s apartment walls may be, that is not a singleton situation what will make it true that the color of my apartment walls are fuchsia. I think it is fairly safe to go out on a limb, then, and say that every situation epistemically compatible with s0 fails to make “it is the case that the walls of my apartment are fuchsia” true. s0 therefore makes the sentence “it is not the case that the walls of my apartment are fuchsia” true.

Note that if nomic-and-initial conditions compatibility were in play, my neighbor’s walls being periwinkle yellow, say would make it true that my walls are fuchsia if the initial conditions at the big bang deterministically led up to these two causal lines. But I think it would violate our sense of what ‘not’ means if my neighbor’s walls being wine red made “it is not the case that my walls are fuchsia” true. This is why I think “compatible” should be taken to mean “epistemically compatible” in the truth condition for negation.

Negation works unproblematically for the dense parts of the lump of swiss cheese, for the mesa surrounded by the fog, for that part of the actual world which is exposed to S. But matters are different inside the holes in the lump of swiss cheese, or down in the depths of the fog surrounding the mesa. Consider the sentence “the walls of my neighbor’s apartment are wine red”.

Consider the situation in the possible world, epistemically accessible for me from my location in the actual world, that has as its key description “the walls of my neighbor’s apartment are wine red”. Within this possible world, the key situation (a singleton) would, if the state of affairs it comprises obtained in the actual world, make the sentence “It is the case that the walls of my neighbor’s apartment are wine red” true. Let this sentence be A at the moment.

As part of a possible world that is epistemically accessible for me, this key situation counts as an epistemic situation. As a situation that is hidden from me, as being attached to a probability less than 1 (but presumably greater than 0), it is not part of my comprehensive actual situation, which comprises only exposed states of affairs with a credence of 1. But being an epistemic situation, it is epistemically compatible with my comprehensive actual situation.

A mere probability does not make a sentence true. The probability that the peanut is under shell #1 is 1/2, as is the probability that the peanut is not under shell #1. But this probability does not make either “The peanut is under shell #1” or “The peanut is absent from shell #1” true. The probability is not the truth-making factor. The assertion “the probability is 1/2” does not answer the question “what makes it true that the peanut is under | is absent from shell #1?” What answers that question is rather the peanut’s presence under | absence from shell #1.

Likewise, the probability that my neighbor’s walls are wine red is something like 1/the number of wall paint colors that are out there. Let’s say the probability is 1/10000. The assertion “The probability is 1/10000” is not the answer to the question “Are my neighbor’s walls wine red?” (Henceforth the “wine red question”.) The proper answer to that question would be “That my neighbor’s walls are wine red”. That the probability is 1/10000 gives one the possibility that my neighbor’s walls are wine red. But wine-red walls are what make the sentence true — if it is true — not the mere possibility of wine-red walls. The sentence “My neighbor’s walls are wine red” corresponds | fails to correspond to whatever state of affairs obtains as to the color of my neighbor’s walls, not to whatever state of affairs might obtain.

[[[To say that a sentence is “merely probable” is not yet to say that it is true, though of course it might also turn out to be true in the end. The mere probability that the peanut is under shell #1 is not a truth-maker.]]]

Every brick that composes this particular epistemically accessible possible world is present and is the same brick that composes the actual world, with the exception of one missing brick. In place of this brick is a piece of paper on which is written “The walls are wine red”. In order to be true, the wine-red sentence has to correspond, so to speak, to the brick, not to the inscription.

Nothing within my comprehensive actual situation makes the wine-red sentence true: not the portobello mushroom color of my walls, not Tiger’s sitting on the mat, not my doorbell’s ringing — no sub-situation in this situation comprising whatever states of affairs does. And as we have just seen, neither does the singleton situation that, outside my comprehensive actual situation, is part of the for-me epistemically accessible my-neighbor’s-walls-are-wine-red possible world. So according to the above truth condition for negation, it is false. A, that is to say, “It is the case that my neighbor’s walls are wine red” is false.

But if bivalence held for my comprehensive actual situation, [[[remember to distinguish between holding in and holding for a situatioun]]], then ~A would have to be true: “It is not the case that the walls of my neighbor’s apartment are wine red”. Then the following sentence would have to be true: “My neighbor’s walls are sea-glass green; or my neighbor’s walls are Navajo white; or my neighbor’s walls are goldenrod yellow; or my neighbor’s walls are Tiepolo pink; or my neighbor’s walls are cornflower blue; or my neighbor’s walls are ultramarine blue; or my neighbor’s walls are Venetian red….” and so on for every wall color except for wine-red.

But using the same reasoning as for wine red, each of these or’d sentences is false for my comprehensive actual situation. That is to say, no situation epistemically compatible with my comprehensive actual situation s0 — no sub-situation in or within s0, and no situation drawn from a possible world epistemically accessible for me from my location in the actual world — makes any of these or’d sentences true. So the entire sentence that comprises them is false. So for my comprehensive actual situation s0 and for all situations epistemically compatible with it, both “It is the case that my neighbor’s walls are wine red” and “It is not the case that my neighbor’s walls are wine red” are false.

Bivalence holds within an epistemic situation (i.e., states of affairs are exposed to a sentient being), in that way my comprehensive actual situation is like a possible world. But bivalence fails for my comprehensive actual situation i.e., when I reach outside that situation to consider situations that are hidden from one and sentences whose truth is merely probable. When considering the truth of A requires a reach into the fog surrounding the mesa, or into one of the holes in the lump of swiss cheese, both A and ~A are false. One’s comprehensive actual situation turns out to be not enough to render at least one of these sentences true. This in turn means that there exist situations for which, for certain sentences A (obviously not all), “If A then A” is false. In this way, Relevant Logic can avoid the Classical Logic paradox: “A –> (B v ~B)”, alternatively, “A –> (B –> B).

Complete, a possible world answers the wine-red question as well as the peanut question as well as the Bigfoot question as well as the Amazonian chartreuse butterfly question. Finite and fragmentary, a situation cannot. A situation is partial. Like Judge Judy’s courtroom being the wrong tribunal for answering the question who is the culprit in the murder of the century, a situation is the wrong tribunal for answering the wine-red and the peanut (is the peanut under shell #1 or under one of the other two shells?) questions.

Partial situations allowing for this can exist because situations are not complete. They do not answer every question. (Does a Bigfoot creature roam the woods of western Washington state or does it not? Is there a butterfly fluttering about in the Amazon at GPS location xyz or not; and if there is, are its wings chartreuse? Is the peanut under shell #1 or does that shell hide only a small expanse of table-top? Are the walls of my neighbor’s apartment wine red or some other color?) My comprehensive actual situation remains the same whichever happens to be the correct answer. The identity of this situation does not depend upon the answer. This sameness leaves room for non-bivalent situations. [[[[Expand on this]]]]

But as we saw above, the identity of a possible world does depend upon the answer. The Bigfoot possible world is not identical with the non-Bigfoot possible world. So possible worlds do not allow for the failure of bivalence. In any possible world, including the actual world, either a Bigfoot creature roams the woods of western Washington state or does it not. Unlike situations, which can be partial, a possible world must be complete. It must answer every question.

All of the above is my attempt to nail down, by putting in my own terms, Mares’ discussion of non-bivalent situations in his RELEVANT LOGIC: A Philosophical Interpretation. This is an exercise in writing to learn. In particular, this is my attempt to nail down the following passage from Mares:

Partiality is straightforward [ahem! Cough cough], so we will start with that. Consider the situation that consists of the information that is currently available to me. The includes what is going on in my study as I write this section of my book, and what I can see through my window. Nothing happening here makes it true that it is currently raining in Toronto (which is on the other side of the globe). But situations in which it is raining in Toronto are compatible with my current situation. So neither ‘It is raining in Toronto’ nor ‘It is not raining in Toronto’ is true in my current situation. Thus bivalence fails for this situation and the situation is partial.

RL, p. 75

Let me turn now to inconsistent situations.

Inconsistent Situations: Dealing with inconsistent situations is a bit more of a stretch. But let me venture (as I try to strengthen my shaking knees) the following attempt to use the concept of ostensible objects to allow for inconsistent situations. There are singleton situations, I claim, that are incompatible with themselves because they comprise impossible objects. ‘Compatible’ is not always a reflexive relation for a situation.

First, some words on the concept ‘incompatible’.

Compatibility Again: Situation s1 is compatible with situation s2 when s1 and s2 can hold at the same time. For example, the situation that corresponds to the description “Tiger is sitting on the mat” (where I have deposited him to keep him off my keyboard) holds at time t. The situation that corresponds to the description “The doorbell is also ringing in my apartment at t” also holds at t. [[[Need to say something about “corresponds” vs. “defines”]]] That the two situations hold at the same time clearly means they are compatible. Their co-holding is dispositive. And as two sub-situations within my comprehensive actual situation s0 (I will be reserving “s0” to name my comprehensive actual situation unless otherwise noted), one would of course expect them to be compatible, as are all sub-situations within s0.

In the actual world, my cat Tiger was born on earth and did not travel here from the planet XZDOIN#K deep within the Orion galaxy. But surely the situation whose description forms the key description of the possible world in which Tiger is from XZDOIN#K (with cats, those creatures of mystery, one always has one’s suspicions) is also compatible with the doorbell’s ringing and with many of the sub-situations within my comprehensive actual situation. Since this is a situation drawn from a spawned-from-the-known possible world, it is perhaps fairly safe — though as we have seen, not absolutely safe — to say that this situation is not compatible with the situation that formed its “raw material”, Tiger’s origins being the planet earth. Surely the situation defined by the situation “Tiger is from planet earth” cannot co-hold with the situation defined by “Tiger is from planet XZDOIN#K deep within the Orion galaxy”. The two situations are different. They cannot be identical situations. The two situations are not compatible. The one precludes the other.

However, the mere fact that two descriptions seem like they cannot be descriptions of one and the same situation does not mean that these two descriptions cannot correspond to the identical situation. What is more, one of the differing descriptions may be a key description of a spawned-from-the-known possible world and the other may be the “raw material” of this description. One may think they have accessed a possible world numerically distinct from the actual world, but end up landing in the same old actual world anyhow.

Above, I have said that two situations s and s0 are epistemically incompatible iff the key descriptions to which they correspond are epistemically incompatible. In turn, a key description d0 is epistemically incompatible with a key description d for an agent S in world w for whose definition d is key iff every world w0 accessible for S by way of a key description d0 is epistemically distinct from w. [[[[Two descriptions d0 and d1 are epistemically incompatible for an agent S (in w) iff, given the body of knowledge possessed by S, d0 and d1 together fail to provide a key description giving S epistemic access to a possible world.]]]]]

Ostensible Objects:

[[[This particular expanse of wall cannot be portobello mushroom or sea-glass green at once. The one situation precludes the other. The two situations are not compatible. And their inability to exist at the same time is dispositive for their being incompatible.

Likewise, the holding of two situations at the same time is dispositive for their being compatible, their not precluding one another. My cat Tiger’s being on the mat (where I have just deposited him to keep him from sitting on my keyboard) at the same time that the doorbell is ringing definitively shows that these two situations are compatible. ]]]

Now I will try to show that ‘compatible’ is not always a reflexive relation. A situation can be incompatible with itself. If a situation is incompatible with itself, it excludes itself. This is a funny position for a situation to be in, a bit like Mark Twain or Groucho Marx or whoever said he refuses to belong to any club whose standards are so low as to have him as a member. But there we are.

Take a look at a depiction of an impossible cube, inspired by the Dutch artist Maurits Cornelis Escher.

https://www.youtube.com/watch?v=UibfZnfNYyc

If you see the cube as if a bit from above, take a look at the vertical bar closest to you, and the bar that gets experienced as running horizontally behind that bar to connect to two lower vertices behind it. Were this an actual object existing in the physical world and you were requested to touch the horizontal bar behind, you would reach past the vertical bar just mentioned and move your fingers along the horizontal bar behind from one vertex to the other. I hypothesize that this potential touching informs your visual experience of the bar behind so that your visual experience is that of a ‘bar behind’. The horizontal bar looks ‘behind’ just as a paper shopping bag looks heavy when one knows it contains groceries and looks light when one knows it is empty. This, I take to be one example of Merleau-Ponty’s contention that the different senses fuse together (see his Phenomenology of Perception), a view not totally dissimilar to George Berkeley’s contention that sight and touch are “entangled” (see his A New Theory of Vision). Those willing to sign a waiver releasing the author from any liability for any resulting brain damage might also want to consult a certain Cliff Engle Wirt’s journal article The Concept of the Ecstasis.

But the (more or less) horizontal bar is also experienced as cutting in front of the vertical bar. If the experience ‘behind’ is salient enough, the eye just accepts this, even though there is no visual experience of the horizontal bar coming from behind to wrap itself in front of the vertical bar then back again on its journey to the other vertex. In the physical world, these two states of the horizontal bar exclude one another. The horizontal bar cannot, at the same time, be behind and in front of the vertical bar. The two situations are not compatible.

But the visual experience of ‘horizontal bar in front’ is compatible with the visual experience of ‘horizontal bar behind’. For the situation comprising the one experience can and does hold at the same time as the situation comprising the other experience. That the two situations should be compatible may seem strange at first, but really should not. For just as, as John Searle constantly points out, the visual experience of a blue object need not be itself blue, and the visual experience of a square object need not itself be square, the visual experience of ‘bar simultaneously in front and behind’ need not itself have the (impossible) property ‘simultaneously in front and behind’.

Now visual experiences are of ostensible objects which usually actually exist but sometimes do not. My visual experience of my cat tiger (with a mouse) at my left as I write this has as its ostensible object a cat that actually exist. Tiger, although he is, yes, an ostensible object of my vision, is not merely an ostensible object of my vision.

My visual experience of the periwinkle-pink rhinoceros grazing peacefully at my feet while I write this … er…. I mean, were I to have such a visual experience … is (or rather, would be) of an ostensible object that does not exist. There is in fact no purple rhinoceros grazing, peacefully or otherwise, at my feet. The purple rhinoceros is merely an ostensible object of my vision. Nonetheless, I can still talk about him and even give him a name (say, “Alfred”). I can even get upset if he seems miffed and refuses to show up for a few days.

Likewise, my visual experience of the impossible cube I “see” on my laptop screen is of an ostensible object that in fact does not exist. It is merely an ostensible object. It no more exists in the physical world than does my periwinkle-pink rhinoceros. Even so, it does seem to make sense to talk about impossible cubes. We talk about ‘this impossible cube’ among different, varying impossible cubes (say, some with differently colored bars). An ostensible object is an object, existing or not existing, actual or possible or impossible, that can be talked about

Now I propose that we regard one’s comprehensive actual situation and sub-situations as comprising at least those objects that one is involved with. From this it follows that my comprehensive actual situation and all sub-situations comprise ostensible objects, both merely ostensible objects and more-than-merely-ostensible objects. My involvement with my cat Tiger consists in my bond with him. My involvement with the impossible cube consists in my preoccupation with this object as a possible avenue to show how inconsistent situations can exist.

Now if we allow the situation ‘Cliff in his apartment typing away at the keyboard of his laptop to produce disreputable screeds’ to comprise the superset ‘ostensible objects of perception’, not just the proper subset comprising ‘physical objects existing in time and space’, that situation will contain, as a sub-situation, the impossible cube I visually experience on my screen. The situation comprising this impossible object is not compatible with itself. It is constantly excluding itself. In a way that sounds very Hegelian, it is incoherent, existing in contradiction.

One would expect such an object to be unstable, and this one in fact is. The horizontal bar behind is constantly threatening to become the horizontal bar in front of the vertical bar, with the result that the cube suddenly switches from being vied from above to being viewed from below.

Now let’s take s to be the singleton situation that comprises the impossible cube depicted on my laptop screen. Let A be the proposition expressed by the English sentence “The horizontal bar is behind the vertical bar”. (That this proposition is about the impossible cube is given.) I venture the claim that no situation compatible with s makes A true. For surely the only situation that could possibly (I will leave ‘could possibly’ undefined) make A true is s itself. But as we have just seen, s is not compatible with itself. And surely no situation that is compatible with s will make A true. For example, that Tiger is to my left is compatible with s (this situation holds at the same time as s), but clearly does not make A true. Nor can I think of any other situation apart from s that would make A true.

Therefore, ~A counts as true according to the truth condition for negation stated above. But it is also the case that no situation compatible with s makes ~A true. Only s could make ~A true. For example, that Tiger is lying to the left of me, always about to press his paw onto the keyboard, clearly does not make true ~A, ie., “The horizontal bar is not behind the vertical bar”. But s is not compatible with itself. So ~(~A) is true, which means of course that A is true. So the truth condition for negation gives us the result that a situation exists for which A ^ ~A is true. Inconsistent situations do exist.

This way we can avoid the Classical Logic paradox (A ^ ~A ) –> B. That is to say, from a contradiction every proposition follows — the Principle of Explosion. This principle would, of course, allow for multitudinous irrelevant implications and therefore cannot hold in Relevant Logic. We can avoid the principle of explosion if we maintain that situations comprise ostensible objects, which include not just actual objects, and not just possible objects, but also impossible objects.

Naturally, a possible world cannot contain impossible objects. (Duh.) Unlike situations, possible worlds must be consistent. So, as Mares notes, a possible world must be both complete and consistent:

First, whereas worlds are complete, situations can be incomplete. To use the terminology of Barwise and Perry … worlds decide every issue. That is, they tell us, for any proposition, whether that proposition is true or false. Situations, on the other hand, do not decide every issue. In some situations, the information whether a given proposition [e.g., “Bigfoot roams the forests of western Washington state”] is lacking. This property of situations is sometimes expressed by saying that at some situations ‘the principle of bivalence fails’. Second, situations need not be consistent. That is, there are some situations that make contradictions true. Possible worlds, on the other hand, are completely consistent.

Mares, RL, pl 27

I have tried to use the concept of possible worlds that are epistemically accessible to a knower S as a way of making sense of the concept ‘partial situation’, that is to say ‘non-bivalent situation’. Partial aka non-bivalent situations are ones in which A v ~A and A –> A may not be true. Trimming out all the possible worlds (that is, all the ones not identical with the actual world) that are epistemically accessible to me leaves me with that portion of the actual world about which I have information — Cliff’s comprehensive actual situation. Defining this situation this way already makes it relative to a knower, i.e., a person in a situation. A situation has a center — this person.

Likewise, I have tried to use the concept ‘ostensible object’ to make sense of the concept of an inconsistent situation. As the center of my comprehensive actual situation, my perceptual apparatus gives me the ability to experience ostensible objects that do not exist as well as ostensible objects that do exist. Some of the merely ostensible objects are possible objects (e.g., the purple rhinoceros gazing peacefully at my feet); others are impossible objects (e.g., the impossible cube inspired by Escher). A ^ ~A is true for at least some situations containing impossible objects.

So now Cliff’s comprehensive situation includes more than just the actual world. As a possible world, the actual world cannot contain impossible objects. Containing a surplus, so to speak, over and above the actual world, my comprehensive situation can no longer be described as just a restriction of the actual world. We need now to speak of just ‘Cliff’s comprehensive situation’, leaving out the ‘actual’.

The falsity of A ^ ~A and the truth of A –> A (aka A v ~ A) still hold for possible worlds. So aficionados of the law of the excluded middle and of the principle of non-contradiction need not freak out. These fail to hold only for situations. On the other hand, situations are better suited than possible worlds to account for natural language:

…the ubiquity of expressions that clearly depend on restricted parts of the world gives us reason to believe that natural language is built primarily to talk about restricted parts of the world and not talk about complete possible worlds.

Mares, RL, pl 40

The End: And here I will end this installment of my writing-to-learn project for Mares’ Relevant Logic A Philosophical Interpretation. My tiny fragile bark is already way out as it is in the open sea, trying, far from the safety of any shore, to cope with some huge and frightening waves. I do not need to go any further at the moment. I leave you, gentle reader, with Emil Nolde’s depiction of a person, in their finitude, in the midst of, in the thick of a situation they must grapple with.

Emil Nolde, The Sea I

Lumber Room:

Compatibility/Incompatibility:

What follows is a series of points that I will be trying to shoehorn into this essay as I revise it now that I have hammered it into at least an initial shape.

[[[Let me say that a situation such as the shell #1 situation is epistemically compatible with another situation — say, my comprehensive actual situation — if it is epistemically possible for it and that situation to combine so as to comprise a larger situation. The situation whose defining description forms the key description of the shell #1 possible world is a possible part of my comprehensive actual situation. It is therefore compatible with that situation, and that situation is compatible with it. ]]]

[[[Conversely, every situation that maps to a key description of a spawned-from-the unknown possible world that is not epistemically accessible for me is epistemically incompatible with my comprehensive actual situation. Above, I gave an example of a spawned-from-the unknown possible world in which there is always the possibility of a rogue peanut in one of Elizarraraz’ shell games, taking its cure from the behavior of electrons as described by my undergraduate chemistry TA, ending up on the nose of the Mona Lisa. Given our knowledge that peanuts (nor any other objects on the post-quantum level) do not behave this way in the actual world, we know something that rules out the identity of this possible world with the actual world. The rogue peanut world is not epistemically accessible for me from my location in the actual world, which means that the rogue peanut situation is not compatible with my comprehensive actual situation. To rule out is to exclude; to exclude is to render incompatible. Conversely, to fail to rule out is to fail to exclude; to fail to exclude is to render compatible.]]]

[End with drive towards inconsistent situations?]

. [[[This knowledge may or may not be knowledge that I can access immediately, or even knowledge that I know I have. ]]]

[[[[[[[That the peanut be under shell #1, for example, fits in with, does not conflict with, is not ruled out by the knowledge I have about the actual world. I make this claim rather confidently — in fact, I am willing to bet my paint brushes on its truth. Nonetheless, unless I can be said to both to know and to have at my fingertips everything I know, perhaps I cannot be absolutely certain that in the body of knowledge I possess, there is an item of information that would rule out the the peanut’s being under shell #1 within the currently unknown-to-me part of the actual world. I might, while sleepwalking, for example, have one-twentienth consciously seen Elizarraraz filling shell #1 with cement, rendering the presence of the peanut there physically impossible. (In no possible world nomically accessible from the actual world can a shell filled with cement hide a peanut. I assume.) ]]]]]

[[[[[This visual information might be stored in my brain, ready to pop out at any time, or perhaps accessible only through deep hypnosis, but at any rate information unknowingly possessed by me. In that case, I would possess information that would rule out the situation’s being part of the actual world. This is so even if at the same time I also have the opposite belief. I can be in two cognitive states at the same time whose contents directly contradict one another. This point will become important later when I consider inconsistent situations. ]]]]]]

[[[[[Nonetheless, I rather doubt that anyone would want to gainsay my ‘I know nothing that would ….’ claim should I make it. And certainly there is a fact of the matter regarding whether this claim is true — whether there is nothing in my body of knowledge that would rule out the presence of the peanut under shell #1 in the actual world. ]]]]]]

I[[[The descriptions “the shell is empty” and “the shell hides the peanut” clearly preclude one another; they are metaphysically incompatible. ]]]]] Likewise, I might discover later that what I took to be the hollow of the shell was in fact filled with concrete, rendering its hiding the peanut physically impossible. The descriptions “the shell is filled with concrete” and “the shell hides the peanut” clearly preclude one another; they are physically incompatible.

But until I know things like this, the possible situation in which the peanut lies under shell #1 could be — in some sense of “could be” — a part of the actual world. The ‘could be’ is epistemic, stemming from the epistemic concept of relative ignorance, or lack of knowledge. So far, I do not know anything that would rule out the situation’s being part of the actual world, so this being a part of is still a possibility. Nothing that “automatically precludes from the get-go its being a part of the actual world and therefore something that could eventually be discovered to form part of my comprehensive actual situation.]]]]]

[Logically/metaphysically accessible possible worlds and nomically accessible possible worlds do not need to refer to an agent S. Possible worlds that are epistemically accessible for S do need to refer to an agent S. These are possible worlds that smuggle in the concept of a center. They invite the notion of a situation.]

Two situations are compatible within a possible world at time t if and only if they can (epistemic “can”) hold at the same time within that possible world. Let me produce some examples. [Non-normal situations can be incompatible with themselves. Satisfied descriptions. ]

[[[If my cat Tiger is sitting on a mat (where I have deposited him to keep him off my keyboard) at the same time that the doorbell rings, the situation ‘Tiger is sitting on the mat in my apartment’ is clearly compatible with ‘the doorbell is ringing in my apartment’. That these two situations hold at the same time t in the actual world is proof positive that they are compatible in the actual world at t. Now I know nothing that would rule out the identity with the actual world of the possible world in which this situation in my apartment holds. Which is good, because the actual world is identical with this possible world. The fact that the two situations do hold at the same time in this one possible world, namely, the actual world, shows that they can hold at the same time in this world; and this possibility is (at least) an epistemic possibility. ]]]]

[[[[[[[[[[[[[[[[[[[[[[[[[[Identity a sufficient condition for compatibility for normal situations. Two descriptions — stable. Not flashing this way then the opposite. Not like “This sentence is false”.Bring up the over/under business much much earlier.]]]]]]]]]]]]]]]]]]]]]]]]]]

Now let me circumscribe the rogue peanut world to just one situation s1 within that world, namely, the shell game the Elizarraraz has set up in the Pilsen neighborhood of Chicago, that somber city, within an easy walking distance of Western Avenue.

To rule out is to exclude; to exclude is to render incompatible. submit, then, the following. Our knowledge (at least I hope that is what that is) that rogue peanuts do not exist in the actual world rules out s1‘s being a situation in the actual world. Therefore, s1 excludes my comprehensive actual situation, s0. s1 and s0 are incompatible situations; and s1 is incompatible as well with any situation within s0. To rule out is to exclude; to exclude is to render incompatible. Therefore, no situation drawn from a spawned-from-the-unknown possible world is compatible with my comprehensive actual situation or with any of its sub-situations.

If it were so ruled out. I submit that, conversely, every situation drawn from a possible world that is If it were so ruled out. epistemically accessible for me from my location in the actual world is compatible with my comprehensive actual situation s0 and with every sub-situation of s0. For every such situation is possibly — in the epistemic sense of ‘possible’ noted above (think: ‘the peanut is possibly under shell #1) — a part of the actual world. The description mapping to every such situation fails to get ruled out by my knowledge (if it were so ruled out, it obviously would not be part of a possible world epistemically accessible for me from my location in the actual world). Failing to get ruled out is failing to get excluded is failing to be incompatible. Failing to be incompatible is to be compatible.

[[[[[An example of failing to believe that an item of knowledge is part of my body of knowledge.]]]]]]] Likewise, I may falsely believe that a situation is compatible with my comprehensive actual situation when it fact it is not. Suppose, for example, that I really have taken the eight graduate courses in quantum mechanics (with a grade no lower than a B+ in any of them and I got that B+ in just one of the courses) came in just one course. Suppose, further, that I acquire the knowledge that peanuts cannot, while obeying the laws of nature, behave like electrons in the way described by my undergraduate chemistry TA. But all my friends are still New Age crystal devotees who think that peanuts can behave like electrons and display the same effects as they. In the midst of a party in which I imbibe too much in their company and watch too many You Tube videos with them about quantum physics and stare at too many crystals, the wages of tribal identity set in for a moment and I come to believe during the party that peanuts, which only the moment before had been contently abiding under one of Elizarraraz’ shells, can suddenly find themselves the next moment on the nose of the Mona Lisa. The morning after, I come to my senses, no longer under the intoxicating tribal influence of my peers, and no longer believe this. But during the party I held a belief that conflicted with my knowledge. Beliefs can conflict with other believes and with knowledge.

Even during the party, however, my belief did not determine my comprehensive actual situation. My knowledge, which although not occurrent at the moment, was nonetheless not wiped away by the party, continued to define my comprehensive actual situation. The rogue peanut situation continued to be incompatible with, ruled out by, my comprehensive actual situation.

Satisfaction of a description is important also in analyzing compatibility/incompatibility. The impossible cube is not satisfied by any object in the actual world. Since situations are descriptions (descriptions which are (were/will be) satisfied by the actual world when they are actual situations and not satisfied by the possible world when they are merely possible situations), two situations are compatible | incompatible in a possible world if their defining descriptions can be | cannot be satisfied in that world.

[Why use epistemic could and not, say nomic could? Because we are talking in finite situations. So at least theoretically two situations could be epistemically compatible but nomically incompatible. I submit, however, that we should stick to the epistemic notion of compatibility in analyzing human language and human coping with the world. That my neighbors’ walls could be cornflower blue is quite live and real for me. And, as I will show in a moment, one can use that ‘could’ to provide the truth conditions for “not” in the sentence “The walls of my apartment are not fuchsia” that can be comprehended by a finite human being. The corresponding nomic ‘could’ cannot be used that way. ]

[Finite situations appropriate for human language. This is a concept of compatibility that is in harmony with the notion of situations, which emphasizes their local and fragmentary character. My situation is a bunch of so far as I knows. My situation does not comprise the entire actual world because I am not omniscient. ]

[Henceforth, unless otherwise noted, I will be using “compatible” to mean “epistemically compatible”.]

Cannot be satisfied at the same time. My apartment walls cannot be portobello mushroom and fuchsia at the same time. The two descriptions are not compatible. They cannot be fulfilled by any situation in the actual world at the same time. The portobello mushroom situation and the fuchsia situation cannot obtain in the same possible world.

Identity in satisfied situations and simultaneity, then, are two sufficient conditions for compatibility. But there is a third sufficient condition that I want to consider, namely ignorance, my not knowing everything. Nomic and initial conditions example.

There is nothing so far in what I know that would render the description ruled out by what I know.

Truth:

[[[Situations whose boundaries are determined by the limits of one’s knowledge would divide the actual world into what is exposed to one — the finite situation itself — the dense part of the lump of cheese, the mesa surrounded by the fog — and everything that is hidden from one — the holes in the lump of swiss cheese; the fog surrounding the mesa.]]]

[[[Is the truth of the sentence a matter of change from the perspective of one’s own situation? If yes, then one’s own situation is not making the sentence true. Truth is never a gamble]]]

[[[There is no doubt that there are three shells before my eyes. This is so even if ‘there are three shells before my eyes’ is defeasible. If 1 through 4 have credences of less than 1, every exposed state of affairs within my comprehensive actual situation has a credence of 1. There is no doubt that there are three shells before my eyes. This is so even if defeasible. A credence is a number that is assigned; and I shall assign a credence of 1 to all those states of affairs in my comprehensive actual situation for which any doubt seems merely theoretical, not to be taken seriously in the course of one’s affairs. David Hume, for example would or at least should, assign a credence of 1 to the billiard table and the balls when he gets up from his skeptical studies to play a game of billiards. Certainly he would know nothing that would rule out the identity with the actual world the possible world in which the pool table actually does exist in space/time. But to discuss this further would take me too far afield.]]]

[[[Now if there is anything that could serve as a conduit of information between the hidden situation and my comprehensive actual situation such that states of affairs in the former could become exposed to me in my position in the latter, the probability-as-credence of this state affairs would increase to 1. If my neighbor is completely reliable, for example, the color of his walls would become exposed to me were he to tell me they are wine red. The probability/credence that they are wine red would increase from 1 divided by the number of paint colors to 1. That the walls are this color then becomes part of my comprehensive actual situation.]]]

[[[For a sentence to “reach out” so to speak to a hidden situation and to be ‘made true’ is for the probability/credence to increase to 1 from a lesser number. For a sentence that is about a state of affairs within a finite situation to be made true, it needs to have already been exposed and the garden-variety credence given to it to be 1 already. For a situation to make a sentence true is for the sentence to correspond to an exposed state of affairs.]]]

Within my comprehensive actual situation, the resources are lacking to make 1 through 4 true. As I will show when I get to the truth condition for negation, the resources are also lacking within that situation to make the negations of 1 through 4 true. Let A be any of the sentences 1 through 4. A is not true. If we adhere to a 2-valued logic, this means that A is false within the situation. But, as I will show, the situation also fails to make ~A true. So ~A is also false within the situation. A v ~A would be false because both A and ~A would be false. Bivalence would fail for my comprehensive actual situation.

And once it is turned over, it will be the case, I am rather inclined to think, that nothing in my body of knowledge rules out the identity with the actual world of the possible world in which the peanut always was (from the time Elizarraraz placed it under one of the three shells) present | not present under shell #1 the whole time. But there is something in my body of knowledge that rules out the identity with the actual world of the possible world in which the peanut was present | not present under the shell the entire time — namely, that peanuts do not behave like electrons. Or at least I hope that I know this. I might add parenthetically, by the way, that whether this really is an item in my body of knowledge is independent of whether I believe it to be.

[[[Need to discussion situation types vs. individual situations]]]

So while nothing in my comprehensive actual situation makes it true that either A (“the peanut is under shell #1”) or makes it true that ~A (“it is not the case that the peanut is under shell #1”), one is likely to be proceeding on the strong assumption that this is so. But if we confine ourselves to just the situation, the situation does not have the resources to make either A or ~A true.

[[[Within the worldly situation of such a being, every sentence would either be made true within the situation or fail to be made true. The probability of 1 through 4 each would be 1.]]]

[[[The epistemic version of truth, i.e., the correspondence to something actual, would then be the correspondence to something known to be actual.]]]

[[[So we have local truth on the one hand and global truth on the other. Local truth is not relative truth.]]]

Bivalence

[[Exposure makes the epistemically accessible situation a part of s0. I asserted above that a situation such as the shell #1 situation is epistemically compatible with another situation — say, my comprehensive actual situation — if it is epistemically possible for it and that situation to combine so as to comprise a larger situation. It could be (epistemic could) the case that the peanut is under shell #1, in which case the presence of the peanut there would become, upon exposure (say by turning over the shell or through a sonar device that is designed to detect the presence of peanuts under shells) a part of s0. Or it could be (again, epistemic could) the case that the peanut is not under shell #1, in which base the absence of the peanut there would be what becomes part of my comprehensive actual situation. ]]

[[That the walls of my apartment are portobello mushroom has already gotten exposed in the situation comprising my apartment in particular and in my comprehensive actual situation in general. Even if somehow I haven’t really noticed the color of my apartment walls (say, when I entered the apartment for the very first time), nothing blocks me from focusing my attention on it for a moment and saying “Aha, portobello mushroom”. The color of my apartment walls lies exposed in my situation. ]]

[[[Situations drawn from possible worlds epistemically accessible to S from their position in the actual world are the graveyard for the truth of A v ~A.]]] [[[Important that on my account the compatibility relation could hold between situations in different possible worlds, as long as the situations could be part of the actual world. I don’t know what trouble this will pose for me. Important to discuss the Tiger from a different planet possible world.]]]

[[[But every other sentient being is situated, that is to say, is plunged into a concrete situation that has them as its center and from whom much is hidden, and to whom some is exposed.]]]


Dissolving The Monty Hall Paradox

My intent in the following is to dissolve the paradoxical character of the Monty Hall game and to explain WHY it may seem so paradoxical at first.

I turn the Monty Hall paradox into a shell game. I do this partly because I loathe cars (the original game was a TV show in which the guest had to choose one of three doors. A car was behind one door; a goat was behind each of the remaining two doors). And partly because the traditional slipperiness of shell games fits my “insight” (or maybe it is a delusion).

Three shells are in front of me, one of which hides a peanut. My task is to select a shell in the hopes of uncovering the peanut. In the normal shell game, that is where the game ends — the selected shell is either hiding the peanut or it is empty. In a shell game of the Monty Hall persuasion, however, the person setting up the game (call him Elizarraraz), instead of turning over the selected shell to reveal what it is hiding | alternatively not hiding, turns over instead a shell which he knows to be empty. I am to choose one of the remaining shells.

None of the three shells is perceptually distinguishable for me from the others, except when Elizarraraz labels them with different sets of tags, a task which Elizarraraz performs honestly and reliably. There will be three sets of tags. Each of these sets defines a different shell game of the Monty Hall persuasion.

Each of the three games is to be repeated in ten million times. (A bit exhausting, perhaps, but this is a thought experiment after all.) In each of these ten million iterations of each game, Elizarraraz uses some randomizing device to choose under which shell to place the peanut.

In the first game, Elizarraraz labels the shells #1, #2, and #3. Each of these shells has an identity based on its trajectory through four-dimensional space/time. As anyone who has read the popular literature on Einstein’s Relativity Theory (by the way, my knowledge of the theory goes no deeper than this) knows, objects with mass are continually moving through at least one dimension, and possibly through all four (the dimensions typically plotted on the graph with x, y, and z axes, plus the temporal dimension) at once. When it is in everyday thought and speech regarded as at rest, it is in fact still moving through time. Each of the three shells has its own trajectory through the four dimensions of space-time which individuates it. I will call this identity of the shells their “natural identity”.

I select a shell. Elizarraraz now turns over a shell he knows to be empty, and tells me to select again. I choose by employing a randomizer through each of the ten million iterations of the game. My strong intuition, which I have yet to prove in at least a quasi-mathematical way, is that out of the ten million iterations, 1/3 of the time, shell #1 is hiding the peanut, and 1/3 of the time shell #2 is hiding the peanut, and 1/3 of the time shell #3 is hiding the peanut. The possibility space (aka sample space) is {shell#1p shell#2p, shell#3p}. This set has a cardinality of 3, and the probability functions assigns a value of 1/3 to each member of this set. (Or so I am confident I will eventually prove.)

In the second game, Elizarraraz labels the remaining shells ‘shell A’ and shell B’. He employs a randomizing device to choose which of the remaining shells to attach which label to. In doing so, he creates two “artificial” objects, shell A and shell B. Each of these shells has during any given iteration of the game a part, which can be either shell #1, shell #2, or shell #3. I will call these parts temporal parts. As befits the slippery character of the classical shell game as one may once have encountered it on the EL in Chicago, in which the maybe-not-totally trustworthy person who sets the game in motion through their sleight of hand relies on the difficulty in perceiving any differences among the shells, the individuating trajectory through space-time described in the first game above is not important here. The temporal parts of shell A | alternatively shell B fuse together, so to speak, to form for shell A | B a single object. This is not a “natural” object, a “natural” identity, but an “artificial” one.

Let me provide another example of an artificial identity. In a certain Pho restaurant in Houston I used to frequent before the pandemic, the employee behind the cash register would take my order, then hand me a sign on which was printed, say; “table #6”. I could place that sign on any non-occupied table I wanted. Those tables are individuated by their trajectory through space-time as various customers place it on various tables, but they also serve as temporal parts for an artificial object, an artificial identity named “Table #6”. Shells A and B are like Table #6.

Or again, suppose that a a population chose each year a different individual to be its king — but for whatever reasons rooted in their mythology thought of the king as ‘the same king’ throughout the years. Or again — one more time — a novelist might emphasize the humdrum uneventfulness of the passage of time for a character by making the character not think of the proprietor of the dry-cleaning shop they frequent as different people even when in fact they are different people. The difference is not salient for the character. Shell A | shell B is an artificial object like the king or the fictional proprietor of the dry-cleaning establishment.

My strong intuition, which I have yet to prove in at least a quasi-mathematical way, is that out of the ten million iterations, 1/2 of the time, remaining shell A will turn out to be hiding the peanut, and 1/2 of the time remaining shell #2 will turn out to be hiding the peanut. Here the possibility space for the game is the set {shellAp, shellBp}. The possibility space has a cardinality of 2, and the probability function assigns to each member of this set a value of 1/2. (Or so I am confident I will eventually prove.)

In the third game, Elizarraraz labels the remaining shells as ‘initial choice’ and ‘switched choice’. This time the choice of label is made on the basis of the shell’s actual history, not on the basis of a randomizing device. Again, each shell so labelled could be shell #1, shell #2, or shell #3 — I am not able to tell because I cannot distinguish the shells without the labels. This is a shell game, remember, the whole point of which is to be slippery.

Out of the ten million iterations, 1/3 of the time the peanut will be under the shell labeled ‘Cliff’s initial choice’, and 2/3 of the time the peanut will be under the shell labelled ‘Not Cliff’s initial choice.’ This has been proven time and time again. The mathematician Paul Erdos (the guy who is a machine for turning coffee into theorems) was at first skeptical, but became convinced once he saw that computer simulations of the game, repeated ten million times, result in the peanut being under the initial-choice shell 1/3 of the time and under the switched-to shell 2/3 of the time. The computer simulations are completely dispositive for this result.

Ten million repetitions of the game in a computer simulation will confront you with the brute fact that the ratios are 1/3 and 2/3. But you can also “see” the with the mind’s eye the necessity of this result using tree diagrams like the one shown here, a method that reminds me a bit of Jorge Luis Borges’ THE GARDEN OF FORKING PATHS.

To get the proof down to a more concrete level, appealing, not just to the mind’s eye of the intellect, but also to the senses of sight, touch, and hearing, picture, then draw a system of branching metal pipes (which one can feel as well as see) oriented downward. A drop of water enters the system at the top. As the drop rolls down, it encounters various junctions, which are either “tridents” or forks. The forks arise when one pipe is closed off, mirroring Elizarraraz’ turning over a shell that he knows to be empty. Which pipe the drop flows down is a matter of chance (either 1 in 3 or 1 in 2). One set of pipes mirrors my sticking to my initial choice of shells. Another set mirrors my switching to the other shell. The first set of pipes leads to a coffee can labelled ‘initial choice’. The second set of pipes leads to a coffee can labelled ‘switched choice’. Were a drop of water to be introduced ten million times (okay, ten million and three times) into the system of pipes that the drawing schematizes, one would hear the drop plop into the ‘initial choice’ can 1/3 of the time, and into the ‘switched choice’ can 2/3 of the time.

These two cans have natural identities. That is to say, their identity is determined by their trajectory through the four dimensions of space/time. That their identities are natural, not artificial, may make the 1/3 | 2/3 result more intuitive.

The possibility space for the cans with their natural identifies is {initial_choice_can, switched_to can} In that set, the probability function assigns the value 1/3 to the initial_choice can, and value of 2/3 to the switched_to can. Likewise, the possibility space for the shells, with their artificial identities, is {initially_chosen_shell, switch_to_shell}. In that set, the probability function assigns the value 1/3 to the initially_chosen_shell, and a value of 2/3 to the switched_to_shell.

The complex character of the artificial identities of the “shells” in the second and third games means that those “shells” are constantly switching out and exchanging, so to speak, their temporal parts. For any given repetition of the second game, the current temporal part of shell A might be shell #1 and that of shell B might be shell #3 [In creating the artificial identities, we are “assembling” them from parts]

With any luck, I have dispelled the drastically counter-intuitive nature of the Monty Hall shell game. By now, it should also be plain why this game seems at first to be so frigging paradoxical. It will only seem paradoxical if one confuses the third game with the second game.

These are two different games, but it is easy to confuse them. Here are the two shells facing me in the moment after all — OF COURSE if I ignore their history I will thinking of them (whether the corresponding labels have been applied to them or not) as ‘remaining shell A’ and ‘remaining shell B’. Under THAT description, the probability really is 1/2. The only way I could decide between the two would be to flip a coin, since the information expressed by the descriptions ‘initially chosen’ and ‘switched to’ is not available to me. So, as the two shells face me in the moment, sneering at me, making me forget the entire tree of tridenting and forking possible paths the shells could be taking during the ten million iterations, OF COURSE I will be thinking (wrongly) that my chances will be 50/50 regardless. But as Judea Perl notes, much as the statisticians (he claims) loathe the idea, the history of an entity counts in determining probabilities. It does so by determining both what is contained in the set comprising the possibility space, and the values assigned to those artificial identities by the probability function. Once the history-determined artificial identities of these two entities ‘shell initially chosen by Cliff’ and ‘shell not initially chosen by Cliff’ becomes available to me, the probabilities are no longer 50/50, but 1/3 or 2/3 respectively.

The three different games are indeed three different games, each with different entities in their sample space and different probability functions assigning different values to those entities. The confusion of the three games with their corresponding possibility spaces is what creates the sense of paradox.

Links

Ternary relation and conditionality


A Question For Edwin D. Mares

This is the question I have at the moment for Edwin Mares: Doesn’t the relevant conditional ‘if p then p‘ (I use ‘conditional’ to include implication and entailment — more on this below) pose as much of a challenge to an interpretation of the relevance-making relation Rstu in which s is an ‘information link’ enabling ‘situated inference’ as it does to the interpretation in which s is a channel of information? What precisely is the difference between an ‘information link’ and a ‘channel of information’ anyhow? [Doesn’t implication require some sort of necessity (This is the insight behind strict implication) even if not logical necessity? We can model different kinds of necessity based on different accessibility relations. One of these surely apply. Is it possible to do this with situations rather than possible worlds? Can we make an implication both necessary and defeasible? (Yes, with epistemically accessible possible worlds)

As I explored this issue in an attempt to make my understanding of Relevant Logic a bit less shallow, I came to the conclusion that ‘information link as part of situated inference’ does better account for the implication than does ‘information channel’. But of course more questions arose in my mind.

Let me start motivating these questions by explaining the interpretation of Rstu in which s is a “situation” serving as an information channel connecting two other situations, t and u. Much of what can be said about s as ‘information channel’ will also hold true for s as information link — or at least to what degree this is so will be one of my questions. And in the course of doing that, I hope to make sufficiently clear what in the heck Rstu is in the first place — sufficiently clear, I hope, to make it reasonable to think — reasonable at least by my own lights even even if by no one else’s — that I have some vague inkling of what I am talking about.

After exploring the ‘channel of information’ and ‘situated inference supported by information link’ interpretations of the relevance-making relation Rstu, I will briefly look at one more interpretation, which I will call the ‘persistent lack of counterexamples’ interpretation.

A Problem That Arises When The Relevance-Making Relation R Gets Interpreted In Channel Theoretic Terms: On pages 54-55 of his RELEVANT LOGIC A Philosophical Interpretation Mares says:

Barwise … and Restall … have suggested treating the Routley-Meyer semantics in channel theoretic terms. We should think of situations as being sites of information. One site, for example, contains the news reporter at Ground Zero. Another site contains me and my lounge and television set.

Some situations are also channels of information that connect sites. On their interpretation, we read Rstu as saying that s is a channel between t and u.

This interpretation motivates a slightly weaker logic than R. Here is the problem: on the channel theoretic reading, Rsss says that s is a channel from itself to itself. It does not seem intuitive to require of all situations that they be [at least?] channels from themselves to themselves, at least not if the notion of a channel is a generalization of cases like that of the link between television cameras and a television.

Edwin D. Mares, RELEVANT LOGIC A Philosophical Interpretation (Cambridge, Cambridge University Press, 2004), pp. 54-55. Henceforth RL.

But if we look at the matter more closely, we end up talking anyway about situated inference enabled by information links. And in the process we come up with an alternative to having to posit a situation that serves as a channel of information to itself. Finally, while I don’t know about you, dear reader, I at least end up with a somewhat less shallow understanding of Relevant Logic.

[Situated inference via information link, channel of information which I take to be a special case of this; lack of counterexamples through time.]

We end up talking anyway about situated inference enabled by information links: What is an information channel anyhow, and why should we identify the situation s with it?

Consider the doorbell to my apartment. My favorite doorbell repairman, who has 25 years experience repairing doorbells, has just given it a health check and declared that it is, without the slightest shadow of doubt, functioning properly. (I feel obligated to ensure that the examples I adduce in my philosophical thought experiments are in good working order.) 100% of the time, when my doorbell rings, someone (so far it had not been — alas! — Channing Tatum) or something (for example, a twig blown onto it with gale force by Hurricane Ike or Hurricane Harvey) is depressing the button outside. The connection between the doorbell’s ringing and the button’s getting pressed seems absolutely reliable. The reliability of the connection seems perfect. (But note the weasel word ‘seems’.)

Because of this reliability, of this 100%, I regard the doorbell’s ringing as telling. It counts as information that someone or something is pressing the button outside. If, somehow, someone or something was pressing the doorbell just 99% of the time when the doorbell rang (say, I had good reason to suppose, on the basis of a camera directed a the button, that no one or no thing was depressing the button during 1% of the times), that the doorbell is ringing would no longer be telling. I would both be in a twilight-zonish mood, and I would have to consider the possibility, when the ringing sound happens, that no one or nothing was pressing the button outside. The doorbell’s ringing would no longer have the status of information that.

If I may be allowed to wade for a moment into some murky waters, the ringing sound has to increase to 1 the probability that the button outside is getting depressed for that sound to count as information that the button is getting depressed. (Yes, I know that it is sometimes supposed that only logical truths can achieve a probability of 1, but bear with me for a while. Maybe for a long while.)

Because the doorbell apparatus is in a state such that, at least when the apparatus is in this state, the ringing sound raises to 1 the probability that the button outside is getting depressed, that apparatus counts as a channel of information linking the former to the latter. Let me call this channel, the situation of the doorbell apparatus, situation s. This situation, I submit, renders true the following implication:

1a) If the doorbell in my apartment is ringing, then someone or something is pressing the button outside

Note that s has to be in place for 1a) to be true. If a different situation were in place the implication would not be true. If, for example, the apparatus were completely disfunctional so that the causal chain from button to sound never occurred, my hearing the doorbell sound, through whatever bizarre, non-normal freak circumstances (say, aliens were producing it somehow), would obviously not raise to 1 the probability the button outside was getting pressed, and would not be information that this is so. Similarly, if the apparatus were completely functional, but someone created a wireless connection to the part that actually creates the sound so that the sound can be produced bypassing the causal chain from the button to that component, a different situation s would be in place, one in which the previous channel of information no longer existed. Likewise, the previous situation s would no longer be in existence were some physical peculiarity of the apparatus to arise (not really imaginable by me) that made it possible for a wave of electricity to form independenttly of the button’s getting depressed and triggering the ringing sound.

In other words, the apparatus has to be in such a condition — s has to be such — that, as long as s is in place, the button’s getting depressed is the only “possible” cause of the ringing sound’s getting produced. Showing how this could be can get a little tricky; one needs, I think, to mark off s from other situations and isolate it, I believe, conceptually from them. (Consider this to be promissary note 1.) But before discussing this, let me turn to the twin brother of 1a):

1b) If someone or something is depressing the button outside, then the doorbell inside is ringing

As long as the doorbell apparatus is in good condition (situation s), pressing the button outside will result in the ringing sound getting generated inside. That is to say, provided that the laws of physics don’t change. Let me venture, then, the claim that the production of the sound is nomically necessary, given s, when the button is pushed. In all possible worlds which obey the laws of physicals of this actual world (now), pressing the button will generate the ringing sound provided situation s holds. If, say, three months from now the laws of physics abruptly change so that, in addition to grand pianos falling upward, pushing the button no longer generates the sound even given situation s, that world would not be nomically accessible from the actual world now. It would be excluded from the set of all possible worlds nomically accessible from this actual world.

Because the production of the sound is nomically necessary in s, that the doorbell is being pushed is, I venture, information that the ringing sound is getting generated inside. This is so even if one does not hear this sound as they push the button. That pressing the button serves as information (given s) that the ringing sound is happening inside renders 1b) a true implication and gives one a license to infer that this is so.

Should s change so that, say, pressing the button generates the ringing sound just 99% of the time, the button’s getting pressed would no longer be information that the sound was happening inside. Let’s say that a wire has come loose in such a way that the connection is made just most of the time, not always. In those 99% of the cases, the causal relation would be there, but, because of the 1% of the cases, the informational relation is not there. Even when the pressing the button does result in the sound’s getting generated, pressing the button is not information that the sound getting generated. To be a channel of information, s has to result in perfect reliability. In the following passage, Mares is talking about informational links, which is, I take it, different from a channel of information (I take the latter to be a special case of the former); nonetheless, what he says here about informational links also applies to channels of information:

To be an informational link [or a channel of information] a relation needs to be perfectly reliable. As Barwise and Seligman argue, causal relations are often not reliable enough to be considered informational links. They use the example of a flashlight with an unreliable connection between the button and the light ((Barwise and Seligman 1997), p. 17). Sometimes pushing the button turns on the light, but sometimes it does not when other factors come into play, such as a wire’s coming loose.

RL, p. 44

Nomic necessity renders 1b) true. I think, however, that more than nomic necessity is needed to render 1a) true. I now turn to 1a) again (making good promissary note 1).

When the doorbell apparatus is not perfectly reliable, 1a) may still be true. For even though pressing the button does not produce the ringing sound 1% of the time, it remains true that 100% of the time someone or something is depressing the button outside when it does ring. This is why affirming the consequent is a fallacy. One cannot go from:

If the doorbell in my apartment is ringing, then someone or something is pressing the button outside

Someone or something is pressing the button outside

Therefore the doorbell is ringing

For if s doesn’t hold because the apparatus is faulty, while there may be a flow of electricity from button to the part producing the sound, there is no flow of information. Certainly in this particular case the 1% failure rate precludes the consequent from being information that the antecedent is true. So no conclusion that the sound is getting produced for you.

fdfsdfdfdd

he circumstance in which the condition of the doorbell apparatus is in this state of the doorbell apparatus situ 100% of the time when the ringing sound is produced, it serves as a channel of information linking the latter to the former. is functioning properly, it is able to serve as a Let me start with an example that, I believe, will make the channel of information conception intuitive.

The following example makes the case for treating s in information-theoretic terms powerfully intuitive, at least for me. When the doorbell apparatus of my apartment is in good condition, pressing the button outside reliably generates a ‘doorbell’ sound inside my apartment; and when I hear the doorbell sound inside my apartment, 100% of the time someone or something is depressing the button outside. Given the good condition of the apparatus, and given the laws of physics, that apparatus serves as a channel of information s through which information flows from the button’s getting depressed (situation u) to the doorbell sound inside my apartment (situation t), such that when I hear the doorbell sound, I have received information that the button outside is getting depressed. This flow of information supports the following two implications (assuming these are implications and not a defeasible natural-language indicative conditional, but I will ignore the difference for now).

1a) If the doorbell in my apartment is ringing, then someone or something is pressing the button outside

1b) If someone or something is depressing the button outside, then the doorbell inside is ringing

It is the information, not the causal relation that supports these two implications. (I will assume they are implications and not natural-language conditionals.) Provided that 100% of the time — provided there are no counterexamples, ever — someone or something is depressing the button outside when the ringing sound is produced inside, the latter is information that the former is occurring. This of course is not a causal relation going from sound to button, though it is supported by a causal relation from button to sound. If a counterexample ever occurs — a ringing sound without the button outside getting depressed — the sound can no longer count as information that the button is getting depressed. This has to be 100%. It has to be 100%. This presupposes that the causal chain starting with the button’s getting pushed is the “only possible” cause of the the sound, in a sense of “only possible” I am about to get to.

1b) because they would not be true if something were wrong with the doorbell apparatus. If, say, some of the wiring in that apparatus came loose so that (full disclosure: I know precious little about doorbell apparati) pressing the button outside resulted in the ringing sound inside just 99% of the time, 1b) would be false because the button’s getting pushed would no longer be information that the doorbell was sounding inside. This is so even though there is a causal relation between the antecedent and the consequent.

But given the good condition of the doorbell apparatus, there is no way — in a sense of ‘no way’ that I am about to explore – that the doorbell can ring and the button outside not get depressed. Likewise, there is “no way” that the button outside can be pressed and the doorbell inside not ring. That is to say, the doorbell must be a reliable conduit of information between u (the outside of my apartment) and t (the inside of my apartment, where I hear the ring sound). This conduit must be perfectly reliable, Mares says:

To be an informational link a relations needs to be perfectly reliable. As Barwise and Seligman argue, causal relations are often no reliable enough to be considered informational links. They use the example of a flashlight with an unreliable connection between the button and the light ((Barwise and Seligman 1997), p. 17). Sometimes pushing the button turns on the light, but sometimes it does not when other factors come into play, such as a wire’s coming loose.

RL, p. 44

[The reader should compare this concept of information with Dretske’s dicussion of that concept in his KNOWLEDGE AND THE FLOW OF INFORMATION. ]

1b above is contingently true because it is true only when and only as long as the apparatus is perfectly reliable in transforming the energy of the button’s getting pushed into the ringing sound. There are possible worlds — say, the actual world three months from now — in which 1b is false. Three months, from now, the apparatus might not be in good condition because some of the wiring has come loose (full disclosure: I am completely ignorant of how doorbell apparati work), so that pushing the button outside resulted in the ringing sound inside just 99% of the time, not 100%. The causal connection would not be perfectly reliable, and therefore could not ground an information-that relation between button and ringing sound.

Reliability is a property attaching (or failing to attach) to equipment and to signals. The doorbell’s ringing is a reliable signal that the button outside is getting pushed because the doorbell apparatus is functioning reliably. Contingency, on the other hand, is a property attaching to propositions and to relations between propositions. That the doorbell is ringing is a contingent proposition because it is possible for it not to ring. The proposition expressed by the English sentence ‘the button is getting pushed’ is related contingently to the proposition ‘a ringing sound is occurring inside’ because the latter’s following from the former hinges upon the reliability of the doorbell apparatus. Because this relation is contigent, 1b is a contingent proposition. Likewise, 1a) is contingent because, even when the doorbell apparatus is functioning reliably, someone could conceivably set up some wireless Rube Goldberg setup producing the ringing sound in such a way that bypasses the button.

But the apparatus also has to be reliable — indeed, it has to be perfectly reliable. If there even one case occurred in which s held (the doorbell apparatus is in good condition), the button outside was pushed, but no ringing sound ensued — and how could that happen if all of a sudden the laws of physics no longer held? — then 1b would not be a true implication. Likewise, if there were even one case in which the doorbell rang (t), and the apparatus was in good condition (s), but no person or thing was depressing the button outside (u) — suppose, quite implausibly, that a poltergeist is at work inside the apparatus, in no way degrading its condition — then 1a) would not be a true implication.

The connection between antecedent and consequent in 1a and 1b) is contingent (RL, p. The antonym for ‘contingent’ is ‘necessary’.

In these cases, 1a) and 1b) would be defeasible. A proposition is defeasible if extra information can come to light showing that it is false. But implication, Mares says, is indefeasible:

A defeasible inference is one that may not hold if extra information comes to light. Implication, in the sense that we mean it here, is a non-defeasible relation between propositions.

RL, pl 45

But isn’t it the case that all contingent propositions (among which empirical propositions occupy pride of place) are defeasible? The proposition expressed by the English sentence ‘every body with mass warps the space around it’ could — in some sense of ‘could’ — turn out to be false if we found a body with mass that does not warp the space around it. That the doorbell’s ringing always means the button outside is getting pushed could turn out to be false if we find out that poltergeists exist after all (maybe it is is some weird quantum entanglement thing), or, a bit closer to home, there is some physical law that would allow for a burst of energy suddenly appearing leading to the production of the ringing sound in a way that bypasses the button. In that case, it would seem, all “implications” would be defeasible, that is to say, there would be no implications at all. At best, whenever it seems extremely implausible that further information would make the falsity of an implication inescapable, we can posit the indefeasibilty of certain if-then propositions as an ideal. Implications in the narrowest sense of the term would exist in an idealized realm in which, in contrast to all the murky uncertainty of the world we live in, there is not even the faintest shadow of a doubt that there are no poltergeists.

Let’s say that non-idealized, defeasible implications are a species of natural language conditionals. Natural-language implications, as opposed to ‘in the strictest sense of the term implications’, have defeasibility as one of their properties. They also have as one of their properties non-transitiveness which make them resistant to regimentation. But these are outside of the scope of this post, and I will not be considering them any further. Henceforth, unless otherwise noted, I will be using ‘implication’ to mean ‘implication in the strictest sense of the term.’ I now turn back to these.

[To be true in the strictest sense of ‘true’, and not just an idealization, aOfn implication has to indefeasible. And of course, ‘indefeasible’ is a synonym for ‘necessary’. As is indeed, to come full circle, ‘perfectly reliable’ — as in there is no way, given situation s, that when the doorbell is ringing there is someone or something outside not depressing the button. Deductive inference also implies necessity]

It would seem that we have fallen into a contradiction. Implication, as opposed to entailment, is contigent, not necessary. But contingent implications — if any exist — are indefeasible. Indefeasibility is necessity. Extra information cannot show up later that would render the indefeasible implication false. How can this apparent contradiction be resolved?

But before I try to resolve this contradiction I want to point out that I have just arrived at situated inference. Implication is inextricably tied to inference because we must be able to infer the consequent from the antecedent.

Mares ties this reliability to deductive inference. If I may add yet one more plate to the stack, I will be exploring the connections between reliability, information, and deductive inference shortly. In a nutshell, implication needs to mirror deductive inference. For now, however, I would like to note the connection Mares wants to draw between deduction, which is non-defeasible, and the ‘perfect reliability’ of the informational connection between sites required by implication.

The problem [posed by unreliable causal connections] is that unreliable connections do not warrant deductive inference. At best, they can be used to justify defeasible inference. A defeasible inference is one that may or may not hold if extra information comes to light. Implication, in the sense that we mean it here, is a non-defeasible relation between propositions.

RL, pp. 44-45

Add to the mix the idea that implication is a contingent, not a necessary conditional (it is not entailment), and we have a number of threads that need to be untangled. Let me start with untangling the threads the ‘conditional simpliciter’ (Mares does not include the word ‘simpliciter’), to implication, to entailment, and to ‘natural-language conditionals. Unless otherwise noted, I will be restricting my discussion to relevant versions of all of these. (Complications, complications.)

Conditionals simpliciter comprise any If Then proposition that is (at least) transitive and obeys whatever other rules which I will leave unlisted for now. Conditionals simpliciter come in two flavors: implications and entailments. The relation of implication is contigent; the relation of entailment is necessary. To say that the relation of implication is contingent is to say (roughly) that things could have been otherwise, for which we can advance as evidence that we can imagine them to have been otherwise. The doorbell apparatus for example is currently in good condition, but we can easily imagine it to be defective such that pressing the button results in the ringing sound just 99% of the time. So the implication relation expressed by 1b) above is contigent — things could easily have been otherwise such that the relation would not have held. Similarly, to use Mares’ example, the following implication held (at least at the time of his writing):

In New Zealand, if one does not pay income tax on honoraria given for presenting seminars at other universities, then one is in violation of the tax code.

Here the relationship between not paying income tax on an honorarium and a violation of the New Zealand tax code is not a necessary connection. We can easily imagine a world in which the tax code were different such that it made honoraria tax exempt income. When we make claims like the one above, we do so assuming other facts that connect the failure to pay tax and a violation of the law, that is, particular facts about the New Zealand tax code. Thus the way in which the violation of the law and the failure to pay tax are connected is contingent.

RL, pp. 10-11

The relations between the axioms of arithmetic and particular true arithmetical propositions are, on the other hand, entailments. They are “necessary” — where for the moment at least “necessary” means metaphysically aka logically necessary, i.e. true in all worlds metaphysically/logically accessible from the actual world. For now, I will be touching metaphysical and logical possibility very lightly, except to say that, like Mares “I follow the mainstream in the metaphysics of modality in equating metaphysical and logical possibility” (RL, p. 14).

We say that the relationship between the axioms of arithmetic and 2 + 2 = 4 is that of entailment and the relationship between the failure to pay tax and the violation of the law is that of implication.

RL, p. 11

So far …. natural language conditionals those implications that refuse to be regimented into the rules governing implications and which are defeasible and no move is made to idealize …. and In some sense of ‘absolute’, this “no way” must be absolute. In some sense of “must” the button outside must be getting depressed if the doorbell is ringing and pressing the button must result in the doorbell’s ringing. In some sense of ‘non-defeasible’, the relation must be non-defeasible. Because of this reliability, this ‘absolute no way’, this ‘must’, this non-defeasible character, the ringing sound serves as information that the button is getting depressed, and the button’s getting depressed serves as information (whether this information gets picked up or not) that there is a ringing sound inside.

But the relation between antecedent and consequent in 1a) and 1b) is a contingent relation. In a possible world — say, the actual world three months from now — in which the wiring in the doorbell apparatus came loose in such a way that pressing the button outside resulted in just an intermittent production of the ringing (say, it produces the sound just 99%, no longer 100% of the time), pressing the doorbell would no longer be information that the doorbell was ringing inside. 1b) would then no longer be a true implication. Likewise, if someone constructed some sort of Rube Goldberg contraption hooked up to the doorbell apparatus such that the doorbell could be gotten to ring without the aid of the button, then 1a) would no longer be true, and the doorbell’s ringing inside would no longer be information that the button outside was getting depressed.

The truth of 1a) and 1b) is hinges upon situation s, the situation comprising the doorbell in its current good condition. It is contingent upon that situation; it holds only given that situation; 1a) and 1b) are true only for that situation. Put a bit more opaquely, 1a) and 1b) are true only “ins.

‘Contingent’ is usually opposed to ‘necessary’. But I asserted above that the relation between situations connected by a channel of information is in some sense a necessary one. What are we to make of this necessity?

The following attempt to say what ‘necessary’ means in the sentence ‘It is necessary for Jones to pay his bills’ may prove illuminating. Consider all the possible worlds that are accessible from the actual world by virtue of having Jones both present in all of them and enmeshed in the same social and economic institutions as he is in the actual world. In all of these possible worlds, unpleasant things happen to Jones if he does not pay his bills. Now restrict these to all those possible worlds in which Jones has the means to pay his bills. Now, again, narrow these worlds down further to those in which nothing impedes Jones’ paying his bills (the internet goes down, the postal workers go on strike, Jones has a masochistic desire to be a martyr to capitalism). Again, remove Jones from those worlds in which, suffering from a crippling psychotic anxiety revealing ala Heidegger the world as a totality, finds that nothing has meaning and finds himself unable to engage in the world as average everyday bill-paying Dasein. Again, remove from those worlds the worlds in which Jones, desiring to be a Sartrean hero gloriously exemplifying the same absolute freedom and spontaneity of consciousness exhibited by someone, who, standing before the yawning expanse of the Grand Canyon, decides to jump for no other reason than that they have the freedom to do so; simply decides not to pay his bills.

In other words, remove from all of these possible worlds anything that might remove Jones from the normal, mundane, boring course of paying his bills in order to avoid unpleasant things from happening. In all the possible worlds accessible this way from the actual world, Jones pays his bills. It is necessary that he pay his bills, meaning, he does so in all possible worlds that are accessible from the actual world in the way just described.

Resolving the contradiction: In a similar, in attempting to say in what sense it is ‘necessary’ that the button outside be depressed when the doorbell is ringing inside, or that the doorbell be ringing when the button outside is depressed, we can try to define an accessibility relation by starting with all possible worlds nomically accessible from the actual world (shares with the actual world the same laws of physics), then paring those down. We remove all those possible worlds in which the doorbell apparatus is not in good working order. We then require that, in addition to being nomically accessible, all the remaining worlds be epistemically accessible. A possible world is epistemically accessible from the actual world for a knower S when S knows nothing that would rule out the identity of the actual world with that possible world. For example, I know nothing that would rule out the identity with the actual world of the following possible world: at such and such GPS coordinates specifying a spot in the Amazon (I leave the production of these coordinates as an exercise for the reader), a violet butterfly is flapping its wings. This possible world is epistemically accessible from the actual world relative to me.

Now at the time of this writing I do not know anything that would rule out the identity with the actual world of a possible world in which poltergeists do not exist. This would rule out any scenario in which a poltergeist disrupted the reliability of 1a) by suddenly causing the doorbell to sound without the button’s getting pushed, or disrupted the reliability of 1b) by cutting off the normal chain of events from button to sound. Likewise, I do not know anything at the time of this writing that would rule out the identity with the actual world of the possible world in which freak occurrences such as a sudden burst of electrical energy coming out of the blue (maybe ultimately caused by that violet butterfly flapping its wings in the Amazon) having no causal link to the button but still in accordance with the laws of physicals causes the doorbell sound. Nor do I know anything that, still in accordance with the laws of physics, would cause the normal stream of causal events to suddenly stop, even though the doorbell apparatus is still in perfect condition, right after the button outside was pushed.

Obviously there is some risk that researchers might someday come to find with certainty that poltergeists do exist and do influence physical events (maybe through some kind of quantum entanglement, if I may be permitted to violate, at least this once, my rule that one should not bring up quantum mechanics in a philosophical discussion unless they have have completed at least eight graduate courses in quantum mechanics with no grade lower than a B+ in any of them, and that B+ is allowed in just one of them.) Were these researchers to impart their knowledge to me, the possible world in which poltergeists do not exist would no longer be epistemically accessible to me from the actual world.

Likewise, I am always at risk of finding out that researchers might uncover with certainty the occurrence of the freak physical events just described (the sudden burst of electricity in the wiring coming out of the blue; the strange sudden halting of the causal chain from button to sound). And were we to discover this could still eliminate from our possible worlds those physical conditions which lead to these freak events. In this way, I hope to define a realm of the normal mundane but still recognize that, since we don’t know everything, there are possibilities at the margin.

In this way, we can define a sense in which the relation between antecedent and consequent in 1a) and 1b) is, in a limited sense, necessary, but also contingent and defeasible. This is the sense that I hope to give to the “must” that is involved.

Jones paying his bills in order to prevent unpleasant things from happening is part of the ‘normal unimpeded stream.’ The stream of causal events going from button getting pushed to doorbell sound is the ‘normal unimpeded stream.’ The opposite of ‘normal’ is ‘freak’.

Entailment is implication plus necessity. Situated inference explains why we need Rsss and not just Rss. Because implication requires some kind of necessity, there is no sharp dividing line between implication and entailment. Conditionals in general are divided into two groups: implications and entailments. Natural-language conditions are broader than implications.

Given the situation s comprising the apparatus in its current state, situation u in which the doorbell is ringing inside my apartment, and situation t in which the button outside is getting depressed, 1) is supported by the relation Rstu. The implication is true given s, or for s. Put a bit more opaquely, 1) is true in s.

Unless it is clear from the context that something else is going on, I will put the ‘relevance-making’ situation (in this case, the situation comprising the channel) in the first position after the ‘R’. Apart from my wanting to maintain a certain convention, the position of the variable naming the situation does not matter.

Rstu is an accessibility relation, where the concept ‘accessibility’ is borrowed from ‘possible world’ talk. Here, one situation (the doorbell’s ringing inside my apartment) is giving me access to another situation (the outside of my apartment) that would otherwise be closed off from me. This particular example, I submit, makes interpreting Rstu in terms of channels of information connecting sites powerfully intuitive.

All of this is very well and good for the implication that the button outside is getting depressed if the doorbell is ringing inside. But what about the implication:

2) If the doorbell is ringing, then the doorbell is ringing

? This of course is the sort of proposition uttering which tends to lead to the exclamation “You do have such a wonderfully intuitive sense for the blazingly obvious” or “You are so smart! What does your boyfriend feed you?”. But this does not make the proposition any less true. It is of course necessarily true in a way that makes it an example of entailment, not of contingent implication. But (at least I am assuming for the present) that it is a case of relevant entailment, which, if I understand correctly, is for Mares relevant implication plus necessity. So we still need to understand what makes p relevant to p here.

The ternary relation seems to be necessary for relevance. Why? Well, be patient — I will get to that eventually. But here there is just a single situation — let’s call it s — namely, the inside of my apartment (in, say, the actual world), where I hear the doorbell sound. If the ternary supports relevance in this case, then, it would have to be Rsss. In other words, the same situation is getting named three times. So the ‘relevance-making’ situation, the information channel, has to be, somehow, a channel from itself to itself. I take it this is the sort of thing Mares means when he says: “…on the channel theoretic reading, Rsss says that s is a channel from itself to itself.”

Well, that certainly sounds weird. There does not seem to be any ‘initial s‘ that can serve as the mediating s in Rsss — nothing that serves the physical function of the doorbell apparatus as a channel of information. But surely the doorbell’s ringing in my apartment is relevant (at the very least in a ‘you have a wonderfully intuitive sense for the glaringly obvious’ sort of way!) to the question whether the doorbell is ringing in my apartment, just as its ringing in my apartment is relevant to the question whether someone or something is depressing the button outside. But if a channel of information does not account for the relevance here, what relation does?

One possible answer that comes to mind is ‘the binary relation that holds between possible worlds in strict implication’. That is what makes the antecedent p relevant to the consequent p in ‘if p then p‘:

The obvious way to model implication in this [Kripke-style possible world semantics] is to use the following truth condition: ‘A –> B‘ is true at a world w if and only if for every world w’ such that w’ is accessible from w, either ‘A‘ is false in w’ or ‘B‘ is true in w’

RL (pp. 26-27

So if we add modal necessity to material implication do we make p relevant to p? Does given ‘A‘ then necessarily we have ‘A‘ makes ‘A’ relevant to ‘A’. Doesn’t necessity automatically give relevance? There is no glue between p and q in Material Implication, which is why p’s relevance to q is of no import in Material Implication. Might necessity provide a strong enough glue? Maybe we don’t need all this information channel stuff to account for relevance after all.

Well, no, certainly not if necessity is interpreted as what holds in strict implication. For the following is true as a strict implication even though the antecedent is clearly not relevant to the consequent:

3) If (if the doorbell is ringing then the doorbell is ringing) then (if the earth has just one moon then the earth has just one moon)

Label the antecedent (if the doorbell is ringing then the doorbell is ringing) as p. Label ‘if the earth has just one moon then the earth has just one moon’ as q. In every possible world accessible from the actual world, either q is true or it is false. In classical logic, which is getting used here to talk about this stuff, this is equivalent to ‘if q then q‘. So in every possible world accessible from the actual world either p is false in that world or ‘if q then q‘ is true in that world. So given the truth condition for strict implication given above, ‘If p then (if q then q)’ is true in the actual world as a strict implication. But as we can see from the example given in 3), this is clearly unsatisfactory,

To switch metaphors from ‘glue’ to ‘sieves’, the necessity in strict implication is not fine-meshed enough to filter out all irrelevant implications. Relevance is one thing; necessity seems to be a different thing.

On the other hand, some sort of necessity seems to be a … well … necessary … condition for relevant implication. For a true implication cannot be defeasible:

A defeasible inference is one that may not hold if extra information comes to light. Implication, in the sense that we mean it here, is a non-defeasible relation between propositions.

RL, p. 45

And non-defeasibility seems to imply some sort of necessity. I hope to explore what this sort of necessity in further writing-to-learn exercises. For now, however, I would like to return to the difficulties that ‘if p then p‘ poses for an interpretation of relevant implication.

Necessity by itself can’t constitute the relevance-making relation. So we are back to the ternary relation — in the case of 2) Rsss, which, interpreted in information-theoretic terms, seems to pose the bizarre requirement that a situation be a channel of information from itself to itself. But why are we stuck with the ternary relation? Why can’t the relation be a binary one, such as the identity relation? Why can’t the identity with itself of ‘the doorbell is ringing inside’ suffice to make ‘the doorbell is ringing inside’ relevant to ‘the doorbell is ringing inside’?

If p then p: The reason the relevance-making relation has to be ternary, not binary is that it supports not just implication, but deductive inference as well. Consider the following deductive chain, which proves the truth of ‘if p then p‘:

  1. p {1} hyp
  2. p {1} 1, reit,
  3. p –> p {1} 1 – 2, –> I

The three subscripts {1} each refer to one situation s (say, the doorbell’s ringing inside my apartment) that gets named three times. Employing the subscripts forces us to make sure that the premises actually are used to reach the conclusion — in other words, that the premises be actually relevant to the conclusion. If we take this deductive argument to be implicit in ‘p –> p‘ — if we take the implication to be a condensed form of the argument — then the relation that makes p relevant to p in that implication has to be ternary. But we get a bizarre result if we try to force an information channel interpretation on this ternary relation in the case of ‘p –> p‘, because we end up with situations having to be information channels from themselves to themselves.

So ‘if p then p‘ poses a challenge for a ‘channel of information’ interpretation of relevant implication. Might another interpretation fare better? Is their an alternative to making a situation a channel to itself? Mares proposes using the concept of an information link. Let’s see how this concept holds up.

An information link is itself contained as information in situations:

On my view, informational links are themselves contained as information in situations, and vary from situation to situation. For example, the information that a particular convention is in place may be contained in one situation, but not in another.

RL, p. 44

For example, if the city council of Salt Lake City, Utah has passed an ordinance requiring the motorists stop at the white line on the road in front of a stop sign facing a certain direction, that law is information that is part of the situation described by ‘in Salt Lake City’ and is available in that situation. It is not information contained in and available in, say, a desert island owned by some eccentric billionaire with artistic pretensions who has set up stop signs there purely as a form of conceptual art.

I take it that Mares’ phrase ‘contained as information in a situation’ means ‘in that situation the information is available’ whether that information has actually been received or not. That the doorbell apparatus in my apartment is in good working order is information contained in the situation described by ‘the doorbell apparatus’ and is available in that situation say, to a competent doorbell repair person, whether any such person has actually received that information or not.

I take it, then, that an information channel would be a special case of an an information link. ‘Information link’ lets us talk about implications going beyond the narrow cases of physical as opposed to, for example, legal or conventional) situations linked together by channels of information that meet whatever requirements are imposed by information theory. Perhaps exposing thereby my vast ignorance of information theory, I will assume that information theory is best suited for purely physical situations. If so, the looser concept of an ‘information link’ would be better suited to elucidate the phenomena than the stricter concept of a ‘channel of information’, which in many cases, I think, would be trying to fit them in a Procrustean bed.

An information link gives us a license to infer. The laws of nature, for example, once uncovered, give us a license to infer that a body will, because it has mass, warp the space around it. The laws of Salt Lake City will, once one has arrived at the ordinary condition of being aware of those laws pertaining to stop signs, give one a license to infer that one is to stop on the white line in front of the stop sign. Packed, so to speak, into an implication is this inference license.

But does the concept ‘information link’ suffice to deal with ‘if the doorbell is ringing then the door bell is ringing’? What would a ternary relation between situations look like in that case? As shown by the proof offered above, ‘if the doorbell is ringing then the doorbell is ringing’ has packed into it, is a condensed form of that 3-step proof. So implicit in the implication is a three-fold structure in which a particular situation gets named three times. A supposition (the hypothesis ‘were the doorbell ringing in my apartment’), a statement (not requiring the ability to track long arguments or hold a complex thesis in one’s mind) of the obvious, a repetition (in that case a doorbell would be ringing in my apartment), and finally a conclusion (so if a doorbell were ringing in my apartment, then a doorbell would be ringing in my apartment’ — I am a very stable genius!). To unpack the implication would be to go through a process in which the same situation gets named three times. Thus Rsss. Of course, as I state all of this I am wondering how dubious or contrived it sounds. Nonetheless, this seems to me to be a more supple way of accounting for the ternary relation in ‘if p then p’ than trying to force on that relation an information channel that is somehow a channel to itself, something that seems (at least to my untutored eyes) appropriate only for physical implications anyhow.

So yes, for now I will go out on a limb and assert that the concept ‘information link’ succeeds in accounting for ‘if p then p‘ whereas the concept ‘information channel’ does not.


Apple Math, Comprising Some Basic (Doubtlessly Ninth-Grade Level) Probability Theory

Nota Bene:  This little bit of math is the keystone in my attempt here (still in draft status)  to provide a sharp, clear articulation of the concept of relevance as that concept pertains to Relevant Logic.  Here I invited members of the online Physics Forum to point out any mistakes in the math should I have made any.  Since no one there pointed out any such mistakes, I will assume that the math is correct.  Naturally, should it turn out that I did make mistakes in the math, I will be royally pissed.  ūüôā

This post belongs to the ‘I invite anyone and everyone to tear this to pieces, should they uncover any missteps’ category.

The subject here isn’t roses (this is an obscure allusion to a movie I saw in my childhood), but wormy and non-wormy red and yellow apples.

In discussing the subject of apples, I will be using the following terms: ‘set’ (which I will leave as an undefined primitive); ‘sample space’ (which term is I think self-explanatory); ‘event’ (which I will be using in an extremely narrow and a bit counter-intuitive technical sense, following the standard nomenclature of probability theory); ‘experiment’ (ditto); ‘state of affairs’ (which I will be leaving as a primitive); and ‘proposition’ (which I will define in terms of states of affairs).

Wormy Red Apple Image courtesy of foodclipart.com

First Situation:  All Of The Red Apples Are Wormy; Only Some Of The Yellow Apples Are:  Let’s start with the following situation (henceforth ‘situation 1’):  There is an orchard in Southwest Iowa, just across the border from Nebraska. In the orchard there is a pile of apples comprising 16 apples.  Eight of the apples are red.  All of the red apples are wormy.  Eight of the apples are yellow.  Of these yellow apples, four are wormy. 

Let’s suppose that the DBA in the sky has assigned an identifying number (doubtlessly using the Apple Sequence Database Object in the sky) to each apple. This lets us write the set of apples in the pile — the Sample Space ő© — as follows:

The Sample Space ő© =

ő© = { a1rw, a2rw, a3rw, a4rw, a5rw, a6rw, a7rw, a8rw, a9yw, a10yw, a11yw, a12yw, a13yw, a14yw, a15yw, a16yw }

where a1…an indicate the numbered apples, and the superscripts r, y, w, and w indicate a red apple, a yellow apple, a wormy apple, and a non-wormy apple respectively.

An ‘event’ is a (not necessarily proper) subset of this set. It represents the set of possible outcomes should one draw an apple from the pile. This particular red apple is drawn; this other particular red apple is drawn; this particular yellow apple is drawn, and so on. Contrary to the ordinary sense of ‘event’, an ‘event’ here is not something concrete, happening in space and time, but abstract — a set.

Eyes shut, someone has randomly drawn an apple from the pile. They have not yet observed its color. Why their having not yet/having observed the apple matters will become apparent later [promissory note]. Following the standard nomenclature, I will call actually drawing an apple — a concrete outcome that has come forth in space and time — an ‘experiment’.

Now I show that….

E is the event ‘a red apple gets drawn from the pile’, which =

E = { a1rw, a2rw, a3rw, a4rw, a5rw, a6rw, a7rw, a8rw }

F is the event ‘a wormy apple gets drawn from the pile’, which =

F = { a1rw, a2rw, a3rw, a4rw, a5rw, a6rw, a7rw, a8rw,a9yw, a10yw, a11yw, a12yw}

And of course the intersection of E and F, E ‚ą© F, the set of apples that are both red and wormy =

{ a1rw, a2rw, a3rw, a4rw, a5rw, a6rw, a7rw, a8rw}

I will be assuming that each apple in ő© has an equal probability of being drawn.

The conditional probability that the apple drawn from the pile is wormy given that it is red is 1, as you can see from the following steps:

P( F | E ) = P( E ‚ą© F ) / P(E)

P( E ‚ą© F ) = |E ‚ą© F| / |ő©| = 8/16 = 1/2

P(E) = |E| / |ő©| = 8/16 = 1/2

So:

P( E ‚ą© F ) / P(E) = 1/2 / 1/2 = 1

So:

P( F | E ) = 1

The conditional probability that an apple drawn from this pile is wormy given that it is red is 1.

Now P(F) = 12/16 = 3/4.  Since P(E) = 1/2, P(E) * P(F) = 1/2 * 3/4 = 3/8.  So in this case P(E ‚ą© F) != P(E) * P(F),  since 1/2 != 3/8.  But two distinct events are independent of one another if and only if

P(E ‚ą© F) = P(E) * P(F)

So in this case E and F are not independent events.   The probability that the apple is wormy given that it is red increases to 1 from the 3/4 probability given just the draw from the pile, before observing whether the apple drawn is red or yellow.  (Conversely, the probability that the apple is red given that it is wormy increases to 2/3 from 1/2 given just the draw from the pile.)

When the probability of an event is 1, that event is certain, as opposed to ‘just likely’. The concept of certainty is, of course, intimately bound up with the concept of knowledge, an entanglement I hope to examine shortly. But whatever the relation is, the event of this apple’s turning out to be red moves the event of its being wormy from a mere likelihood to a certainty. And whatever the relation of certainty to knowledge is, this certainty surely provides a foundation for knowing that this apple is wormy. In this limited situation (“situation 1”), the apple’s turning out to be red is potentially telling — namely, that it is wormy. It increases our (potential) knowledge.

When this apple drawn at time t0 (the experiment that takes place at that time) turns out to be red , the state of affairs ‘this apple is red’ obtains at t0. I will label this state of affairs ‘p’. Similarly, I will call q the state of affairs that obtains at t0 when this apple is wormy. In situation 1, the fact that the probability of F given E is 1 means there is no way that p can obtain at t0 and q fail to obtain at t0. For the moment, at least, I will refrain from unpacking ‘cannot fail to obtain’, except to link this notion to the probability of an event being 1.

I like to identify propositions with states of affairs that obtain at a particular time. So p is the proposition that the apple is red, and q is the proposition that the apple is wormy. States of affairs obtain or fail to obtain; propositions are true or false. So I am now moving from talking about states of affairs obtaining (failing to obtain) to propositions being true or false. If, gentle reader, you would rather not identify propositions with states of affairs obtaining at some time, just add whatever verbiage is necessary to identify a proposition that corresponds to the state of affairs just mentioned.

In situation 1, whenever p is true q cannot fail to be true. This means that the proposition If p Then q is true, for it satisfies the truth table in Classical Logic for If Then propositions. In situation 1, If p Then q remains true even when p is false (the apple is yellow) and q is false (the apple is not wormy); when p is false and q is true (the apple is wormy); and of course the proposition is true when p is true and q is true. The only time the proposition is false is when p is true and q is false.

What is more, in situation 1, p is relevant to q. For p maps to the event E given which the probability of F, to which q maps, [talk some more about this mapping business] increases from 3/4 to 1, i.e., from mere likelihood to certainty. p inherits this ‘increasing q to certainty’ property. That one proposition/state of affairs (that the apple is red) p increases the probability of another proposition/state of affairs (that the apple is wormy) q surely renders p relevant to q. It is a sufficient condition for p’s relevance to q. It therefore renders If p Then q true in both Relevant Logic (which demands that the antecedent be relevant to the consequent) and in Classical Logic.

I submit, then, ‘increasing the probability of q to 1’ as a candidate for the relevance-making relation that p bears to q when p is relevant to q. This relation is a candidate, that is, for those If Then propositions that can be treated in a probabilistic manner. It is not a candidate for the relevance of the antecedent to the consequent in the proposition ‘If the length of side A of this right triangle is 2 and the length of side B is 3 (neither A nor B being identical with the triangle’s hypotenuse), then 13 is the length of the hypotenuse.’ For even though the antecedent here excludes any other possibility other than the hypotenuse having a length of 13 (just as the apple’s turning out to be red excludes in situation 1 the possibility of it’s not being wormy), there is nothing in the mathematical proposition that invites treatment in terms of chance and draws.

That the probability increases to 1 renders the proposition ‘If E then F’ true — at least in this circumscribed situation (this particular pile in this particular orchard for this particular stretch of time, which stretch of time will come to an end should a non-wormy red apple happen to roll into the pile). Within this situation, the apple will always be wormy should it turn out to be red. The ‘all’ in ‘all the red apples are wormy’ guarantees the truth of the conclusion as long as this ‘all’ lasts. Taking the increase in probability combined with the guarantee (the increase is to 1) together suffice to make ‘If this apple is red, it is wormy’ a true proposition in relevant logic, since the conclusion meets the truth-table standard of classical logic and meets the additional condition demanded by relevant logic, namely, that the antecedent be relevant to the conclusion. F will never fail to be true should E turn out to be true, a state of affairs that is a sufficient condition for the proposition ‘If E then F’ to be true.

I submit, then, that at least in those states of affairs that allow for a probabilistic treatment, the relevance of p to q consists in p’s increasing the probability of q to 1. [tie p and q to E and F.] Naturally, not all p’s and q’s will allow for a probabilistic treatment. Mathematical propositions don’t allow for such a treatment, for example. We should perhaps not assume that what makes p relevant to q is the same in all cases of IF THEN propositions is just one type of relation. But at least in the case of those propositions that do allow for a probabilistic treatment, we can see that increasing the probability of q to 1 given p is a strong candidate for the relevance-making relation, given that this increase suffices to render p relevant to q.

At least in those cases that do admit of a probabilistic treatment, increasing the probability of q to 1 is also a necessary condition for p’s being relevant to q.

Second Situation:  All Of The Red Apples Are Wormy, As Are All Of The Yellow Apples

When all the apples are wormy, the color, either red or yellow, of the apple becomes independent of its worminess. Thus the aforementioned sufficient condition for relevance is absent. Maybe some other relation could render p relevant to q here, but I am at a loss for what it could be. So until someone can point out such a relation, I will therefore go out on a limb and say that dependence is a necessary, as well as a sufficient, condition for the relevance of p to q in cases similar to the wormy apple case. This provides support — though clearly not support achieving the level of certainty — for the original intuition. vvggggg

A paradox or at least weirdness comes to the fore. I deal with this by examining the nature of probability. Assuming a deterministic universe (at least on the post-quantum level) probability is perspectival — on either a global or a local level. The example can seem paradoxical because one is assuming the position of someone who knows everything about the apples. A local orchard god, so to speak. But that is just one perspective. Thus the original intuition is vindicated.

If just a credence, there are no relevant IF THEN propositions from a God’s-eye’ point of view. (Actually, no perspective at all). Possible worlds (complete) vs. situations (partial).

Today’s homage to Plato’s SYMPOSIUM is this image of a young boxer appearing on the cover of a computer book.

Boxer_XML_OnlyComputerBookBoughtJustForTheCover_

I have to admit that this is the only computer book I have ever bought just for its cover.

How can anyone get anything done, much less study computer science and ninth-grade math, with beauty like this walking the earth?

Update 11/12/2018:  Made one revision for the sake of clarity.


The Role Of Informational Content In Establishing Relevance In Relevant Logic

“This current version of the notes is not yet complete, but meets I think the
usual high standards for material posted on the internet.”¬† (Link. ¬†No, I have not read the paper apart from this snippet.) ¬†Please feel free to comment if you have any corrections or objections to the disquisition below, or email me at cliffengelwirt@gmail.com.¬†

 

Logic first became interesting to me when I entered the DBA field and started reading the works of C.J. Date, Hugh Darwen, and Nikos Lorentzos on the foundations of relational databases. ¬†While reading in logic, I became intensely interested in Edwin D. Mares’¬†book RELEVANT LOGIC A PHILOSOPHICAL INTERPRETATION, which seemed to tie in — I am apparently not the first to notice this! — in a very natural way with Fred I. Dretske’s classic work, KNOWLEDGE AND THE FLOW OF INFORMATION. ¬†As an exercise in writing to learn¬†Mares’ book I have been for a while entering posts on this blog on the topic of Dretske’s theory of informational content as it relates to Relevant Logic.

Up until now, these posts have been nothing except an effort to decide what my position is on the topic.  They pretend to be nothing more than efforts to get my own thoughts in order.  As a result, I have not been terribly afraid to be (just occasionally, I hope!) simply mistaken and (worse) unclear.

Basically, I was thinking out loud in order to decide what I do think about the topic. ¬†Even though these exercises in thinking out loud were both tentative and preliminary, I have found it to be a useful discipline in performing them in public, where there is always the possibility that someone actually engaging with the posts (in other words, someone who is not merely a troll) may legitimately, pointing to specifics, exclaim ‘THIS IS SIMPLY WRONG!!!!’ or ‘THIS IS CONFUSED!’

Lions And Trolls Oh My!¬†But now that I am suffering under the delusion that I do have my thoughts on the topic in something vaguely resembling order, I am now actively throwing them out to the lions in order to see what survives intelligent, informed criticism. ((I am assuming there are lions out there who are not only hungry, but also intelligent and informed. As regards lions I keep thinking about Ned Rorem’s LIONS (A DREAM)¬†which I once heard on WFMT in Chicago… but I digress.)) Please consider this post and the the posts linked to here as a request for comment.

As each section of this disquisition takes (almost) final shape, the link to it will become active. Each section will be kept as short as possible partly as an troll-control device: the brevity of each piece makes it easier to force the troll to state a specific objection to a specific assertion ((has the troll misstated the assertion (most of the time intentionally but sometimes not)? If so, challenge them to state it in their own words — honestly this time. Once the troll has correctly stated it, do they think the assertion is wrong? If so, why? ¬†Does the troll think the assertion is unclear?)) rather than allowing the troll to rely on abusive innuendo.

The Problem

What Is Relevance Anyhow?

The Relevance-Making Relation Is Not The Causal Relation

The example that at least initially makes treating the relevance-making relation in terms of Dretske’s notion of informational content attractive: Dretske’s Doorbell Example.

This seems to run aground on the tautology IF p THEN p. The revisions needed to accommodate this tautology.

The ‘peanut is under which shell’ example. Will this example end up making Relevant Logic at least as weird and bizarre as Classical Logic by making the truth of implication statements relative to what one knows?

The measles and wormy red apples example.

******

No post of mine can do without an homage to Plato’s SYMPOSIUM. Here the homage will take the form of Channing Tatum.

ChanningTatumTotalBeauty_0

Edit Log: June 04, 2017: Made some minor changes.

June 10, 2017: ¬†Made some minor changes. ¬†Removed a joke I think wasn’t ¬† ¬† ¬† ¬† ¬† ¬† ¬† ¬† ¬† ¬† ¬† ¬† ¬† ¬† ¬† ¬† ¬† ¬† ¬† ¬† ¬† ¬† ¬† ¬† ¬† ¬† ¬† ¬† ¬†working.

June 14, 2017:  Added quote at the top.


My Attempt To Identify The IF-THEN Relation With The INFORMATION-THAT Relation Ignominiously Bites The Dust

Here is yet another challenge to the idea that ‘If p Then q’ is true when the occurrence of p is information that q. ¬†Unfortunately, I think this challenge nails the matter. Consider Dretske’s shell game example. ¬†The peanut is under shell #4. ¬†So the following statement is true (given that my visual faculties are in sufficiently good working order, and that I am looking in the proper direction with my eyes open):

If I turn shell #4 over now (t0), I will see a peanut at time t1

(t1 being one millisecond or whatever later than t0.) ¬†Is my turning shell #4 over at time t0 information that I see a peanut at t1? Certainly the situation largely fits Dretske’s definition of ‘information that’:

Informational content: ¬†A signal r carries the information that s is¬†F = The conditional probability of s‘s being F, given r (and k), is 1 (but, given k alone, less than 1)

Fred Dretske, KNOWLEDGE AND THE FLOW OF INFORMATION, Stanford, CSLI Publications, 1999, p. 65

(k represents what the receiver already knows about the source.)  The conditional probability of my seeing the peanut at t1 is certainly 1 given my turning the shell over at t0 (and given the other conditions mentioned).  So the IF-THEN statement above certainly fits that part of the definition of informational content.

But is my turning the shell over at¬†time t0¬†a signal that at¬†time t1 that I see the peanut? ¬†A signal is¬† “…any event, condition, or state of affairs the existence (occurrence) of which may depend on s‘s being F.” ¬†(Dretske, p. 65.) ¬†Does¬†my turning the shell over now depend upon my seeing the peanut one millisecond in the future? ¬†How can a present event depend upon a future event? ¬†Clearly not.

A signal cannot occur before the event or thing or state of affairs the occurrence (existence, obtaining) of which it signals. ¬†The smoke does not occur before the fire (or the smoldering). ¬†The doorbell does not ring before the button is pushed. ¬†The deer tracks in the snow do not appear before the deer show up. ¬†Were the watchman in Aeschylus’ play AGAMEMNON in the ORESTEIA trilogy¬†to light his¬†fire before he spots Agamemnon’s ships, his fire would not be a signal informing Clytemnestra of the appearance of those ships on the scene: ¬†Clytemnestra would be receiving false information. ¬†Something cannot be announced before it occurs (exists, obtains).

“But the dark clouds signal the rain that is about to fall; the sports official signals the race that is about to start in one millisecond by firing the pistol into the air.” ¬†Someone may object in this way to my (seemingly obvious) claim that a signal cannot occur before the thing it signals. ¬†Yet, although we can doubtlessly “round up” the dark clouds and the firing of the pistol to the status of signals, they are not so in the very strictest sense of ‘signal’ that I intend to use here. ¬†For the conditional probability that, given the dark clouds, rain will fall is perhaps only 99%, while the probability that the race actually will start given the firing of the pistol is perhaps only 99.9999999999% (the supernova that will hit us eventually may choose that exact millisecond to intervene by making its presence glaringly, searingly obvious, or a huge earthquake might strike at that very moment….).

A signal is¬† “…any event, condition, or state of affairs the existence (occurrence) of which may depend on s‘s being F” and therefore cannot occur before the occurrence (existence, obtaining) of s‘s coming to be F. ¬† The examples I’ve just given are not signals because they occur after what they “signal”, and — surely not coincidentally — they do not depend upon what they “signal.” ¬†Let me dwell a moment, perhaps a bit obsessively/compulsively, on this notion of dependence. ¬†Let me say that an event, object, or state of affairs p depends upon an event, object, or state of affairs q when, given a condition c, ¬†p would occur (exists, obtain) only because q occurs (exists, obtains).

Consider, for example, a doorbell whose wiring is defective in such a way that, 99% of the time when the button outside is getting depressed by someone or something, the doorbell rings. ¬†But 1% of the time the doorbell does not ring when the button outside is getting depressed. (I state the example this way to make it mirror the fact that p does not follow from If p Then q; q.) ¬†Also, there is no poltergeist inside the wiring that sometimes generates the ringing sound even when no one or nothing is pressing the button outside; likewise, there is never, ever any freak burst of electricity ultimately caused by a butterfly flapping its wings in the Amazon that generates a buttonless ringing sound. ¬†Nor (somewhat more plausibly) is there any defect in the wiring that would ever cause a buttonless ringing sound to occur. Let c be the condition of the defective wiring as just described (including the absence of ring-generating poltergeists). ¬†Given c (which I will call the non-poltergeist condition), the doorbell would ring only because the button outside is getting depressed (even though the button’s getting depressed does not necessarily result in the doorbell’s ringing)*. ¬†Given c, the doorbell’s ringing depends upon someone or something’s depressing the button outside and is therefore a signal. ¬†(A signal, moreover, carrying the information that someone or something is depressing the button outside, because the conditional probability of this is 1 given the doorbell’s ringing under condition c. ¬†Another way to put this is to make the perhaps obvious/tautologous point that to be a signal is to carry information.)

Consider another example, one which is perhaps belongs more to the realm of probability than to causality. ¬†One has turned over shells #1 and # 2 and verified that both are empty. They have information that the peanut is located in one of the four shells. ¬†So c is now the condition that either the peanut is located under shell #3 or under shell #4. ¬†Given c, shell #3 would be empty only because¬†it is shell #4 that is covering the peanut. ¬†It is, in fact, difficult to come up with any clear idea of anything else that could be the reason why shell #3 is empty. ¬†Shell #3’s being empty therefore depends upon the peanut’s being located under shell #4, and the former would be a signal carrying information that the latter. ¬†(Conversely, given that there is only 1 peanut at play in the game and given the rest of c, shell #4’s turning out to have the peanut would be a signal carrying information that shell #3 is empty. ¬†Shell #4 would have the peanut only because shell #3 is empty. )

Now consider again the turning over shell #4 example given above as an instance of an event, object, or state of affairs that very definitely is not a signal carrying information.  It would be difficult to give any meaning to the assertion:

my turning shell #4 over at time t0 occurs only because I will see a peanut at time t1

Such an assertion would not, I submit, make any clear sense, since the dependency aka only because relationship is a vector traveling forward (to speak metaphorically) in time.

Also consider yet one more doorbell example: ¬†suppose that the doorbell’s wiring is screwy in such a way that every now and then little bursts of electricity get generated which produce the ringing sound even when no one or no thing is depressing the button outside. ¬†(Or, if you prefer, there is a poltergeist residing inside the wiring that every now and then gets agitated by a freak burst of air pressure inside the contraption that is ultimately caused by a butterfly flapping its wings in the Amazon….) ¬†Nonetheless, the condition of the wiring is such that the doorbell always rings when the button is getting pushed. ¬†100 percent of the time the doorbell rings when the button outside gets pushed, but 1% of the time the doorbell is ringing buttonlessly. (I state the example this way to make it mirror the fact that q¬†does not follow from If q¬†Then p; p. ¬†And I am making it mirror this because, of course, the whole point of these interminable disquisitions is to dig into the nature of IF-THEN statements.)¬† Let me call this condition of the wiring c, as usual. ¬†(In a moment I will be calling it the ‘poltergeist condition.>) ¬†Given c, it would be difficult to give any sense to the following assertion:

My pressing the button outside occurs only because the doorbell is ringing.

Clearly, my pressing the button outside does not depend upon, and is not a signal for, the doorbell’s ringing. ¬†Again, the pressing of the button does not depend upon the doorbell ringing because the dependency aka only because relationship is a vector traveling forward, not backward, in time.

“Feel free to come to the point when you finally have one,” my (possibly non-existent) reader may want to say. ¬†Well, the point of all of the above is the following. ¬†Given their respective condition c’s, each of the following IF-THEN statements is true:

1) If I turn shell #4 over now (t0), then I will see a peanut at time t1

2) If I press the button outside [given the poltergeist condition], then the doorbell will ring.

3) If shell #3 is empty, then the peanut is located under shell #4.

4) If the doorbell is ringing [given the non-poltergeist condition], then someone or something is depressing the button outside.

Although the antecedent p is a signal carrying the information that q in the last two examples, it is not such a signal in the first two examples.

These examples, I think, nail it: ¬†IF-THEN statements cannot be identified with an information relation. ¬†My attempt to identify the IF-THEN relation with the INFORMATION-THAT relation has ignominiously bitten the dust. ¬†(Sob, sob.) Does this mean, then, that we are stuck after all with Classical Logic’s paradoxes of Material Implication, whereby both of the following statements are true?

If Cliff lives in Houston, Texas, then the earth has just one moon.

If Cliff lives in Orange County, California, then Paris, Texas is the capital of France.

(Please God, please God, please don’t let these statements be true.) ¬†Well, maybe we aren’t forced to accept these horribly ugly statements as true after all. ¬†For in each of the 4 numbered examples given above, the conditional probability of the consequent (given the antecedent plus the relevant condition c ((plus the relevant knowledge k))) remains 1. ¬†It is just that in the first two examples the antecedent does not depend upon the consequent, and therefore is not a signal carrying the information that the consequent. ¬†It is not a p only because q relationship. ¬†Perhaps, then, we can identify the IF-THEN relation with a different (but similar) relation, which I will call ‘the conditional probability is 1‘ relation. If so, it would remain true that in examples 3 and 4 above, the antecedent p is a signal carrying information that q. ¬†So whenever p does depend upon q in such a way as to be a signal for q¬†the corresponding IF-THEN statements would, possibly, have the (at least to me) weird properties mentioned in a previous post:

Third, the informational relation is both intentional and relative, as described by Fred Dretske in his KNOWLEDGE AND THE FLOW OF INFORMATION. ¬†Treating If p Then q as an information relation would make implication both intentional and relative. ¬†The very same If p Then q statement would be true inside some frameworks and false inside others. ¬†Rather than accept this, some would perhaps rather accept Classical Logic’s paradoxes of Material Implication.

(Sidenote: ¬†Dretske’s measles example displays the intentional character of information. ¬†By pure chance, all of Herman’s children happen to have the measles; moreover, one does not know this. ¬†So when one discovers that a particular person is a child of Herman’s, they do not have information that this person has the measles.) Or are we truly stuck with this weirdness? Can we find a way to make implication non-relative and non-intentional even in those cases in which p happens to be a signal carrying the information that q?

Today’s homage to Plato’s SYMPOSIUM is this gorgeous young Asian Man: GorgeousAsianGuy

It is hard to understand how anyone can get any work done at all with Beauty like this walking the earth, but somehow we do. How sleek all those black, white, and gray tones are!

Post Updated on June 27, 2015 to make the temporal vector nature of the dependency/only because relation clearer. (Or, if my reader is particularly suspicious, they are free to think I made the update in order to cover up some totally obvious mistakes, not simply to make a somewhat muddy post slightly clearer.)


Measles, Wormy Red Apples, And God (And Peanuts)

In his Knowledge and the Flow of Information, Dretske argues that what information a signal carries is relative to what the receiver already knows about the possibilities at the source:

To illustrate, suppose that there are four shells and a peanut is located under one of them. ¬†In attempting to find under which shell the peanut is located, I turn over shells 1 and 2 and discover them to be empty. ¬†At this point, you arrive on the scene and join the investigation. ¬†You are not told about my previous discoveries. ¬†We turn over shell 3 and find it empty. ¬†How much information do you receive from this observation? ¬†How much do I receive? ¬†Do I receive information that you do not receive? ¬†… [Dretske goes on to argue that the answer is ‘yes’ because the amount of information and what information is received depends upon the reduction in possibilities achieved in each case. ¬†Information is all about reduction in possibilities.] … This constitutes a relativization of the information contained in a signal because how much information a signal contains, and hence what information it carries, depends on what the potential receiver already knows about the various possibilities that exist at the source.

Fred Dretske, KNOWLEDGE AND THE FLOW OF INFORMATION, Stanford, CSLI Publications, 1999, pp. 78-79

The third shell’s proving to be empty when it is turned over is, for me, information that the peanut is hidden under shell 4. ¬†But for you, it is not information that the peanut is hidden under shell 4. ¬†What information a signal carries (here the signal is the third shell’s proving to be empty when turned over) is relative to what one already knows.

Let’s apply this conclusion to the measles and wormy read apple examples.

Suppose that one has received information that all of Herman’s children have the measles. ¬†Should one then discover (say, a friend tells them this) that this layabout in front of one’s shop is a child of Herman’s, that this person is a child of Herman’s is now, all of a sudden, information that this person has the measles. ¬†Before one knew that all of Herman’s children have the measles, that this person is a child of Herman’s was not information that the person has the measles.

The same reasoning applies mutatis mutandis to the wormy red apple example. ¬†If one has information (say, received from a person who has previously examined all of the apples in the pile) ¬†that all of the red apples in the pile are wormy, then that the apple in one’s hand drawn from this pile is red is information that the apple is wormy. ¬†Before one has received the information that all of the red apples in the pile are wormy, a signal that the apple in one’s hand is red is not information that it is wormy. ¬†In both the measles and the wormy red apples examples, what information a signal carries depends upon, is relative to, what one already knows.

So if one claims that If p Then q is true only when the occurrence of p is information that q, then the truth of these sentences (henceforth the ‘measles’ and ¬†‘wormy red apple’ statements)…

If this layabout loitering about on the front of my shop is a child of Herman’s, then this person has the measles.

and

If this apple (drawn from this particular pile) in my hand is red, then it is wormy

…is relative to what one already knows. ¬†They will be¬†true relative to the person who already knows that all of Herman’s children have the measles (without necessarily knowing that this particular person in front of their shop is a child of Herman’s) and that all of the red apples in this pile happen to be wormy. ¬†They will be false relative to the person who does not already know these things.

In previous posts, I noted as an autobiographical fact that I had the strong intuition that both statements above are true, regardless of what one already knows. ¬†But perhaps this intuition, in spite of its being my intuition, should not be regarded as totally sacrosanct. ¬†For I will venture that most people would not be bothered by the relativity of this statement (henceforth the ‘third shell proves empty’ statement):

If the third shell proves to be empty, then the peanut is located under the fourth shell

Clearly (although I say ‘clearly’ with some trepidation, in the spirit of ‘let me throw this piece of spaghetti onto the wall, and see if it sticks,’ or, alternatively, ‘let me see if I can get away with this statement without too many screams of protest’), this statement would be true in the situation occupied by the person who already knows that the first and second shells are empty, and false in the situation occupied by the person who does not already know these things.

What can be learned from, inferred from, concluded from the third shell’s being empty, the apple’s being red, the layabout’s being a child of Herman’s, depends upon the situation one is in that is defined by what one already knows. ¬†There isn’t, I think, anything controversial or counter-intuitive about this. ¬†IF-THEN statements have everything to do with what can be learned from, inferred from, concluded from a given situation. ¬†So the truth/falsity of the corresponding If p Then q statements is also relative to the situation one is in as defined by what one already knows.

And if one is still bothered by this, would one rather return to the paradoxes of Material Implication?

(Begin aside:  Remember that what is motivating this entire attempt to argue that If p Then q is true only when p is information that q is to escape from the paradoxes of Material Implication, which would count both of the following statements as true:

If Calypso music originated in Wisconsin, then the earth has two moons

and

If Paris is the capital of France, then the earth has one moon

To escape these paradoxes, we need to find a way to make p relevant to q in some way.  And the most plausible way to do this, I assert, is to insist that p be information that q.  End Of Aside.)

To undermine my initial intuition further, suppose that one has obtained information that all of the apples in the pile — both yellow and red — are wormy. ¬†In that case, should one (blindfolded) handle each apple in turn and say ‘If this apple is red then it is wormy’, his statement would be (I venture) false. ¬†For the redness of the apple is, in this situation, no longer what excludes the possibility that it is not wormy, or, put another way, is no longer the factor that renders as 1 the probability that the apple is wormy. ¬†That factor is now the fact that the apple is from this pile, not that it is red. ¬†Since the apple’s being red is no longer relevant to its being wormy (is no longer what makes the probability the apple is wormy 1), one cannot learn from, conclude from, infer from its being red that it is wormy. The apple’s being wormy no longer hinges on its being red. The statement is now false for exactly the same reason that ‘If Paris is the capital of France then the earth has one moon’ is false.

One might try to preserve a version of the intuition that the measles and wormy red apple statements are true regardless of anyone’s knowledge by proposing that these are true independently of what any finite intelligence knows or doesn’t know. ¬†What if there were an infinite intelligence — a God who knows everything in general, and the measles status of Herman’s children, the worminess status of the red apples in the pile, and the location of the peanut under the fourth shell in particular. ¬†One could then accurately say the ‘measles’, ‘wormy red apples’, and ‘the third shell proves empty’ statements are true objectively, that is to say, sub specie aeternitatis, even if they are true or false as the case may be, from the subjective standpoints of this or that finite intelligence.

The analogy would be with Galilean motion studied in High School physics. ¬†An object may be moving at 10 miles per hour given one reference frame and 60 miles an hour given another reference frame; nonetheless, there was to be some absolute reference frame embracing all of them which would let one give an absolute, non-relative value to the object’s speed.

But the intuition cannot be rescued this way. ¬†For clearly, nothing could ever be a signal, could be information-that, for an infinite intelligence that knew everything. ¬†Such an intelligence with its penetrating x-ray vision would already know, for example, that the peanut was located under the fourth shell. ¬†Given this knowledge, the third shell’s proving empty would not reduce to 1 for this intelligence the number of possibilities regarding the location of the shell. ¬†For the number of such possibilities was already 1 for this intelligence. ¬†Likewise, for this all-knowing intelligence, that this particular layabout is a child of Herman’s would do nothing to reduce to 1 the probability that this person has the measles. ¬†Nor would the fact that this particular apple is red reduce for this intelligence the number of possibilities regarding the worminess status of the apple from 2 (the apple is wormy or non-wormy) to 1 (the apple is wormy). ¬†With no reduction of possibilities, there is no signal carrying information-that in any of these cases.

God’s knowledge cannot serve as the equivalent in logic of the Galilean absolute reference frame.

Not only is information-that relative to what one already knows, it also requires finitude. ¬†No limitation on one’s knowledge — no hiddenness — no information-that. ¬†And if the truth of If p Then q statements requires that the occurrence of p be information that q, the truth of these statements also require finitude.

One final note: ¬†how can one account for the illusion (if it is that) that both the measles and the wormy red apply statements are true regardless of what one already knows? ¬†I think the answer lies in the fact that, after completely talking through one’s hat at time 1 with the statement “If this apple is red, then it is wormy,” one were later at time 2 to examine all of the red apples and discovered they were all wormy (and that just some of the yellow apples were), it would seem that, since the statement is true at time 2, it would have to have been true at time 1. ¬†The truth value of a statement like this can’t change, can it? ¬†Maybe we would prefer to accept the paradoxes of Material Implication after all. ¬†But it seems to me that one should accept that, at least in the case of the ‘third shell proves empty’ statement, the truth value of that statement can change with time as one obtains more knowledge (you later get information that the first and second shells also proved to be empty). ¬†So the truth value of the measles and wormy red apples statements changing over time should not prove to be an absolute obstacle.

     *****

The entire point of this exercise is not just to make grandiose metaphysically-existentialist-sounding statements such as ‘logical implication requires finitude’ (although I must admit this is one of my aims), but also to escape from Classical Logic’s paradoxes of Material Implication by insisting that there must be some relation between p and q that makes p relevant to q, and that this relation consists in p‘s being information that q.

In the previous post, I noted two apparent counterexamples (the measles and wormy red apple statements) that would seem to preclude identifying this hoped-for relevance-making relation with information-that. ¬†These statements seem to be true even though in these cases p is not information that q. ¬†Also, identifying this relation with information-that would make the truth of IF-THEN statements relative to what one already knows, an implication that may make one prefer the paradoxes of Classical Logic’s Material Implication.

In this post, I employ the ‘third shell proves empty’ statement, as well as the close connection (I claim) that IF-THEN statements have with what one can learn from, infer from, or conclude from a situation to remove whatever counter-intuitiveness might adhere to the notion that the truth of IF-THEN statements is relative to what one knows. ¬†(Of course what one can learn, infer from, conclude from a situation depends upon what one already knows. ¬†Of course the truth/falsity of ‘the third shell proves empty’ statement depends as well upon what one already knows.) ¬†If one can accept the relativity of IF-THEN statements, they will be in a better position to accept the idea that those cases in which p is not information that q (the redness of the apple sometimes fails to be information that the apple is wormy; that this person is a child of Herman’s sometimes fails to be information that this person has the measles) ¬†are also cases in which If p Then q is false.

This leaves the third difficulty mentioned in the previous post:  what to do about the statement If p Then p?  Is a channel of information supposed to exist between p and the self-same p?

Do I have a song and dance that will eliminate this difficulty?

*****

Today’s homage to Plato’s SYMPOSIUM is the soccer player James Rodriguez.

James_Rodriguez

From math teachers to soccer players…How can anyone get anything at all done with beauty like this walking the earth?


Measles And Wormy Red Apples: IF-THEN Statements And INFORMATION THAT (An Apparent Counter-Example)

It would seem that there are some clear counterexamples to the idea that If p Then q is true when p is information that q.

Consider the following (somewhat gruesome, in the light of the irresponsibility of our contemporary anti-vaxxers) measles example from Fred Dretske.  Dretske, by the way, does not discuss this example in the light of IF-THEN statements.

…an exceptionless uniformity … is not sufficient for the purposes of transmitting information. ¬†Correlations, even pervasive correlations, are not to be confused with informational relations. ¬†Even if the properties F and G are perfectly correlated (whatever is F is G and vice versa), this does not mean that there is information in s’s being F about s‘s being G (or vice versa). ¬†It does not mean that a signal carrying the information that s¬†is F also carries the information that s is G. ¬†For the correlation between F and G may be the sheerest coincidence, a correlation whose persistence is not assured by any law of nature or principle of logic. ¬†All Fs can be G without the probability of s‘s being G, given that it is F,¬†being 1.

To illustrate this point, suppose that all Herman’s children have the measles. ¬†Despite the “correlation,” a signal might well carry the information that Alice is one of Herman’s children without carrying the information that Alice has the measles. ¬†Presumably the fact that all Herman’s children (living in different parts of the country) happened to contract the measles at the same time does not make the probability of their having the measles, given their common parentage, 1. ¬†Since this is so, a signal can carry the information that Alice is one of Herman’s children without carrying the information that she has the measles despite the fact that all Herman’s children have the measles. ¬†It is this fact about information that helps to explain (as we will see in Part II) why we are sometimes in a position to see that (hence, know that) s is F without being able to tell whether s is G despite the fact that every F is G. ¬†Recognizing Alice as one of Herman’s children is not good enough for a medical diagnosis no matter what happens to be true of Herman’s children. ¬†It is diagnostically significant only if the correlation is a manifestation of a nomic¬†(e.g., genetic) regularity between being one of Herman’s children and having the measles.

Fred Dretske, KNOWLEDGE AND THE FLOW OF INFORMATION, Stanford, CSLI Publications, 1999, pp. 73-74

Myself, I would rather choose a less gruesome (given the sometimes horrific consequences of measles), even if still somewhat gross, example. ¬†Suppose that there is a pile comprising red and yellow apples in my grandfather’s orchard. ¬†By pure chance, some of the yellow apples happen to be wormy, while all of the red apples are so. ¬†Given his measles example, Dretske would surely claim that just the fact that a given apple from the pile is red would not constitute information that the apple is wormy. ¬†But suppose that, blindfolded, I handle each apple in the pile one by one, saying each time:

If this apple is red, Then it is wormy.

In my mind’s inner ear, my intuition is shouting to me: ¬†“This is TRUE TRUE TRUE TRUE TRUE!!!!!!”

Likewise, surely the following statement is also true:

If this person loitering here in front of my shop among all these other disreputable-looking lay-abouts is a child of Herman’s, Then she has measles.

This statement would be true, it (strongly) seems to me, even if the person uttering it is talking completely through their hat, even randomly, and has absolutely no evidence that ‘this person’ has the measles, or that she is a child of Herman’s, or that there is any connection at all, even an accidental one, between Herman’s children and the measles.

Therefore, there would seem to be clear cases in which an If p Then q statement is true even when the occurrence of p is not information that q.

Nonetheless, I (at least as of this writing) think I can show in a later post that Dretske’s discussion of the relativity of information drastically undercuts what he thinks his measles example shows. ¬†(I am also thoroughly confident, by the way, that if my doubts are valid, they have already been discussed a thousand times already by everyone and their uncle.) ¬†So the idea that what¬†makes p relevant to q in any true If p Then q statement is an informational relation . . . this idea might find a rescuer after all.

 *****

I hope that today’s homage to Plato’s SYMPOSIUM has never suffered from the measles. ¬†This gorgeous hunk is a math teacher in Great Britain (perhaps hailing ultimately from Italy) who moonlights as a model.

pietronew

I am confident that this math teach will inspire many of his students, both male and female, to start the ascending the platonic ladder whose lowest rung consists in the contemplation of the Beauty of Gorgeous Guys, whose next rungs consist in the contemplation of the Beauty of Math and Logic, and which finally leads to the contemplation of the Form of Beauty Itself.

For now, however, I will linger a bit at the lowest rung, the Contemplation of the Beauty of Gorgeous Guys.  I will get to the Form of Beauty Itself sometime.


IF-THEN Treated As INFORMATION THAT

Relevant Logic tries to resolve the following paradoxes of Classical Logic’s Material Implication by insisting that for any If¬†p Then¬†q statement, p must be relevant to q:

If Cliff Wirt resides in Houston, Texas, Then the earth has just one moon.

If Calypso music originated in Wisconsin, Then the earth has two moons.

According to Classical Logic, both of the above statements are true because they fulfil the truth-functional requirements of true IF-THEN statements. ¬†(T T and F F. ¬†According to Classical Logic, F T also yields a true IF-THEN statement; the only truth-table combination that yields a false IF-THEN statement is T F.) ¬†Nonetheless, one may be excused if they think that regarding the two statements as true is a bit paradoxical, to put it mildly. ¬†One cannot conclude, infer, or learn from Cliff Wirt’s residing in Houston that the earth has just one moon. ¬†Even less can one conclude, infer, or learn from the “false fact” that Calypso music originated in Wisconsin the equally “false fact” that the earth has two moons. ¬†One would think that both IF-THEN statements are false because in both, the antecedent, p, is irrelevant to the consequent, q.

So the truth-functional account of the IF-THEN statement has to go, I am thoroughly persuaded, because it can take into account only the truth or falsity of the antecedent and consequent, leaving completely out of view the relevance of the antecedent to the consequent.

What, then, would make the antecedent relevant to the consequent?  What is the relation between p and q when we say If p Then q?  I am partial to the hypothesis that the relation is informational.  If p Then q is true when the occurrence of p is information that q.  If the doorbell is ringing, then someone or something outside has depressed the button; that the doorbell is ringing would be information that someone or something outside has depressed the button.  The first is information that the second because there is a channel of information extending from the button to the ringing sound, such that, when that channel is in good working order, the probability that the button is being depressed is 100% when the ringing sound occurs.

Because this informational relation exists between the ringing sound and the button’s being depressed, one can conclude from, infer from learn from the doorbell’s ringing that someone or something is depressing the button outside. ¬†So — oh my god! — there is a close affinity between If p Then q and p’s being information that q.

There are, however, several obstacles in the way of treating the IF-THEN statement as an informational relation.

First, how would one deal with If p then p?  Is there somehow supposed to be a channel of information between p and itself?

Second, there are (seemingly) clear cases in which If p Then q is true when p is most definitely not information that q.

Third, the informational relation is both intentional and relative, as described by Fred Dretske in his KNOWLEDGE AND THE FLOW OF INFORMATION. ¬†Treating If p Then q as an information relation would make implication both intentional and relative. ¬†The very same If p Then q statement would be true inside some frameworks and false inside others. ¬†Rather than accept this, some would perhaps rather accept Classical Logic’s paradoxes of Material Implication.

*****

Today’s homage to Plato’s SYMPOSIUM takes the form of a very kalos Bruno Mars. ¬†According to Plato, one ascends a ladder whose first rung consists in the beauty of gorgeous young men, whose middle rungs consist in the beauty of things like Classical and Relevant logic, and whose final rung consists in the Form of Beauty Itself.

bruno-mars-promo

I will get to adoring the Form of Beauty Itself eventually.  For now, I will content myself with adoring the Form of Bruno Mars.


Aristotle’s Sea Battle Argument

In a rough draft of a blog post at work whose real topic had, of course, precious little to do with Aristotle, I playfully tried to explicate his Sea Battle argument to an audience of techies as follows:

The following statement (call it p) is necessarily true:

Either there will be a sea battle tomorrow [at location l], or there will not be a sea battle tomorrow [at location l].

At least one, or possibly both of the constituents of an OR statement must be true if the statement is true. ¬†If the OR statement happens to be¬†an ‘A OR not A’ statement, at most one of ¬†the constituent statements can be true. ¬†What is more, since ‘A OR not A’ must be true, one of the constituent statements must be true. ¬†So either

There will be a sea battle tomorrow [at location l]

is true, or:

There will not be a sea battle tomorrow [at location l]

is true.

But which one?

Suppose that ‘There will be a sea battle tomorrow [at location l]’ is the constituent proposition that is true. ¬†(Call this constituent proposition c1.) ¬†One may already have been struck by the Aha Erlebniss that the sea battle will not ¬†fail to happen (and in fact cannot fail to happen) tomorrow at location l. ¬†(henceforth ‘at location¬†l‘) will be understood.) ¬†But my mind and my imagination feel the presence of a gap between c1 and ‘the sea battle cannot fail to happen tomorrow.’ ¬†When I try to jump from the first to the second, I feel a bit as if I were plunging into a void. ¬†The following thought experiment is an attempt to bridge that void,

Start Of Thought Experiment:   To avoid complications involving indexicals, suppose that today, at time t0, I say:

A sea battle happens at time tn.

From the standpoint t0, tn is a point in time that will roll by tomorrow. ¬†Could my statement stop being true at t0+1 (0+1 < n)? ¬†Don’t be silly — of course not. ¬†Someone’s statement ‘The cat (Sylvester, with CAT_ID 347434395) is on the mat (the medieval Persian mat with MAT_ID 84541) at 12:01 pm, October 31, 2014’ never ceases to be true, assuming it was true at¬†12:01 pm, October 31, 2014. ¬†Ditto my sea-battle statement. ¬†Could my sea-battle statement suddenly stop being true at to+2? ¬†No, of course not. ¬†And so on for every time point starting from t0 and going up to tn. ¬†My statement will be equally true¬†at tn – 1 as well as at time tn. Throughout, it remains true that the sea battle will happen at tn. ¬†There is no room left for the sea battle NOT to happen at tn.

In fact, what would it mean for that statement suddenly to become not true, at some point between to and tn? ¬†Well, suppose — doubtlessly per impossible — that the chain of one set of causes leading to a set of effects serving as causes for yet another set of effects ceases — say, at tn-1 — to be deterministic. ¬†That chain continues unbroken until, abruptly at tn-1, it becomes a flip of nature’s coin whether the sea battle happen or not. ¬†Then, it seems to me, the truth of ‘A sea battle happens at time tn’ ceases to be defined. ¬†The statement is neither true nor false. ¬†Therefore, the statement would be not true, though it would not be false either.

Or again, suppose that the chain continues unbroken until suddenly, at time tn, we end up with (again, per impossible, I am sure) with a weird quantum Schroedinger’s sea battle: ¬†the sea battle is simultaneously in a state of happening and not happening at tn. ¬† ¬†In this case, my intuition is, the truth value of my sea battle statement would be undefined at t0 as well as at tn. ¬†End Of Thought Experiment.

So assuming there is a chain of causes working deterministically from t0 to tn, my sea battle statement is definitely true at t0. ¬†And there is no way that the sea battle will fail to happen at tn. ¬†The chain of deterministic causes (assuming this exists) is what gives sense to the idea that my sea battle statement has a definite truth value at t0 — that is is true (false) at that time-point.

This is Fatalism. ¬†Fatalism is often thought to entail that we have no Free Will. ¬†Aristotle comes to this conclusion, and panics. ¬† (At least according to my explication of this stuff to my fellow geek colleagues.) ¬†“Oh my god!!!!!!….er….I mean….oh my Zeus!!!!! ¬†If there is no Free Will, then that loud sucking sound you hear is my ETHICS going down the drain!!!!! ¬†Quick!!! Quick!!!!! Think of something!!!!!!’ ¬†(I have to admit that my translation of the ancient Greek here is a trifle free.) ¬†So to save his ethical theory Aristotle decides to assert that while the total original proposition, p, is necessarily true, the truth value of both of its constituents is undefined. ¬†Neither of its constituents is either true nor false.

But I do not see how this (the constituents’ not having a definite truth value) could be so unless the sea battle’s happening (or failing to happen) tomorrow is a matter of nature’s flipping the coin. ¬†Aristotle cannot be right.

I say ‘Aristotle cannot be right’ in full confidence, as a matter of black and white. ¬†Nonetheless, just a little shade of gray, a tiny sliver of doubt, does enter here. ¬†The laws of nature are supposed to be deterministic on the level of apples and triremes, but non-deterministic on the level of protons and electrons (and for all I know on the level of quarks as well). On the micro level, nature is (if I understand this stuff correctly) constantly tossing a coin. ¬†Although one is not supposed to mention quantum physics in a philosophical discussion unless they (intentional use of ‘they’ as a singular gender-neutral pronoun) have completed at least 8 graduate courses in quantum physics (with no grade lower than a B+ in any of them), I do have to at least wonder quantum weirdness might invade the causal chain leading to the sea battle’s occurring (failing to occur) tomorrow in such a way as to make it only 99.9999999999999999999999999999999999999999999999999999999999% probable, not 100% probable, that the sea battle will happen (fail to happen) tomorrow. ¬†Is this enough to blast away the bridge that leads from the present to the future that lets us say that a statement about the future uttered now is either true or false? ¬†I will leave that as a nagging question leaving in its wake just the tiniest whiff of doubt.

* * * * *

If Aristotle were right, then either p is not in fact an OR statement (it only looks like one), which seems rather counter-intuitive to me), or normal classical logic fails to hold for the future.  Contrary to normal, classical logic, it would not be the case that an OR statement is true if and only if at least one of its constituent statements is true.  This would hold only for statements about the present.

But in that case statements such as ‘If this apple drops from the tree under which I am sitting, this apple will splat onto my head in one second’ (call this the ‘apple if-then statement) will not have a defined truth value. ¬†Reducing to ‘Either this apple does not drop from the tree under which I am sitting, OR this apple will not splat on my head in one second’ (‘if p then q’ is the same as ‘not p or q’), ¬†So the truth value of the total apple if-then statement will be undefined because, being a statement about the future, the truth value of ‘this apple will splat on my head in one second’ is undefined.

So if we restrict normal, classical logic to just the present, the number of interesting statements it rules over will become awfully restricted. ¬†Normal, classical logic will become a parlous affair, just as pitiful as the crowning of John Cantacuzenus ¬†and Irene, Andronicus Asen’s daughter in the waning days of the Byzantine Empire. ¬†As related in C.P. Cavafy’s poem Of Colored Glass:

As they had very little in the way of precious stones

(our wretched dominion’s poverty was great

they wore artificial ones.  A heap of bits of glass,

scarlet, green or blue.

I always end my philosphical/logical posts with an homage to Plato’s SYMPOSIUM, for which purpose I will use Ashton Kutcher (swooning, rapturous sigh) yet one more time:

Ashton_Kutcher_0

Look at those stunningly beautiful brown eyes!!!  How can anyone get any work done with beauty like this walking the earth?