Category Archives: Relevant

The Monty Hall Paradox And Borges’ GARDEN OF FORKING PATHS

Nota Bene: this is still very much a work in progress. I have not yet achieved that mental state at which I can indulge, at least for a while, in the delusion that I have achieved the maximum point of crystalline clarity.’ I am not responsible for any brain damage anyone reading this stuff may incur.

Here in outline form are the points I intend to make (assuming I succeed — success if not totally guaranteed at the moment).

  1. Assuming a deterministic universe, ignorance is a necessary condition for randomness (i..e, probabilities less than 1 and greater than 0). Normally this is a “mix” of ignorance and knowledge. Ironically, my own ignorance comes into play in trying to argue this. There will be various lacunae in my argument which, I hope, will steadily decrease over time.
  2. There are (at least) two components to randomness: the sample space and the likelihood of each element will ‘come up’ in an “experiment”. (I am using the standard, unfortunately misleading terminology one can find by googling ‘probability’. This terminology distinguishes between ‘events’ — subsets of the sample space set whose members are possible outcomes (and, I will argue, possible realizations) — and ‘experiments’ — actual outcomes/realizations appearing in time and space.)
  3. In the normal shell game, Morgenstern’s increase in knowledge suffices to decrease the size of the sample space. Her ignorance increases that cardinality. The likelihood of winning the peanut changes accordingly. In the Monty Hall shell game, Elizarraraz’ knowledge also decreases the sample space.
  4. The cardinality of the sample space also depends upon the background of common sense which specifies that certain things count as belonging to the set and other things do not. Sample spaces are “subjective” and highly perspectival in character.
  5. Normally the actual outcome of an experiment is something that can be taken in at once. It is restricted to the present. This particular card was drawn. The flipped coin came up heads. The die came up 6. The peanut appeared when the shell was turned over. This is the realm of what could be. The shell Smith is about to turn over could be hiding a peanut, or it could be hiding nothing but empty air.
  6. But I would like to expand members of a sample space to include what I will call possible realizations. This particular possible path through time, when realized, resulted in Smith’s now turning over the shell with the peanut. Before it was just one of 13 possible paths that could have threaded its way through time; now it is an actually accomplished, actually realized path. It is the path that made it into actuality among the 12 other paths that could have been.
  7. The idea of calculating the probability via a tree (shown on countless Google-able sites) came to be via the Borges short story GARDEN OF THE FORKING PATHS; but also the idea of treating all the possible paths as members of a set came to me from that story.
  8. I try to define likelihood as an idealization of the numbers one gets after repeating an experiment a very large number of times. (Flip a fair coin 10 million times using a particular standard method; it comes up heads 5 million and 1 times and tails 4 million and 999 times; we idealize that to ‘the coin is equally like to come up heads as tails.) The likehood depends partly upon the method one uses (using Emo’s particular technique of cheating; always switching one’s choice in the Monty Hall shell game; always taking the path on the right at the end); it also depends partly upon the cardinality of the sample space. So likelihood is partly perspectival and “subjective”, partly anchored in the objective world. The more subjective, the less the weight of the anchor. So no sharp distinction between “natural probability” and “subjective probability”.
  9. If one does not “carry over” the knowledge from the past that would let one identify one shell as the ‘initially selected’ one, there is no method one can use that would raise | alternatively lower one’s chances from 50/50 to 2/3 | alternatively 1/3. The probability in this case is 50/50, so making sense of the strong intuition that switching would not make any difference. The ability to give oneself the higher chances depends upon carrying over into the present information from the past. It requires the depth of the past. 50/50 is what you get when you are stuck in the depth-less present. This point is going to require some finessing. From this perspective, the chances are 50/50. From this other perspective, the chances are 2/3 | 1/3.
  10. Two pictures: First, I am staring at the two remaining shells with no way to distinguish them. I am restricted to the present and to the near future (what could be). I cannot remember which was the shell I had initially chosen — the shells are too much alike. (This is a shell game, after all.) My chances are 50/50. Second, I do remember and can identify the shell I had initially chosen. The past and its “possibilities” — its could have beens — is opened up to me. My chances are 2/3 | 1/3, depending on how I use the information.
  11. Want to end with a contrast with Searle’s illustration.
  12. The upshot: assuming a deterministic universe, a ratio or “mix” of ignorance/knowledge is required for the existence of randomness. Randomness is perspectival in character through and through.

[Different games/practices will have different rules/constraints which will determine what the sample space is and its cardinality. To play a probabilistic game/engage in a probabilistic practice one must have a certain amount of knowledge (‘a peanut is hidden under one of the shells’; this shell is the one initially selected by Smith); for there to be a game/practice at all there must be this item of knowledge. Without the knowledge there are only latent games — latent probabilities. And within the confines of the game/practice there must be an ignorance that poses an obstacle to one’s achieving the point of the game/practice. Rules/constraints plus knowledge/ignorance determine the sample space and what game is being played. For it to be a game/practices of chance, ignorance is required. This outline is currently a mess; I hope to clean it up later.

The picture of a set of forking paths in the forest is useful because it encapsulates Smith’s moving from past to future; preserving knowledge of which shell was initially selected is necessary in this movement. Path dependence. So not just a matter of which shells are staring one in the face at the moment.

Some examples: Smith knows that there is a peanut under one of the shells. He turns over the shells one by one until he uncovers the one hiding the peanut. The probability that he will find the peanut is 1. Even here there is ignorance.

It must make sense to say of the game/practice: ‘were the game repeatable.’

“Uncertainty” aka randomness is a “mix” of knowledge and ignorance.

Do this and you will win ~10 million times | alternately 5 million times out of 15 million times. The probability given this practices is 2/3 | alternately 1/3. The probability is relative to the practice (which could be a game). The practice is taking place within an arena (horizon, world) that defines what counts as an item that is eligible to be “in play”. This arena makes the item relevant to the practice (the shells piled up to the side don’t count in the game). One has knowledge, but one is also ignorant. Theoretically, one might be so dumb as to not be able to exploit that knowledge in the practice one is engaged in … unlikely in the case of knowing that shell #1 is empty. One knows things about the shells. Some of these things will affect one’s chances; others will not. Shell #1 is empty | alternatively hides the peanut. Of the two paths you or I (second or first person will be important) comes across at the end, one is on the left and the other is on the right. I or you may or may not be able to exploit either item of knowledge, but both will affect one’s chances if exploited. ]

What is the point of the arguments that are about to follow? These arguments are one snippet in an attempt to get clear in my mind regarding the nature of probability. (Yes, I know, this is absurdly ambitious. You may be a bit less inclined, gentle reader, to break out in raucous laughter if you keep in mind I am just trying to arrive at the point at which, in a doubtlessly delusional state, I suffer from the strong conviction I have gotten clear in my own mind regarding the nature of probability. Once achieved, this strong conviction will doubtlessly evaporate like a mirage as I increase my knowledge of the field. Or if I take my meds.)

The reason I want to get clear in my own mind about the nature of probability because I think this is necessary in order to uncover at least one relation that makes the antecedent relevant to the consequent in relevant indicative conditionals. I expect to be making changes to this post as time goes on.

What is the conclusion I am heading towards with all the verbiage below? This: the existence of a probability greater than 0 but less than 1 has as both its necessary and sufficient condition a ratio of ignorance/knowledge within a given perspective, itself limited by a background of common sense. Probability within these two limits is perspectival down to the very root for this reason; it could not exist within the “perspective” of an infinite mind that does not suffer any ignorance at all, partly because such a Mind would not enjoy any perspectives at all. Given a deterministic universe, this is the only way there can be probabilities between 0 and 1 noninclusive. This places me in the camp of those who, like Keynes, regard probability as “subjective”, but I hope to do justice as well to the rather hard-edged intuition expressed by ‘just don’t take your subjective probabilities to Vegas.’

In the clearest cases, the role knowledge/ignorance plays in determining such a probability is easiest to see in the case of independent events; but dependent events, as in the case of the Monty Hall puzzle, can increase/decrease the probability of a given event.

The Scene. A Shell Game Is Set Up. Let me begin by describing the scene. In an apple and cherry orchard in Iowa, a table has been set up. The sky above is clear. Unknown to and hidden from the people in and about to enter the orchard, but within view should one occupy the right vantage point, a tornado is touching down intermittently across the Missouri River, in Nebraska. I describe the scene this way because it is a situation. A situation is partially defined by what is hidden from one and unknown to one, and by the information that is available to one. Situations will become important in later posts because some versions of Relevant Logic rely on them rather than on possible worlds. I describe this particular one now because I will be returning to it later.

Elizarraraz (although this is not relevant to the example, the name, taken from the name of my landlord in Chicago from whom I was renting a studio for some years, is Ladino for ‘poor king’. Ladino is the Sephardic counterpart to Yiddish, and in Elizarraraz’ case the name, and his paternal ancestry, comes from Mexico. Although they were not officially allowed to, a number of conversos managed to emigrate to Latin America in order to place a more comfortable distance between themselves and the Spanish Inquisition. Just thought I would provide my made-up characters with concrete backgrounds. But I digress) sets up on the table a shell game with three shells and a single peanut.  The shells are labelled in order 1, 2, and 3. Employing a randomizing device of some sort (say, he throws a die), Elizarraraz places the peanut under the shell selected by his randomizer. Naturally, he knows under which shell the peanut is hidden.

[At least for now, I will leave the concept ‘randomness’ as an unanalyzed primitive, explicated, not by a real, concrete example, but by a (vaguely described) ideal one. A fair 6-sided die would be suitably random if, after a very large number of throws, the average ratio of the times each number came up, divided by 6, remained sufficiently close to 1/6. And yes, I will leave ‘sufficiently’ undefined. ]

[Information/absence of information determines the sample space, along with, obviously, what the information is about. The probability function is derived from an idealization of a large number of experiments. An experiment occurs when an outcome in time and space is obtained.]

Information/absence of information determines the sample space, along with, obviously, what the information is about: Smith (although this is not relevant to the example, the name is English for ‘smith’ as in ‘blacksmith’. But you knew that already) enters the scene. He knows that there is a peanut hidden underneath one of the three shells. (Elizarraraz, who is a reliable conduit of information, has told him this.) Smith is about to play what I will call, for reasons that are about to become clear, the ‘normal’ shell game. He is to select a shell and turn it over to see if it is hiding the peanut. I think it would be uncontroversial to say that the probability there is a peanut underneath that particular shell is 1/3, and the probability that there is not is 2/3.

This number is the result of two factors: first, the sample space, and second, the likelihood that any of the members of that sample space will become an actual, and not just a possible outcome (Smith selects the shell that is hiding | alternatively not hiding the peanut). In this particular case (the normal shell game), the sample space is the set of possible outcomes of turning over any of the shells in play on the table. Shell #1 hides the peanut, or shell #2 hides the peanut, or shell#3 hides the peanut. So the sample space Ω is :

{ shell#1p, shell#2p, shell#3p }

or, to show explicitly that if, say, shell #1 happens to be hiding the peanut, the remaining shells are perforce empty:

{ { shell#1p, shell#2p, shell#3p }, { shell#1p, shell#2p, shell#3p }, { shell#1p, shell#2p, shell#3p } }

where the superscript ‘p’ means the shell is hiding the peanut and the superscript ‘p‘ means the shell is not hiding the peanut.

Let me get some terminology out of the way. I will be relying on the standard google-able terminology of probability theory. The sample space is the set of possible outcomes of an “experiment”. An “event” is a subset of this superset, i.e. the sample space. For example, the subset ‘shell #1 hides the peanut and shells #2 and #3 do not’ is an event. The term is a bit unfortunate, because ‘event’ usually connotes — at least to my ears — a concrete happening occurring (or having occurred or occurring in the future) in space and time. Here, however, an ‘event’ is an abstraction — a subset, and not a concrete happening occurring in space and time. But whatever. An “experiment” is by contrast a concrete action, such as turning over shell #1 and discovering it to be hiding the peanut | alternatively hiding just empty air.

There are at least two factors that determine the “size” or cardinality of a sample space — the number of members it has. These factors are 1) what I will call ‘the shared background of common sense’ and 2) a person’s individual ignorance/knowledge. Both of these factors are, in one sense or another, “subjective” and perspectival.

1) Shared background of common sense: A sample space is, I have said, a set of possible outcomes of a given activity. But what determines what is eligible to count as a ‘possible outcome’? The answer to this question will help determine the “size” or cardinality of a sample space. There are a number of factors that contribute to answering this question.

This shared background of common-sense comprises social practices, rules, deeply-ingrained dispositions to count certain things as relevant and other things as not relevant, the stability and predictability of physical objects (at least on the post-quantum level). I address these sub-factors in an order suggested to me by Wittgenstein’s ON CERTAINTY, going from the most vulnerable to change to the least vulnerable, from the least deeply taken for granted to the most deeply.

1a) Rules of the game: Taken-for-granted rules govern practices in general and games in particular and help to define these practices and games. Drawing a standard recognized card from a pack of cards normally counts as an eligible outcome; drawing a scrap of paper that may strayed into the pack normally does not. The sample space for the cards has a cardinality of 52, not 52 plus the one scrap of paper. Flipping a coin has two possible outcomes, heads or tails. The coin’s landing on its edge is not a possible outcome, at least not if the normal rules that apply to the practice of flipping a coin are in force. It might be a possible outcome in a different game. In the normal practice of flipping a coin, the sample space is the set with two members: coin lands heads or it lands tails. In a non-normal practice, the sample space might have three members: The same holds mutatis mutandis for throwing a die. In the normal practice, the sample space comprises six members. But should anyone be skilled enough to make the die land on one of its edges invent a new, non-normal practice, the sample space would comprise 12 members. Winning the peanut is the point of the normal shell game set up by Elizarraraz, not uncovering a particular grain of sand — no matter how exquisite that grain is. So if none of the three shells covers the peanut, but one covers a grain of sand, the cardinality of the sample space will be 0, not 3.

Extrapolation of the rules to form a different game: It may make sense to talk about the cardinality of a sample space of a game that is merely possible, and not actual. Suppose, for example, that no game — call it the non-Monty-Hall shell game — currently exists with the following rules. The three shells, one of which is hiding the peanut, are placed on the table as before. But now the player has two chances, not just one, to try to win the peanut. As before, the player’s selection(s) are made without any action taking place that depends upon knowledge of the peanut | peanut-less state of the shells. Even in the absence of an actual game like this, one can, clearly, see that after the first selection the sample space would now be restricted to two members. Merely possible rules suffice to determine a sample space and its cardinality.

1b) What counts as an eligible item in play is determined by the background of common sense: What items that are ‘in play’ in a practice are also taken for granted. The shells Elizarraraz has placed on the table are the ones that are in play in the normal shell game he is setting up. The shells that are in a pile a few feet away from him are not in play. Nor is this or that shell on the beach 1,500 miles away to the east, or 2,000 miles to the west.

1c) As part of the background of common sense, the stability of the physical world plays a role: Even more deeply taken-for-granted is the stability of the physical world — at least on the post-quantum level. We assume for example — except for a few Twilight Zone moments — that the peanut stays under whichever shell it is under and is not going to behave like the electron which (according to my undergraduate chemistry TA), for all one knows, might be on the nose of the Mona Lisa. Additional to the three shells in the sample space that might be hiding the peanut, there is not also a fourth shell, sometimes one underneath the Mona Lisa or stuck on her nose that might be hiding the peanut, sometimes one on the floor of the Farnese Palace underneath the Carracci ceiling.

1d) Cases in which ‘What size is the sample space?’ does not have a clear answer: For the moment, dear reader. let me refer you to Graham Priest’s treatment of Sherlock Holmes’ induction that Jabez Wilson is likely to do a great deal of writing given the smooth patch on Wilson’s coat at the elbow. (LOGIC A Very Short Introduction, Graham Priest, Oxford, Oxford University Press, 2000, pp. 78-85).

All of these examples concern what is relevant to a particular problem, practice, or game. Naturally, this should raise some concern in the non-comatose reader that I may be going in a circle. For what I eventually hope to gain at the end of these ruminations is an account of at least one relation that makes p relevant to q in a relevant IF p THEN q conditional. For the moment, I will beg for mercy by pointing out that there circles and there are circles; some circles are narrower than others; some wider circles give one a more expansive view of the territory and constitute the scenic route.

2) A person’s ignorance/knowledge:

Let’s return now to Smith, who is about to turn over the shell he has selected. He is ignorant of which of the three shells is hiding the peanut, but he knows that the peanut is lurking behind one. Let’s stipulate that each shell is equally likely to be hiding the peanut. What ‘equally likely’ means I will be turning to shortly; for now, let’s just take it as a primitive. In this situation, Smith’s chances of winning the peanut are clearly 1 in 3; his chances of turning up just empty air are 2 in 3.

Gun on the shelf that will fire later in the story [1]: Obviously, Smith does not know if the shell he is about to turn over hides the peanut or not. Whatever method or non-method he uses to select the shell (he likes the slightly tawnier sand-color of this one; he rolls a 3-sided die; he just picks one), he does not choose based on any knowledge, or on any evidence of what might be | might not be lurking underneath the shell.

Let’s be Smith for a moment — he is, after all, Any Person/Every Person.

Might and could be defined by ignorance.

Smith knows that a peanut lies hidden underneath one of the three shells. (And of course it follows from his knowing that the peanut is under one of the shells that it is under one of the shells.) But Smith does not know under which That shell could be, might be shell #1, or it could be /might be shell #2, or it could be/might be shell #3. Were Smith come to know that the peanut is under, say, shell #3, it would no longer be the case that the peanut could be/might be under shell #1, and it would no longer be the case that the peanut could be/might be under shell #2. These would no longer be possibilities, that is to say, possible outcomes, could be’s or might be’s. [Currently unredeemed intuition] A possibility’ requires a combination of knowledge and ignorance. Remove the ignorance, and the possibility no longer exists. And since, in this particular case, it is Smith’s ignorance that obtains, the possibilities are such, are possibilities only from Smith’s point of view. ffffffffffffffffffffffff

Smith turns over the shell — say, shell #1. It was hiding nothing except empty air (plus a certain stretch of table wood). Smith returns the shell to its previous position (carapace side up).

Now were Smith allowed to repeat the game, but this time with the two remaining shells, #2 and #3 — the two shells the contents of which he is still ignorant — his chances of winning the peanut would surely be 1 in 2. That this is so should be clear intuitively. Of course, the Monty Hall shell game which I will be discussing shortly, tends to generate the exact same false (fasle yes — but with certain caveats) intuition. Can we rely on our intuition in this case? Spoiler: yes. But I will get to that after the long, boring disquisition on the Monty Hall shell game that will ensue shortly.

Now Morgenstern (German for ‘morning star) enters the scene.  (She hails from Brooklyn and she was in MY COUSIN VINNIE.) She does not know that shell #1 turned up empty. She does know, however, that one of the shells hides the peanut, because a reliable source of information, Elizarraraz, told her. The peanut is still under one of the remaining shells. Using a randomizing device, Elizarraraz has shell #2 selected for him. He points to that shell #2 and asks both Smith and Morgenstern what are the chances the peanut is under that shell. For Smith, surely, the answer is 1 in 2. For Morgenstern, the answer has to be 1 in 3. For Elizarraraz, who knows where he put the peanut, the answer has to be either 0 or 1. Were Elizarraraz to point to shell #1, the answer for both him and Smith would have to be 0. What the probabilities are differs from the perspectives of each of the three because the sample space differs for each given what each knows.

[Likelihood: Naturally, both the ‘let’s say’ and the ‘equally likely’ cry out for some finessing, given that the whole point of this exercise is to state what probability/randomness is. Explaining randomness, a probabilitistic concept, in terms of other probabilistic concepts (‘likelihood’) does seem a bit unpromisingly circular. But before I turn my attention to this threatening circularity, I want to focus for a moment on the concept of a sample space. ]

Elizarraraz is not ignorant of which shell the peanut lies. He knows that it is under shell #3, since he was the one who placed it there. From his point of view, it is not the case that the peanut could be/might be under shell #1, and it is not the case that it could be/might be under shell #2. From his point of view, it is certain that the peanut is under shell #3.

Future outcomes. Elizarraraz is ignorant, however, of which shell Smith is about to select. That shell could be, might be shell #1, or it could be /might be shell #2, or it could be/might be shell #3. Were Elizarraraz none of these could be’s/might be’s would be the case. Were Elizarraraz to know that at some future time tsubn that Smith will choose, say, shell #1, then it would be certain that at tsubn Smith will choose that shell. Knowing that p implies the truth of p. But of course it could be true that that Smith will select shell #1 at tsubn and Elizarraraz not know that. In that case, Smith’s selecting that shell remains a possibility from Elizarraraz’ point of view. But what if Elizarraraz does not know because it is not certain — it is not a 100% probability — that Smith will select shell #1 at time tsubn? Suppose it makes sense to say that even from the “view from nowhere,” that of an an entity that knows every true proposition, this entity does not know that Smith will select shell #1 because there is no fact of the matter — there is only a certain probability. Time tsubn comes along, either Smith selects shell #1 or he does not — he selects another shell or no shell at all. Smith just selects the shell, say, even though no previous cause establishes a 100% probability. The event just pops up. nd then Elizarraraz cannot know that Smith will select shell #1 at that time, and Smith

Suppose no peanut was lurking under that shell — say, shell #1. Smith now knows that there was no peanut under shell #1. In at least some sense of the term ‘certain’, he is now certain that shell #1 was not the one hiding the peanut. He has the information that shell #1 was not hiding the peanut. But he knows that (is certain that, has the information that) there is a peanut lurking under one or the other of the remaining shells, #2 and #3. I have, and I think most people will have, the strong intuition that the probability the peanut is under shell # 2 (alternatively shell #3) is 1/2. For the original sample space of 3 has been reduced to 2, and each outcome is, we say, equally likely.

At this point, Smith confronts two possibilities. A possibility is a possible outcome. Possibility #1: the peanut lurks under shell #2 and shell #3 is empty. Possibility #2: the peanut lurks under shell #3 and shell #2 is empty. To talk about ‘a possibility’ here is to say the following: because Smith knows there is a peanut under one of the shells (he just doesn’t know which one), there is a peanut under one of the shells. For if one knows that p, then p is a true proposition (or, better, a state of affairs that obtains [I follow Chisholm in identifying propositions with a proper subset of states of affairs]. From Smith’s point of view, the peanut could be under shell #2 or shell #3; that is to say, he doesn’t know which one. So, at least in cases like this one, [yes, I know, this needs to be more sharply defined] ‘a possibility’ requires a combination of knowledge and ignorance. Remove the ignorance, and the possibility no longer exists.

From Smith’s point of view, it is no longer the case that the peanut could be under shell #1. Its being under shell #1 is no longer a possibility for Smith. And the probability that it is under shell #1 is now 0. Were Elizarraraz to turn over the shell that does hide the peanut (say, shell #3) (and were Smith to see the peanut that had been hiding there, and were nothing at fault in Smith’s visual apparatus), it would no longer be the case that, from Smith’s point of view, the peanut could be under shell #3. It is under shell #3. Its being under shell #3 is no longer a mere possibility, but a certainty. Again, remove the ignorance, and the possibility no longer exists. From Smith’s point of view, the probability that the peanut is under shell #3 is now 1.

When Smith turned over shell #1 and discovered it to be empty, he decreased the size of the sample space from three possibilities (the peanut is under shell #1 and shells #2 and #3 are empty; the peanut is under shell # 2 and shells #1 and #3 are empty; the peanut is under shell #3 and shells #1 and #2 are empty) to just two (the peanut is under shell #2 and shell #3 is empty; the peanut is under shell #3 and shell #2 is empty). A sample space is a set of possibilities; the cardinality or “size” of the space is the number of possibilities it has as members. The metaphor of ‘a space’ is apropos here because a given space, a room, for example, can contain items, just as set “contains” its members. If a sample space contains n possibilities and each possibility is equality likely, then the probability of each event (subset of the sample space) must be expressible as a ratio with n as the denominator. If the size of the sample space is six, for example, the probability of each event must be expressible as 1/6, 2/6, 3/6, 4/6, and 5/6.

When the possibilities involve physical entities, such as a number of shells one of which hides a peanut, it is easy to think of the size of the sample space as equal to the number of those entities. Later, however, I intend to show that the sample space can include possible as well as actual entities.

Now Morgenstern arrives on the scene.

A likelihood is assigned to the sample space: A likelihood is a number that can find | alternatively fail to find an anchoring in the real word — and this to varying degrees. One can anchor a likelihood by repeating an experiment a very large number of times. If one flips a coin ten million times, for example, and the coin comes up heads five million times and tails five million times, one may perhaps be allowed to say that the coin’s turning up heads and its turning up tails are equally likely events. If one flips the coin twenty million times and the coin lands heads 10,000,001 times and lands tails 9,999,999 times, one may perhaps be allowed to say ‘this is close enough for government work — I will say the two events (landing heads and landing tails) are equally likely’. I regard as highly credible the idea that, no matter how often the coin if flipped, the numbers will rarely be completely even. At the time of this writing I have no idea — I lack the mathematical depth — whether the numbers will converge on some 50/50 limit which we could then use to assign a likelihood to the sample space in an objective manner; or whether the numbers will vary randomly, with upper control and lower control limits ala Taiichi Ohno (in which case I will be stuck in a circle trying to define ‘probability’ in terms of concepts that are themselves probabilistic), or whether the numbers themselves just vary randomly (oh my Gosh, there is that concept ‘randomness’ again) without being subject to a statistical analysis like this. I propose that the way to get out of this circle is to treat ‘equally likely’ as a concept that has vague boundaries but is nonetheless not empty.

But maybe we are not entitled to be confident about this intuition. The Monty Hall paradox shows rather clearly that our intuition in these matters cannot always be accepted at face value. Let me briefly describe the Monty Hall paradox.

The name of the paradox comes from a television game show hosted by a certain Monty Hall. The show employed doors hiding cars and goats, but I prefer to stick with shells hiding either a peanut or empty air. The game proceeds as it does with the non-Monty-Hall shell game, but with this difference. After Smith has selected a shell, he does not turn it over to see if it hides the peanut. Instead, Elizarraraz turns over one of the peanuts. The peanut he turns over has to meet two criteria: first, it cannot be hiding a peanut; and second, it cannot be the shell (initially) selected by Smith. Elizarraraz then gives Smith the choice of either sticking with his initial selection, or switching to the remaining shell (that has not yet been turned over).

One can be forgiven for having the strong intuition that neither strategy has any advantage over the other. As one pictures the two remaining shells with the mind’s eye, may seem completely obvious that Smith’s chances of winning the peanut are 50/50 if he sticks with his initial selection, and 50/50 if he switches. The sample space, after all, would seem to comprise just two possibilities, just as does the sample space of the non-Monty Hall game. Possibility #1: the one shell either hides the peanut, in which case the other shell hides just empty air; or (possibility #2) the former shell hides empty air, and the latter shell hides the peanut. This is what could turn up, what could be very shortly in the near future.

But, as it will turn out, this is not the sample space of the Monty Hall shell game. And Smith’s chances of winning the peanut are not 50/50 regardless of his strategy, but 1 in 3 if he opts to stick with his initial selection, and 2 in 3 if he opts to switch. As if that were not (at least initially) counter-intuitive enough, it remains true that Smith’s chances of winning the peanut are 50/50 if he chooses by flipping a coin which of the remaining two shells to select; and his chances of choosing his initial selection |alternatively| choosing the shell that was not his initial selection are also 50/50. How can all of these propositions be true at the same time? How can the ‘2 in 3′ be true at the same time the ’50/50’ is true? And what can we learn about the nature of probability from the co-truth of these propositions?

Taking my cue, first from Judea Pearl, then from Luis Jorge Borges, I will prove the ‘1 in 3’ vs. ‘2 in 3’ probabilities for sticking with the initial choice vs switching. Then, after proving the 50/50 cases, I will show how these are compatible with the 1 in 3 and the 2 in 3.

Computer simulations of Monty-Hall-type games (for example, the one available online here or here) show definitively that Smith’s chances of winning the peanut are 1 in 3 if he sticks with his initial choice and 2 in 3 if he switches. One of the simulations I linked to repeats the game ten million times. Few, I think, would dispute that these simulations show that the chances are 1 in 3 | 2 in 3. But they won’t suffice to give one any intuitive sense why those are the chances. No Aha Erlebnis will be coming from just observing the simulations.

A table listing all of the possibilities, all the possible cases, goes some way, I think, towards giving one this intuitive sense. As shown in the table below (a modification of the table presented by Judea Pearl in his BOOK OF WHY (BOOK OF WHY, p. 191), which in turn is taken from Marilyn vos Savant’s column from the 90’s), there are nine distinct possibilities, nine possible cases. Each of the nine cases is equally likely. One can then start to see why the computer simulations would give Smith a 1/3 chance of selecting the shell with the peanut if he sticks with his initial choice, and a 2/3 chance if he chooses the remaining shell.

Shell #1Shell #2Shell #3If SameIf DifferentWhich Means That
peanut, initial selectionempty, not initial selectionempty, not initial selectionSmith winsSmith loseseither shell #2 was turned over, leaving shell #3 to be select should Smith opt to change his selection; or shell #3 was turned over, leaving shell #2 to be selected should Smith opt to change … in either case, Smith loses if he opts to change his selection
empty, initial selectionpeanut, not initial selectionempty, not initial selectionSmith losesSmith winsshell #3 is the only shell eligible to be turned over, which means that Smith will choose shell #2, and win, if he opts to change his selection
empty, initial selectionempty, initial selectionpeanut, initial selectionSmith losesSmith winsshell # 2 is the only shell eligible to be turned over, which means that Smith will choose shell #3, and win, if he opts to change his selection
peanut, not initial selectionempty, initial selectionempty, not initial selectionSmith losesSmith winsshell # 3 is the only shell eligible to be turned over, which means that Smith will choose shell #1, and win, should he opt to change his selection
empty, not initial selectionpeanut, initial selectionempty, not initial selectionSmith winsSmith loseseither shell #1 was turned over, leaving shell #3 to be selected should Smith opt to change his selection; or shell #3 was turned over, leaving shell #1 to be selected should Smith opt to change. In either case, Smith loses if he opts to change his selection
empty, not initial selectionempty, initial selectionpeanut, not initial selection Smith losesSmith winsshell #1 is the only shell eligible to be turned over, which means that Smith will choose shell #3, and win, if he opts to change his selection
peanut, not initial selectionempty, not initial selectionempty, initial selectionSmith losesSmith winsshell #2 is the only shell eligible to be turned over, which means that Smith will choose shell #1, and win, if he opts to change his selection
empty, not initial selectionpeanut, not initial selectionempty, initial selectionSmith losesSmith winsshell #1 is the only shell eligible to be turned over, which means that Smith will choose shell #3, and win, if he opts to change his selection
empty, initial selectionempty, initial selectionpeanut, initial selectionSmith winsSmith loseseither shell #1 was turned over, leaving shell #2 to be select should Smith opt to change his selection; or shell #2 was turned over, leaving shell #1 to be selected should Smith opt to change. In either case, Smith loses if he opts to change his selection

The table, however, is not perfect as a device for generating the desired Aha Erlebnis giving one to see that Smith’s chances are only 1 in 3 if he sticks with his initial choice. One may want to see rows 1, 4, and 7 in the table as each comprising two possibilities, not one, rendering problematic the math that gives us the 1/3 and 2/3 probabilities. One would be wrong, of course; nonetheless, it remains true that the table is burdened as an Aha-Erlebnis-generating tool by this complication. Also, the table does not show why the 50/50 chances (initially and perhaps even non-initially) seem so powerfully intuitive.

Listing out all the possibilities in the form of a tree, gives us a picture, another way of showing the 1/3 and 2/3 probabilities without the burden of this complication. We can picture repeated plays of the Monty Hall shell game as a trunk branching off into a number of branches. Doing so will nail down the 1/3 and 2/3 probabilities quite conclusively, though perhaps without generating an Aha Erlebnis, a concrete intuition.

Picturing the game this way will also provide at least a start at an explanation why the conclusion that the chances are not 50/50 seems so paradoxical. The idea of treating the game this way came to me in a flash of insight after reading Jorge Luis Borges’ short story THE GARDEN OF FORKING PATHS. (“You are so smart!” at work, though sometimes I suspect they mean this in a ‘you have a wonderfully intuitive sense for the blindingly obvious’ way), but, of course, essentially the same idea has occurred to other people, as one can see here and at numerous other places on the internet. I would like to think, however, that I have my own twist on the idea. Anyway, onto the chart shown below and an explanation of what it shows.

The Monty Hall Shell Game Considered As Conceptual Sleight Of Hand: In the chart shown below, Elizarraraz (employing a randomizing device) chooses which shell to place the peanut under (tanned orange). In order to make the chart readable, I show just Elizarraraz’ choice of shell #1. The possible choices that ensue from the “space” that would open up if Elizarraraz placed the peanut under this shell are, I claim, canonical. That is to say, they comprise a piece (shell #1) of the larger picture that enable one to draw conclusions about the larger picture (all three shells).

A moment later, Smith comes into the scene and, employing a randomizing device, makes his initial selection of a shell (pink). Elizarraraz then turns over one of the shells, employing, not a randomizer, but his knowledge of which shell Smith has selected and which shells are empty (baby-aspirin orange). Those shells Elizarraraz cannot turn over are crossed out by red lines.

Finally, using a randomizer, Smith decides either to switch shells or stick to his initial choice. The decision to switch is shown (for reasons that will become clear when I get to the ‘forking paths’ metaphor) by the bolded arrow. The winning shell (Smith gets the peanut) is shown by the darker viridian or “sea-glass” green color of the oval symbol picturing the shell. The losing shell is shown by the lighter viridian green, which looks like a light blue.

[Each oval represents a possible outcome (for example, Smith initially selects shell #1). Until we get to the culminating possibilities (represented by the green ovals), each possible outcome opens up (and sometimes closes down) what I will call a ‘possibility trail’, i.e., a “trail” in which one possible outcome follows another. Smith’s initial choice of shell #1, for example, opens up a path in which Elizarraraz turns over shell #2, which in turn forks into two paths, one leading to Smith’s winning the peanut and the other leading to his losing the game; and opens up another path in which Elizarraraz turns over shell #3, which path in turn forks into…; and results in a dead end, in which Elizarraraz is constrained by the rules of the game from turning over shell #1. ]

[Each fork opens up what I shall call a “cone” of possibility paths. Elizarraraz placing the peanut under one of the shells opens up three such cones, not labelled here. Smith’s choosing a shell opens up three cones, which I label A, B, and C. The paths in cone A culminate in four different possible outcomes; the paths in cone B and cone C each culminate in two possible outcomes. ]

[Cones A, B, and C match with rows 1, 2, and 3 respectively in the table shown previously. Each cone/row constitutes a wider sample space whose “places” or “slots” are themselves narrower “sample spaces” whose “places” are still narrower samples spaces defined by the forks and, ultimately, by the possible ending outcomes. These narrower sample spaces would (note the subjunctive mood) succeed one another in time; one such sample space, one set of possibilities would open up for example were Smith to initially select shell #1. There are two final sample spaces in cone A. These sample spaces begin, respectively, at Elizarraraz’ possibly turning over shell #2, or his possibly turning over shell #3, and include their ending “leaf” possibilities: shells #1 or #3; or shells #1 or #2 respectively. Both of these final sample spaces are included as places in the sample space comprising cone A. The sample space that is cone A is defined by the fork that gets generated by Smith’s possibly making the initial selection of shell #1. Cone A in turn, along with cones B and C, are included in the sample space that is generated by Elizarraraz’ possibly placing the peanut under shell 1.]

If Elizarraraz has placed the peanut under shell #1, then of course Smith has only a 1 in three chance of winning if he sticks by his initial choice. For in this case he will win the peanut only if that initial choice was shell #1. But the chances shell #1 was his initial selection are just 1 in 3. So his chances of winning by sticking with his initial choice are also just 1 in 3. It follows that his chances should he switch will be 2 in 3. If this conclusion is not already already intuitive to you, gentle reader, I think it will become more intuitive once I start laying out the forest of forking paths picture.

Suppose that Smith, compulsive gambler that he is, plays the Monty Hall Shell Game ten million times. At the end of each game, he is presented with just two shells. One was initially selected by him, the other not. Now suppose that the shell that was initially chosen is marked as such; ditto the shell that was not initially chosen. If Smith sticks to a strategy of of chosen the shell he did not initially select, he will win 2/3 of the time and lose 1/3 of the time. Conversely, if he sticks to a strategy of sticking to his initial choice, he will lose 2/3 of the time and win 1/3 of the time.

Now suppose the markings ‘initial choice’ and ‘not initial choice’ are removed from the shells — and, because the shells looks so similar, Smith cannot remember which one he had initially selected. No labels ‘shell #1’, ‘shell #2’, ‘shell #3’ have been applied to help guide him. Smith has to flip a coin to decide on which shell to select. I think it is clear from the chart that Smith will win the peanut 1/2 the time by flipping a coin. This 50/50 probability is, I think, what makes the Monty Hall Shell Game so drastically counter-intuitive. One looks at the two shells, each of which could be hiding the peanut, and (correctly) sees a 50/50 chance should they flip a coin.

But notice that in the game, Smith is not asked to flip a coin to decide between the two remaining shells. Instead, he is asked either to stick with his initial choice or to switch. That is the Monty Hall Shell Game, which presents Smith with a 2/3 (alternatively, 1/3) chance of winning. He is not asked to flip a coin to decide between the two remaining shells. That is a different game altogether, one that results in a 50/50 chance of winning. Let me call this other shell game the ‘Monty Hall With-A-Final-Coin-Toss-Added-In-At-The-End-For-Good-Measure Shell Game.’

We base the figure 2/3 | alternatively 1/3 on what WOULD happen were the Monty Hall Shell Game played 10 million times, adopting one or the other of the two available strategies. This provides confirmation. But it does more than that, because it provides a way to define randomness that does not rely on the concept of ‘equal likelihood’ or some other ‘probability function’. It gives us a way to define it in a non-circular fashion. So: ignorance/knowledge in the context of what WOULD happen plus idealization.

If Smith is to be able to play the Monty Hall Shell Game, he needs to know which of the two shells remaining in the penultimate step was his initial selection and which shell was not — the actually or possibly switched-to shell. Smith needs to have this information in order to play the game. The rules require keeping track of what happened in the past — there has to be a trail, a path, so to speak, leading from the past to the present. If Smith loses this trail — say, all shells have the tendency to look alike to him, and no one — Elizarraraz or anyone else — bothers to inform him which is which — then Smith has no available evidence to base his choice on except for flipping a coin. The ‘Monty Hall With-A-Final-Coin-Toss-Added-In-At-The-End-For-Good-Measure Shell Game’ is the only one he can play. Not exactly the same as the original game described above, the Non-Monty-Hall shell game, but now has the same 50/50 chance of winning the peanut.

Information has to leak, so to speak, from the past to the present and be available to Smith in the present. It has to exist, has to be available, and has to be picked up and used by Smith. This means that there is a dependency between events (‘event’ here used synonymously with “experiment”) that happened in the past (which shell Elizarraraz turned over) and the probability of possible events (‘possible event’ here is used synonymously with the standard probability term ‘event’) in the present. The 1/3 alternatively 2/3 probabilities inherent Smith’s sticking with his initial choice alternatively switching shells in the final step of the Monty Hall Shell Game depend upon Elizarraraz’ having turned over a shell in the past. This is completely unlike the standard coin flipping scenario, in which any later coin flipping event is independent of any earlier one because no earlier event affects the probability of any later event.

Even if everyone playing either game has gotten completely confused by the similarity of the shells, the information is, I will assume, still present. It is just much less available — much harder for anyone to pick up — much harder to the point of practical impossibility. And even if the information regarding which shell Elizarraraz turned over is still present in Smith’s mind, Smith is not likely to be playing the game armed with the chart below in his mind. It is a rare person who would be able to do so. Thus Smith is likely to think of the two remaining shells as a situation calling for a coin flip yielding a 50/50 chance. [Point of these paragraphs: one more case of the perspectival character of probability. The probability depends upon the information present — or at least available.]

Dependency trails. The present — what could be now — vs. what could have been, which includes all these possible dependency trails. The sample space as including the trails. “Room” made by the constant shell-game like shifting. What could have been as in a way “present” now ala Borges. Evidence. Turning over the shell changes the probability just as it does in the original shell game. Perspectival character — what the probability is depends upon whether one takes the present ‘could be’ perspective or the past ‘could have been’ perspective. Paradoxical because one tends to take the could be perspective.

Which is which will differ frequently as Smith makes his ten million plays of the game. In the case in which Elizarraraz has placed the peanut under shell #1, the initial choice will sometimes be shell #1 and the switched-to shells either shell #2 or shell #3; sometimes the initial choice will be shell #2 and the switched-to shell will be shell #1; sometimes the initial choice will be shell #3 and, again, the switched-to choice will be shell #1. If we imagine labels getting applied each time to the initial-choice shell and the switched-to shell, those labels will be constantly moving back and forth between the three shells. They will be “orthogonal” to the labels ‘shell #1’, ‘shell #2’, and ‘shell #3’, should those labels also be applied to the shells.

So which game is being played — and what the rules are for each — matters for what the probabilities are. ffff

[Since in both these games the designations ‘shell #1’, ‘shell #2’, ‘shell #3’ drop out of the picture, one may get the feeling that these are similar to the shell game as traditionally played, in which a slick operator switches the peanut between hard-to-distinguish shells by slight of hand. But here, of course, one is not trying to force their eyeballs on three actual shells in an attempt to keep from getting fooled within a single playing of the game. Shell stays the same; peanut surreptitiously moves. Instead, one is dealing with labels which stay the same even as the shells they apply to change with each new playing of the game. [How come 2/3 probability when only 2 shells remaining?]]

Under one description for the shells, the chances of winning the peanut are 50/50. Under another description (shell not initially chosen; shell initially chosen), the chances are, respectively, 2 in 3 and 1 in 3. But these are (at least at any given time) the same shells. What accounts for the difference? The difference, I think, lies in the history of how the shells got there. And in explaining this, Borges short story THE GARDEN OF FORKING PATHS will prove useful.

Enough of the shell games. Let me now apply a completely different picture, one inspired by Borge’s short story THE GARDEN OF FORKING PATHS. This picture will be of a forest containing within it a multitude of forking paths. It will, I propose, make it easier to articulate certain aspects of the paradox I am trying to make sense of.

Monty Hall Game Considered As A Tree/Forest Of Forking Paths

The chart above was originally drawn as a graphic tree depicting the Monty Hall Shell Game. But now lets draw it as depicting thirteen forking paths in a forest. Smith will be walking the paths fifteen million times (he is an indefatigable hiker).

Here the sample space comprises paths cut into the forest. Just as Smith’s overturning one of the shells in the Non-Monty-Hall Shell game reduced the sample space from 3 to 2 (should the shell prove empty), the ten paths that lead to dead-ends (the clearings marked with a red X) reduce the sample space from 18 paths to just 8. Information in the shell game corresponds to dead ends in the forking paths. Certainty one will not go any further in the forest case. In the shell game the shells still in play are met by Smith’s ignorance; here where the remaining forks lead to is what meets Smith’s ignorance.

These paths are in a parallel universe which mirrors our universe, in which Smith is playing the Monty Hall Shell Game. The ovals in the chart above, which used to represent choices (Smith’s or Elizarraraz’), now represent clearings in the forest. The arrows, which used to represent ‘go on to the next step’ now represent paths leading from one clearing to the other. Which clearing Smith ends up in, and which path he takes, is determined by the choices he and Elizarraraz take in the shell game in our universe. So the forking paths picture will be a bit science-fiction-y; nonetheless, my hope is that it will result in a gain in intuitive clarity (certain points will be easier to make) which will make up for its contrived character. Think of it as like the Mercator projection which serves as the standard in maps of the world. In this projection, certain features are captured at the expense of distortions in the areas of the land and water masses mapped.

Each oval represents a clearing in the forest. Each arrow represents a path leading from one clearing to the next. There are three different starting clearings which map to Elizarraraz hiding the peanut under shell #1 alternatively shell #2 alternatively shell #3; above, only the clearing corresponding to his hiding the peanut under shell #1 is shown, since I take this to be canonical. Three paths fork of, or, more precisely, trident off from the starting clearing. If Smith takes the path to the left, These of course map onto Smith’s initially selecting one of the three shells. If Smith takes the path on the left, hink of the arrows in the chart above as depicting Let me first describe the forking-path interpretation in just enough detail to let me state the two points I want to make. Then I will lay out the interpretation in more adequate detail. We will be having Smith walk the paths…maybe ten million times would be cruel and unusual punishment, but enough times that a frequency becomes apparent. The paths end in a forest clearing which contains something stupendous which I will leave to the reader’s imagination. Maybe it is a glorious vision of a topless Channing Tatum clearing brush. Maybe it is seeing Edward in full shining resplendent crystalline display. Maybe it is seeing a gorgeously feral Jacob — another graceful son of Pan! Or maybe it is just an extra-special peanut that outshines any other peanut. Whatever.

When Smith, walking down the path for the x number of times, comes to the final fork in the path, he can do one of two things. First, he can select the path by flipping a coin. Or, second, he can adhere to a right-hand/left-hand strategy: always choose the path on the right (alternatively the left).

I think it is plan from the graph that if he chooses by flipping a coin, he will arrive at the clearing with the special prize (a view of Channing Tatum, or the extra-special peanut) one half the time. If he adheres to the right-hand/left-hand strategy, he will arrive at the clearing with the special prize two thirds of the time if he always takes the path on the right, or just one-third of the time if he always takes the path on the left. Always taking the path on the right corresponds, in the Monty Hall Shell Game, to Smith’s switching, and always taking the path on the left corresponds to his sticking to his initial choice.

The different strategies lead to different probabilities. In a short while, I will relate these differing probabilities to those of the Non-Monty-Hall Shell Game played by Smith and Morgenstern. I intend to show that just as knowledge (or lack of knowledge) accounts for the difference in probabilities in the Smith and Morgenstern case, the related concept of evidence (or lack thereof) accounts for the difference in probabilities in the forking path case (and in the Monty Hall Shell Game).

But given the difference in the probabilities established by the different strategies, one can explain why the Monty Hall Shell Game seems so paradoxical to about everyone, at least at first. For when one imaginatively confronts the choice faced by Smith (stick to the initial choice of shells or switch), one surreptitiously thinks of the choice in terms of a ‘let’s flip a coin’ scenario. This scenario is, after all, easy to picture imaginatively. The alternative is to have the choice guided by something like the graph above. This graph is, naturally, not at all easy to picture.

Let me now turn to a fuller explanation of the above chart, interpreted either as a tree (the Monty Hall Shell Game) or as a set of forking paths.

I think I have fulfilled my promise to use the forking paths picture to nail down even more firmly the 1/3/2/3 stick with the initial choice/switch probabilities. Now let me show how this picture helps explain why this result seems, at least initially, so counter-intuitive.

Now after Smith has traveled down one or another of the paths in one or another of the three possibility cones, he is presented with two shells (in cone C, for example, either shell #1 or shell #3). The peanut could be under either of those shells. At the time of this writing (September 8, 2019 — I note the date because particular pieces of my autobiography have in the past turned out, somewhat surprisingly, to be philosophically fruitful), it seems absolutely clear to me from looking at the chart that Smith’s chances of winning the peanut are 50/50. Later I may try to nail this intuition down more firmly by coding my own simulation of the Monty Hall shell game.

But note that what I am ascribing a 50/50 chance to is the peanut’s being under (for example) shell #1 or shell #3. I am not ascribing a 50/50 chance to the peanut’s being under the Smith’s initial choice of shells or his switched choice. The descriptions ‘initial choice shell’ or ‘switched choice shell’ have no meaning in this narrow sample space delimited by what could be, i.e., by the present and the potentialities of the (presumably) near future.

To get these descriptions, we have to go deeper than what could be and move into what could have been. We have to move into the past. Smith could have chosen shell #2, but he has chosen shell #3, which in turn made shell #2 the only possible choice of shells for Elizarraraz to turn over, which in turn left Smith with a final choice of shells #1 and #3. Were Smith to go back in time multiple times to his initial choice of shells but with his randomizer determining different choices — or, less science-fictionally, were he to repeat the Monty Hall shell game a large enough number of times, he would end up winning the peanut 1/3 of the time by sticking to his initial choice, and 2/3 of the time by switching.

The probabilities are determined by the sample space. When the descriptions ‘initial selection shell’ and ‘switched choice shell’ make sense, the sample space embraces three possibilities, the three possibility cones, one of which culminates in Smith’s winning the peanut should he stick to his initial choice, and two of which culminate in his winning the peanut should he switch choices. That’s the sample space that counts when those descriptions are meaningful. When those descriptions don’t make sense because we are restricted to what could be, that is, to the present because the sample space is restricted to the present, to what is facing Smith now, and to a narrow snippet of the near future, the sample space comprises only two possibilities: the peanut is under this shell or under that other one.

Were Smith told, when confronted with the two shells, to choose one of two strategies: switch or stick with the initial choice, neither strategy would make any sense at all unless he had access to enough of the past to let him identify which shell was his initial choice; or unless someone who was keeping track told him. And even then his adopting one strategy or the other would be incompletely rational unless he had plotted out all the cones with the possible paths that could have been, including both the paths that led to the present situation and the paths that ended up as dead ends. He would be better off not worrying about which shell was his initial choice and just flipping a coin.

What the sample space is, and therefore what the probabilities are, depends upon which game is being played — flip a coin, or stick-with-the-initial-choice-or-switch. Different sample space, different game; different game, different sample space. Although Pearl’s point in the following may be a bit different from what I have just described, his actual words still fit with my point. (Maybe there is another Borges story about something similar.) Pearl notes:

The key element in resolving this paradox is that we need to take into account not only the data … but also … the rules of the game. They tell us something about the data that could have been but has not been observed.

BOOK OF WHY, p. 192

When confronted with just the two remaining shells in the present, it is easy to forget that these are two different games.

Thinking about the the different cones containing different possible paths requires a certain amount of time, patience, and wetware power and bandwidth. Considering the possibilities when confronted (perceptually or imaginatively) with just two shells requires much less time, patience, and wetware power and bandwidth. This fact, plus the fact that it is perhaps not so obvious when staring at the shells that the descriptions ‘initial choice’ and ‘switching choice’ cannot be applied to the shells if one’s time horizon (and the resulting sample space) are too narrow are, I submit, at least one reason the actual probabilities of the Monty Hall shell game seem at first so drastically counter-intuitive.

As Pearl notes, there are probably 10,000 different reasons, one for each reader, why the actual probabilities of Monty Hall game seems so counter-intuitive. To return for a moment back to cars, goats, and doors:

Even today, many people seeing the puzzle for the first time find the result hard to believe. Why? What intuitive nerve is jangled? There are probably 10,000 different reasons, one for each reader, but I think the most compelling argument is this: vos Savant’s solution seems to force us to believe in mental telepathy. If I should switch no matter what door I originally chose, then it means that the producers somehow read my mind. How else could they position the car so that it is more likely to be behind the door I did not choose?

BOOK OF WHY, pp. 191-192.

The specter of mental telepathy is doubtlessly one reason the result seems so counter-intuitive; one’s tendency, resulting from the limitations on human mental power, to be perceptually/imaginatively restricted to what could be as opposed to what could have been is another. I won’t try to judge here whether one is more compelling than the other, especially since I have not yet wrapped my head around Pearl’s account of causality.

Now back (finally!) to the point of bringing up the Monty Hall puzzle in the first place. Regarding the non-Monty-Hall shell game, I asked what makes us so sure the probability is now 1/2 that the peanut is under one of the remaining shells after Smith has turned over one of the shells which turned out to be empty. Why should we trust our intuition in this case, when our intuition regarding the Monty-Hall case were initially so far off? Well, let’s provide a table of the possibilities.

Shell #1Shell #2Shell #3Shell Uncovered by SmithFormer Possibility Converted to Actuality
peanut empty empty 1 yes
empty peanut empty 1 no
empty empty peanut 1 no
peanut empty empty 2 no
empty peanut empty 2 yes
empty empty peanut 2 no
peanut empty empty 3 no
empty peanut empty 3 no
empty empty peanut 3 yes

There are two independent events a work here: Elizarraraz randomly placing the peanut under one of the three shells, and Smith’s randomly turning over one of the shells. Neither event affects the probability of the other. If we then eliminate the rows in which Smith happened to turn over the shell containing the peanut (as marked by ‘yes’ in the column ‘Possibility (that the shell hides the peanut) turned into actuality (yes, the shell did hide the peanut), we get 6 rows. Each of the three pairs of rows describes a probability: if Smith finds that shell #1 was hiding nothing except empty air, then row 2 (the peanut is under shell #2) and row 3 (the peanut is under shell #3) describe the situation. Since both rows describe equally likely possibilities, the chances are 50/50 that shell #2 hides the peanut, and the chances are 50/50 that shell #3 hides the peanut.

Our initial intuition is therefore vindicated. Smith’s turning over one shell and finding it empty changes the probability the peanut is lurking in any one of the remaining shells from 1 in 3 to 1 in 2. (It sure is nice to have a wonderfully intuitive sense for the obvious.) The probabilities changed because the sample space changed, just as changing the Monty-Hall game from ‘switch or stick with the initial choice’ to ‘flip a coin’ changed the probability of winning the peanut from 2/3 (if Smith switches) to 50/50. The probabilities in the Monty Hall case changed because the sample space relevant to the game Smith was playing changed. Having the ability to describe one of the remaining shells as ‘the initial choice’ expanded the sample space needed to support this description from two possibilities regarding each shell’s hiding or not hiding a peanut to three possibility cones each containing one or more possible paths to the current situation.

Now Morgenstern (German for ‘morning star) enters the scene, after Smith has put back the shell he turned over.  (Say, this is shell #1) She does not know that shell #1 turned up empty. The peanut is still under one of the remaining shells. Elizarraraz points to shell #2 and asks both Smith and Morgenstern what are the chances the peanut is under that shell. For Smith, surely, the answer is 1 in 2. For Morgenstern, the answer has to be 1 in 3. For Elizarraraz, who knows where he put the peanut, the answer has to be either 0 or 1. Were Elizarraraz to point to shell #1, the answer for both him and Smith would have to be 0. What the probabilities are differs from the perspectives of each of the three because the sample space differs for each given what each knows.

From Elizarraraz’s perspective, there is no hiddenness, no ignorance given how things stand with regard to the peanut under shell situation, because his knowledge is complete regarding that situation. Obtaining within that perspective is certainty: either a probability of 1 or of 0. I will go out on a limb and say that within that perspective there is no sample space at all.

Uncertainty, a probability greater than 0 but less than 1, can exist only given a particular ratio of local ignorance and local knowledge. If one’s local knowledge of the peanut under shell affair is 0 (one does not even know if there is a peanut under one of the shells) and even Elizarraraz has forgotten if he has placed a peanut under one of them or not, one can appeal to a (possibly hypothetical) infinite (or at least extremely large) Mind that does know, in which case the probability is either 0 or 1. Or one can appeal to a brute, currently unknown fact of the matter, in which case, again, the probability that the peanut is under any given shell is either 0 or 1.

But if there is to be a probability greater than 0 or less than 1 within anyone’s perspective — including the Infinite (surely impossible for that one) or at least Extremely Large Mind’s — there has to be some ignorance, some hiddenness as well as some knowledge. For an omniscient God, everything has either a probability of 1 or 0. Ignorance/knowledge is a necessary condition for such probability in between 0 and 1.

It is also a sufficient condition for there being, within a particular perspective, for there being such a probability. All that Morgenstern needs to know is that there is a peanut under one of the shells, and all she needs to be ignorant of is which one, for there to be, within her perspective, of a probability of 1 in 3 that the peanut is under this shell, or that one, or the one remaining one. The probability is 1 in 3 within this perspective because Morgenstern’s ignorance/knowledge determines the sample space.

Knowledge/ignorance suffices for the existence of a probability between 0 and 1. But other factors help determine what exactly that probability is. In the non-Marty-Hall shell game, we need only to take into account the increase in Smith’s knowledge in determining the size of the sample space when he turns over one of the shells and discovers it to be empty. The probability the peanut is under one of the shells increases from 1 in 3 to 1 in 2 because the two events — the placement of the peanut under one of the shells and Smith’s turning over one of the shells — are both random and independent.

But in the Marty Hall shell game, Elizarraraz’s turning over one of the shells doubles the probability that switching will win the prize from 1 in 3 to 2 in 3. It therefore constitutes evidence that the peanut is likely to be under the shell that wasn’t Smith’s initial choice, whether Smith is in a position to utilize this evidence for not. Since, prior to the final step in the Monty-Hall shell game, the only difference between it and the non-Marty-Hall shell game is that in the former Elizarraraz’s turning over one of the shells is, because of his knowledge, not random and is independent of neither his placement of the peanut under one of the shells nor of Smith’s initial selection of one of those shells, it follows that this lack of independence is another factor in addition to Smith’s knowledge/ignorance helping to determine the specific probability of Smith’s finding a peanut if he switches (sticks with the initial choice). By itself, all his knowledge/ignorance does by itself is guarantee a probability of at least 1 in 2 should he switch (stick with the original choice) ; given the additional factor of a lack of independence in the event of choosing which shell to turn over, that probability increases to 2 in 3 (decreases to 1 in 3) should he switch (stick with his initial choice).

At the time of this writing, however, I am unable to say anything more succinct and more sophisticated regarding why this should be so other than ‘look at the chart shown above; given the all the ovals crossed out because Elizarraraz’s choice of shells to turn over was neither random nor independent of the other events, this is how all the possibilities panned out — all three of the possibility cones, and all of the possible trails within those cones. Stay tuned.

[Present circumstances. A sample space is a set of possible outcomes of a given activity governed by a set of definite rules, or at least limited by certain definite conditions. These rules or conditions determine what is eligible to count as a possible outcome. Flipping a coin has two possible outcomes, heads or tails. The coin’s landing on its edge is not a possible outcome, at least not if the normal rules that apply to the practice of flipping a coin are in force. It might be a possible outcome in a different game. In the normal practice of flipping a coin, the sample space is the set with two members: coin lands heads or it lands tails. In a non-normal practice, the sample space might have three members: The same holds mutatis mutandis for throwing a die. In the normal practice, the sample space comprises six members. But should anyone be skilled enough to make the die land on one of its edges invent a new, non-normal practice, the sample space would comprise 12 members. Drawing a standard recognized card from a pack of cards normally counts as an eligible outcome; drawing a stray scrap of paper normally does not. Uncovering the peanut in the normal shell game counts as a possible outcome; uncovering a grain of sand does not — not even if this were a truly extraordinary grain of sand. ]

Different games (say, not realizing the point of the normal shell game) different sample space because what is eligible to count as a possible outcome differs. Assume for example — except for a few Twilight Zone moments — that the peanut stays under whichever shell it is under and is not going to behave like the electron which, for all one knows, might be on the nose of the Mona Lisa. ffff

[Present circumstances. A sample space is a set of possible outcomes of a given activity governed by a set of definite rules, or at least limited by certain definite conditions. These rules or conditions determine what is eligible to count as a possible outcome. Flipping a coin has two possible outcomes, heads or tails. The coin’s landing on its edge is not a possible outcome, at least not if the normal rules that apply to the practice of flipping a coin are in force. It might be a possible outcome in a different game. In the normal practice of flipping a coin, the sample space is the set with two members: coin lands heads or it lands tails. In a non-normal practice, the sample space might have three members: The same holds mutatis mutandis for throwing a die. In the normal practice, the sample space comprises six members. But should anyone be skilled enough to make the die land on one of its edges invent a new, non-normal practice, the sample space would comprise 12 members. Drawing a standard recognized card from a pack of cards normally counts as an eligible outcome; drawing a stray scrap of paper normally does not. Uncovering the peanut in the normal shell game counts as a possible outcome; uncovering a grain of sand does not — not even if this were a truly extraordinary grain of sand. ]

Different games (say, not realizing the point of the normal shell game) different sample space because what is eligible to count as a possible outcome differs. Assume for example — except for a few Twilight Zone moments — that the peanut stays under whichever shell it is under and is not going to behave like the electron which, for all one knows, might be on the nose of the Mona Lisa. ffff

Today’s homage to Plato’s SYMPOSIUM is Channing Tatum. Again. Who would want anything more?


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.  🙂

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 Ω — as follows:

The Sample Space Ω =

Ω = { 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 ∩ 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 Ω 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  F ) / P(E)

P( E  F ) = |E  F| / |Ω| = 8/16 = 1/2

P(E) = |E| / |Ω| = 8/16 = 1/2

So:

P( E  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  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  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.


Doorbells, Rubies, Shell Games, And Implication: An Example That Makes Treating Implication As An Information-That Relation Attractive

The Problem:  What Does Relevance Consist In?  Following Relevant Logic, we can avoid Classical Logic’s paradoxes (or at least weirdnesses) of Material Implication, according to which the following statements are true…

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

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

…by insisting that the antecedent p be relevant to the consequent q.

Two questions immediately becomes pressing:  first:  what does ‘relevance’ mean?  Second, what is it that makes p relevant to q?

First Question:  What Does ‘Relevance’ Mean?  As I intend to use the term, ‘relevance’ in general is a relation/connection that exists between one situation/state of affairs and another and is important to our concerns.  In the case of relevant implication, the aforementioned relation is important to us because it underwrites a guarantee that we can infer q from p. That we can legitimately make inferences is one of our concerns.

Implication is a relation between propositions.  One infers one proposition from another. Following Roderick Chisholm, I will be identifying propositions with states of affairs.  For example, the proposition that this cat, Felix, is sitting on this Persian mat with MAT_ID 1123581321 is identical with the state of affairs consisting in Felix sitting on the Persian mat with MAT_ID 1123581321.  So I will alternate between referring to p and q as propositions and as states of affairs.

By ‘situation’ I mean, roughly, ‘a site comprising one or more connected states of affairs which are available from a possible perspective.  A perspective is always limited and therefore does not have available to it other states of affairs.   The room in which I am typing this constitutes one situation.  In this situation the doorbell’s ringing, when it occurs, is available to me.  The button which, when pushed, causes the doorbell to ring is on the wall outside.  This state of affairs is hidden from me in my current situation.  The immediate vicinity of a person who is about to press the doorbell button outside is another situation.  The states of affairs comprised by the room inside are not available to this person.

Second Question:  What Is It That Makes p Relevant To q?  One at least initially attractive answer to the second question is the following:  p is relevant to q when p is information that q.

Here is one issue that I want to bring out into the open from the very start.  The careful reader will notice, as they go along, that I am vulnerable to the charge of circularity.  I will be analyzing implication in terms of  information and information in terms of situations, which in turn I analyze in terms of perspectives.  But it would seem that perspectives need to be analyzed in terms of information.  You, my gentle reader, my fearsomely implacable  judge, will decide later whether I am successful in defending myself against the circularity charge.

In what follows, I will first state what makes treating relevance that way attractive.  After dealing with a counter-example that, at first sight, seems completely devastating, I will argue that the INFORMATION THAT relation remains the basis for understanding relevance as it pertains to implication — at least for the examples that I present or link to in this post.

To state the matter a bit abstractly at first, p is information that q when a channel exists through which information flows from a source site, an at least partially-obscured situation s0 (which includes the state of affairs that q), to a reception site, a situation s2 (which includes the state of affairs that p), making the information that q available in s2. Such a channel exists when some state of affairs that c in a situation s1 renders the conditional probability that q given p 11.

The channel may open up between s0 and s1 because s1 is a physical situation comprising states of affairs whose obtaining during a certain stretch of time makes it impossible without violating physical laws for that p to obtain without that q‘s obtaining.

To bring up an example into which I am about to go into much greater detail shortly, during the time that the wiring to a doorbell is in a certain physical condition, it would be impossible for the doorbell to ring without the button outside getting pushed by someone or something.  Suppose (as is surely the case) that the doorbell could ring without the button’s getting pushed only if a defective physical condition of the wiring, given the physical laws of the universe, could allow for events x, y, or z occurring (for example, an unwanted electrical pulse caused by a short the wiring).  Currently, the wiring is not in this defective condition and will not be so for a stretch of time.  (Nothing, for example, could cause a short, given the physical laws of the universe.)  Given this current condition of the wiring, the doorbell could ring without the button outside getting pushed only were the physical laws of the universe violated.

In the case of the doorbell, the channel is opened up by the physical condition of the wiring, a condition that functions as a constraint disallowing any doorbell ringing occurring without the proper cause — the button’s getting pushed.  This physical constraint underwrites, so to speak, a guarantee that the doorbell will never ring without the button outside getting pushed.

This is a physical, causal constraint.  There may be other constraints as well.  [Including knowledge, perhaps?]

Another factor that will turn out to be pertinent to p’s being information that q is one’s state of knowledge cum ignorance regarding q.  I will be asking later whether this factor poses a problem for regarding p‘s being information that q as the relation that makes p relevant to q by making the truth of an IF THEN statement relative to one’s knowledge.

 

Initially, the following doorbell example, taken from Fred Dretske’s KNOWLEDGE AND THE FLOW OF INFORMATION2 made this account of p‘s relevance to q highly attractive to me.  Warning:  what follows will be a veritable operatic doorbell aria.  Those who are not fans of operatic arias are advised to go elsewhere.

The Doorbell Aria:  You are in a room (s2 ) in which you are able to hear the doorbell.  The wiring of the doorbell comprises situation s1The state of affairs c regarding this wiring is such that in all possible worlds in which the laws of physics of this actual world hold, the doorbell will never ring without someone or something depressing the button outside.  (Situation s0  is the ‘outside’, including the button.)  This never happens, ever, no matter how much time goes by.

The doorbell’s ringing guarantees that someone or something is depressing the button.  There are no poltergeists inside the wiring, no sudden bursts of electrical energy ultimately caused by a butterfly flapping its wings in the Amazon, or anything like that, that will cause the doorbell to ring without the button outside getting depressed.  If one takes each occasion on which the doorbell rings, rolls back the clock, then lets the clock roll forward again, but this time with just one tiny change in the world they find themselves in (say, the butterfly flapping its wings in the Amazon has an orange dot on its wings rather than a maroon dot), and if they repeat this exercise for each possible world whose physics is the same as our actual world, someone or something will be depressing the button outside each time.  Rinse and repeat for each time the doorbell rings.  100% each time.

100% of the time, when the doorbell rings, the button outside is getting depressed by someone or something. Given the doorbell’s ringing, the conditional probability that the button outside is getting depressed is 1.

The wiring is burdened by a defect, however, that results in the doorbell’s occasionally failing to ring even when the button outside is getting depressed.  Let’s say that this failure to ring occurs in 0.001 percent of all the possible worlds in which the laws of physics are identical with those of this actual world.  Suppose that each time the button outside gets depressed the clock gets rolled back, then rolled forward again, but into a another possible world whose physics is the same as our actual world but has just one tiny change (for example, in the color of the spot on the wings of the butterfly in the Amazon).  In 0.001 percent of these possible worlds, the doorbell fails to ring.  Rinse and repeat for each time the button gets pushed.  0.001 percent each time.

0.001% of the time, the doorbell fails to ring when someone or something depresses the button outside.  The conditional probability that the doorbell will fail to ring even when the button outside is getting depressed is 0.001.   The button’s getting depressed does not guarantee that the doorbell will ring.

If we follow Dretske’s definition of informational content, we will see that the doorbell’s ringing is information that the button outside is getting depressed.  We will also see that the button’s getting depressed is not information that the doorbell is ringing inside. This (to anticipate) mirrors the situation in which 3) is true, and 4) is false.

3) IF the doorbell is ringing, THEN someone or something is depressing the button outside.

4) IF someone or something is depressing the button outside, THEN the doorbell is ringing.

Back to Dretske’s definition of informational content:

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

Let me linger a bit on “but given k alone, less than 1”.  k must be your knowledge cum ignorance of the source situation s0 outside.  At the moment, the doorbell is not ringing.  You have zero knowledge of how things stand out there with regard to the doorbell’s getting pushed.  The value of k is therefore zero.  With just this “knowledge” aka ignorance, and in the absence of a signal that the doorbell is getting pushed, the conditional probability that this is happening will be the probability that the doorbell is getting depressed at any given time of the day multiplied by 0.001.  This figure, whatever it is, will be considerably less than 1.

Now the doorbell is ringing.  All of a sudden, the conditional probability that the button outside is getting pushed has leapt to 1.  The doorbell’s ringing is therefore information that the button outside is getting pushed by someone or something.

Correlatively, when I am pushing the button, my knowledge of what is happening inside is zero, provided I am not able to hear the doorbell ringing in any case.  Given this knowledge alone, the probability that the doorbell is ringing is 0.999.  Given my knowledge plus the button’s getting pushed, that knowledge stays 0.999.  Therefore, according to Dretske’s definition of informational content, my pushing the button in this case is not information that the doorbell inside is ringing.

If the INFORMATION THAT relation is what makes for the relevance of p to q in true IF p THEN q statements, then 3) is true because this relation exists between p and q, and 4) is false because this relation does not exist.  Likewise, 1) is false because ‘Cliff lives in Houston’ is not information that the earth has just one moon, and 2) is false because even if Cliff moved to Orange County, California, that item would still not be information that Paris, Texas is the capital of France.  1), 2), and 4) are all false because in each statement the antecedent is not relevant to the consequent.

— “Wait a second!” I hear someone objecting.  “You mean that ‘someone or something is depressing the button outside’ is not relevant to ‘the doorbell is ringing?”  I do think that the notion of degrees of relevance — a relevance spectrum — needs to be introduced here.  The truth of ‘Cliff lives in Houston, Texas’ presumably adds exactly 0 to the probability that the earth has a single moon.  The truth of ‘I am pushing the button outside’ adds 0.999 to the probability that the doorbell is ringing inside.  The truth of the former statement lacks any relevance at all to its consequent.  The truth of the latter statement … well, it is not exactly completely irrelevant to its consequent.  But I do think this is a matter of ‘close, but no cigar’.  The truth of ‘I am pushing this button outside’ is not relevant enough to ‘the doorbell is ringing inside’ to make 4) a true statement.

Assume that an INFORMATION THAT relation exists between p and q in the following truth table except, of course, when the truth value of q makes it impossible for such a relation to exist.  (When this happens, of course, the IF THEN statement is also false.)  In that case, we would get a truth table for implication that is exactly like the one set forth by proponents of Classical Logic.  Except now the truth table makes intuitive sense — even the last row.  This is the row in Classical Logic’s truth table for implication that seems absolutely counter-intuitive to anyone sane.

Truth Table For Implication
p q IF p THEN q
T T T
T F F
F T T
F F T

 

Let’s consider the rows one by one:

  1.  ‘Doorbell is ringing’ is true, as is ‘the button outside is getting pushed’  IF p THEN q is obviously true in this case provided that p really is information that q.
  2. ‘Doorbell is ringing’ is true, while ‘the button outside is getting pushed’ is false. That q is false while p true guarantees that p is in fact not information that q, so IF p THEN q is guaranteed to be false.
  3. The doorbell is not ringing even though the button outside is getting pushed.  p remains information that q when that p is the case, so IF p THEN q is true.
  4. The doorbell is not ringing, and the button outside is not  being pushed.  Nonetheless, p would be information that q should that p obtain.  So IF p THEN q is true because the INFORMATION THAT relation still exists between p and q.

In short, provided this treatment of relevance is correct (which it is not quite — but I will get to that later), IF p THEN q is true if and only if p is information that q.  When (on this treatment of relevance) p is not information that q, then IF p THEN q is false no matter what the truth values of p and q are. This means of course that IF p THEN q cannot be treated in relevant logic as equivalent to NOT p OR q, as it is in Classical Logic.

This, then, is what makes treating relevance as consisting in INFORMATION THAT initially so attractive. First, the INFORMATION THAT relation at work in the doorbell example mirrors in a satisfyingly intuitive way the truth of 3) and the falsity of 4). Second, this treatment provides an intuitive explanation for the fourth row of the truth table for implication given above.  Proponents of Classical Logic are notorious for coming up with nothing more satisfying in this regard than ‘If you believe a false statement, you will believe anything’.

As side note, I would like to add that what I discussed in this post is the INFORMATION THAT relation as stemming from physical laws.  Here (but this needs to be re-worked) I discuss the INFORMATION THAT relation as stemming from what at first looks like logical principles but which, I think, may be more aptly described as the laws of probability.  I do want, after all, to base logic ultimately on something similar to INFORMATION THAT in a non-circular way.

[To sum up:  the relevance of p to q is a relation — a connection — between the state of affairs p and the state of affairs q which is important to us because it underwrites inference by guaranteeing q given p.]

Incidentally, the shell-game example discussed in the post just linked to clearly shows that the relevance-making relation cannot be the causal relation, at least not in all cases.  Turning over shell #3 to reveal a peanut is a signal carrying information that the peanut is under shell #4, but this action does not cause the peanut to be under shell #4.

However, there is a fly in the intuitive ointment. How is one to deal with statements like the following:

5) IF there is a ruby exactly 2 kilometers underneath my feet, THEN there is a ruby exactly 2 kilometers underneath my feet

or, more generally, with:

6) IF p THEN p

?

It would be a bit strange to suggest that a channel exists between the situation s0 (the way things stand exactly two kilometers underneath my feet) and the exact same situation s0.  It would seem, then, the relevant-making relation cannot be identical with the INFORMATION THAT relation after all.3  Although an identity relation clearly exists “between” s0 and s0, it would seem there is never an INFORMATION THAT relation between “them”.

However, while there are clearly cases in which no INFORMATION THAT relation exists between s0 and s0, adopting Roderick Chisholm’s notions of direct evidence and self-presenting states of affairs suggests that, in some other cases, we can treat that p as information that p.  I won’t be implying that Chisholm is correct in thinking that there is such things as direct evidence and self-presenting states of affairs.  If there is such a thing, however, it would suggest that sometimes INFORMATION THAT is not always a three-place relation(source, channel, receiver), but sometimes a one-place relation.

Let’s look at Chisholm’s (simpler) statement of what direct evidence consists in:

What justifies me in thinking I know that a is F is simply the fact that a is F. 

Roderick Chisholm, THEORY OF KNOWLEDGE, SECOND EDITION, Englewood Cliffs, New Jersey, Prentice-Hall, Inc., p. 21.  Henceforth TOK. 

For example, when I suffer a sharp pain in my shoulder to which I point and say ‘here’, what justifies me in thinking I know I am suffering a sharp pain here is simply the fact that I am suffering a sharp pain here.

Likewise, if someone asked me the (somewhat strange) question ‘how can you tell there you are suffering a sharp pain there?” I could only answer:

7)  I can tell I am suffering a sharp pain here because I am suffering a sharp pain here.

But information consists in what one can tell.  It follows, then, that:

8)  My suffering a sharp pain here is by itself information that I am suffering a sharp pain here.

A knock at the door (to use something other than the doorbell example for once) announces that someone or something outside is impacting the door.  Something not identical with this person or thing does the announcing.  The pain, by contrast, is self-announcing.  The information in this case doesn’t travel or flow from a source site to a reception site because the source and reception sites are identical.

If one insists that information has to travel from a source site to a reception site, so that self-announcing information cannot really be information, we still have something that is very much like information.  For to have information, or at least something that is like information, it suffices that one be able to tell something (that someone or something is depressing the button outside, that the peanut is under shell #4, that I feel pain here). One is able to tell something in all these cases, including the self-announcing case.

This gives another twist to:

9)  IF I suffer a sharp pain here, THEN I suffer a sharp pain here.

Here p is relevant to q because q (alternatively p) is a self-announcing state of affairs that is either a case of INFORMATION THAT, or is something very much like INFORMATION THAT.

….

Let me turn now back those cases in which s0 clearly is not information that s0. ….

I argue, however, that 5) (and, to generalize, 6) are true because, were a ruby to exist exactly two kilometers underneath my feet, the conditional probability that there is a ruby exactly two kilometers underneath my feet would be 1.

Compare with:  were the doorbell to ring (given c described above), the conditional probability that the button outside is getting pushed is 1.  The doorbell example describes a case of a signal carrying information that because two distinct situations are in play, a source situation that is at least partially concealed from those inhabiting a reception situation.  The (at least partial) concealment of a source situation from the perspective of a reception situation concomitant with the (at least partial) ignorance that is inherent in k is required for an INFORMATION THAT relation to exist.  Without this, any signal arising from s would be “old information”, that is to say, not information at all.

So I would like to revise Dretske’s definition of informational content to the following:

Informational content:  A signal r in reception situation s2 carries the information that t in source situation s0 is F = Because c is G in situation s1, the conditional probability of t‘s being F, given r (and k in s2), is 1 (but, given k alone, less than 1)

This guarantees the truth of IF p THEN q when p is information that q. When there is only a single situation, s0, knowledge (ignorance) k drops out of the picture because there is no longer any situation s2 from whose perspective one has (at least partial) ignorance of what is happening in s0. The signal r also drops out of the picture because we are no longer talking about INFORMATION THAT. What remains, however, is:

The conditional probability of t‘s being F in situation s0 is, given t‘s being F in situation s0, 1.

I think it requires only a moderately keen grasp of the obvious to grasp this point.

So what is common to both the doorbell and the ruby examples is a conditional probability of 1.  You get the ‘conditional probability is 1’ feature of the ruby IF p THEN p example by removing features from the INFORMATION THAT relation existing in the doorbell IF p THEN q (where p and q are about states of affairs in distinct situations).

I submit, then, that the two-place relation4 that makes p relevant to p in the statement IF p THEN p is a derivativedegenerate case of the INFORMATION THAT relation.  It is what you get by removing features from the IF THEN relation in order to accommodate the drastic simplification of a richer, complex situation s comprising s0, s1, s2 (and k in s2 ) into a more impoverished, simple situation s comprising just s0. This relation is degenerate enough to no longer count as INFORMATION THAT; all that remains of the INFORMATION THAT relation is the ‘conditional probability is 1’ feature;  nonetheless, INFORMATION THAT remains the touchstone for understanding all the cases of implication presented or linked to so far — the doorbell case, the shell-game case, and the ruby case.

Or so I am thinking at this moment.  We will see if this conclusion will survive consideration of further examples of implication.

 

 

 

 

1 I think this is identical with the theory of relevance developed by Jon Barwise and later by Greg Restall, as presented in Edwin D. Mares, RELEVANT LOGIC A Philosophical Interpretation, Cambridge and New York, Cambridge University Press 2004, pp. 54-55. Henceforth RELEVANT LOGIC.  

I mention situations because I have in mind the Routley-Meyer truth condition for implication, to wit:

AB‘ is true at a situation s if and only if for all situations x and y if Rsxy and ‘A‘ is true at x, then ‘B‘ is true at y. (RELEVANT LOGIC, p. 28.)

What I, at least, am calling a situation is what comprises one or more states of affairs available to one (or more, if the situation is shared) sentient creatures whose limitations prevent them from having direct access (in the absence of a signal) to other states of affairs.  The room inside which a person is able to hear the doorbell ringing is situation s2 — the reception situation.  The area immediately outside, where another person may be pressing the doorbell, is situation S0. — the source situation.   The wiring to the doorbell, which perhaps a gremlin or poltergeist is inhabiting, is situation s1 — the channel situation.

Of course, the fact I am bringing both situations and possible worlds into the discussion is probably a signal, that is to say, a dead-giveaway that I do not yet sufficiently understand the distinction between situations and possible worlds. Keep in mind that this post is an exercise in writing to learn.  So I want to issue a warning to non-experts in the field:  I probably know less about this stuff than may at first seem to be the case.  Needless to say, the actual experts, won’t be fooled.

2 Fred Dretske, KNOWLEDGE AND THE FLOW OF INFORMATION, Stanford, CSLI Publications, 1999, pp. 54-55.

3 cf RELEVANT LOGIC, p. 55.

4 — “Wait”, you say. “This is a two-place relation? Isn’t p identical with p?  So why isn’t this a one-place relation?” Yes, p is identical with p.  But the relation in question is a two-place relation because p is getting stated twice.

****************************

Today’s homage to Plato’s SYMPOSIUM is Channing Tatum, who is welcome to fix my pickup truck anytime. (In fact, I think I will buy a pickup truck just so that I can invite him to fix it.)

To distort Plato’s SYMPOSIUM just a little bit, pining after Channing Tatum is the first step on the ladder of Beauty that leads shortly thereafter to appreciation of the beauty of Classical Logic and Relevant Logic, and then, finally, to the form of Beauty — Beauty itself. Of course, my enemies say that I should avoid logic altogether and stick to pining after Channing Tatum.


Shells And Peanuts Again (And Again…And Again…In A Never-Ending GROUNDHOG DAY)

peanutshell_03

So one more time — but this time with feeling:  following Relevant Logic, we can avoid Classical Logic’s paradoxes of Material Implication, according to which the following statements are true…

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

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

…by insisting that the antecedent p be relevant to the consequent q.  The question of course now is:  what is the relation that makes p relevant to q?  In my previous post, one can, if they are sufficiently drunk, just barely make out the answer:  ‘whatever condition c along with (in the case of subjective probability) knowledge k makes the conditional probability of q equal to 1 given p is what makes p relevant to q.   Sometimes this ‘whatever’ is identical with an INFORMATION THAT relation (p is information that q); sometimes it is not.

( When the relation is identical with the INFORMATION THAT relation, c is the channel of information that allows p to be information that q. When the relation is not identical with the INFORMATION THAT relation, c consists in background conditions, especially causal laws, which, just as in the channel-of-information case, make the conditional probability of q given p 1. My current claim is that even when the relation is not identical with an INFORMATION THAT relation, it has a structure in common with the INFORMATION THAT relation.)

What I propose to do now in the next several posts is go through the> various examples I’ve mentioned previously (shell games, children with measles, wormy red apples, the ringing of defective doorbells, and so on) and a) work out when, in the example, the IF-THEN relation is identical with an INFORMATION-THAT relation and when it is not, and b) see what strange conclusions arise from this account of the relevance-making relation.  Maybe some of these will be so awful that one would prefer Classical Logic’s paradoxes of Material Implication.

In this post I propose to work through Dretske’s famous shell game example.  In that example, one will remember, a peanut is hidden under one of four shells.  I know from whatever reliable means that there is a peanut under 1 of the shells.  This knowledge reduces the probability that (a | the ) peanut is under shell #4 from 1 in whatever billions to just 1 in 4. Maybe my waffling here between ‘a’ and ‘the’ opens up a can of worms; I am unsure. I turn over shell #1.  There is no peanut under that shell.  The conditional probability that the peanut is under any given one of the remaining shells is now 1 in 3.  I turn over shell #2.  Empty.  The conditional probability that the peanut is under any given one of the remaining shells is now 1 in 2.  I say:

If shell #3 is empty, Then the peanut is located under shell #4

And what I say is surely true!  True, true, twue!!!!!  For if shell #3 turns out to be empty, then the conditional probability that the peanut is under shell #4 is 1.  The condition c that makes this conditional probability 1 given p is the characteristic that objects have — at least those objects large enough to be immune to whatever quantum weirdness — of persisting in one place unless molested.  The electron (at least according my remembered ((and almost certainly garbled in my memory)) pronouncement of a chemistry TA I had as an undergraduate) one finds orbiting this or that particular atom could have been on the nose of the Mona Lisa before getting observed, and might be there again a moment later.  But the peanut is not going to jump around like that, leaping to shell #1 one moment while unobserved, and onto the nose of the Mona Lisa the next moment.  It is going to stay placidly and inertially where it is — under shell #4 — while one turns over shell #3 and observes it to be empty.  Given this background fact about objects the size of peanuts, shell #3’s proving to be empty rules out the possibility that the peanut is not under shell #4.

Here the relevance-making factor — what makes the IF-THEN statement I uttered true — is also that factor that would make shell #3’s turning out to be empty INFORMATION THAT the peanut is located under shell #4.

But let’s turn back the clock.  I am now back at the point at which I am turning over shell #1.  Empty.  If I now jumped the gun and said (as if this were the movie GROUNDHOG DAY ((which I have not seen, by the way)), in which one atrocious day gets repeated again and again so that…”The phrase “Groundhog Day” has entered common use as a reference to an unpleasant situation that continually repeats, or seems to.”):

If shell #3 is empty, Then the peanut is located under shell #4

what I say would surely be false. Or at least it must be false if what I said in my first paragraph is true.  For were I to turn over shell #3 and discover it to be empty, the conditional probability that the peanut is located under shell #4 would not be 1, but 1/2.  So the same IF-THEN statement would be true at one time, and false at another.  And it would be true relative to my knowledge k at one time (I know that shells #1 and #2 are empty), and false relative to my lack of that same knowledge at a different time.

Not coincidentally, the (possible) emptiness of shell #3 being information that the peanut is located under shell #4 is something that is true at some times and false at other times, and is relative to one’s knowledge (or lack thereof) in exactly the same way.  In this particular case, what makes the If p Then q statement true is identical with what makes p information that q.

Now turn back the clock yet one more time (I warned you that this is another iteration of GROUNDHOG DAY).  This time I already know from a reliable source of information, even before I have turned over any shells, that the peanut is located under shell #4.  I turn over shell’s #1 and #2 as before.  Both are empty, as before.

But now, shell #3’s proving to be empty upon turning it over would no longer be INFORMATION THAT the peanut is located under shell #4.  This is so for at least two reasons.  First, according to Information Theory, “old information” is an oxymoron.  It is not information at all.  Shell #3’s turning out to be empty is not going to tell me, inform me, show me, that the peanut is under shell #4 because I already have this information.

Second, to generate information is to effect a reduction in possibilities.  In Dretske’s example of an employee selected by a succession of coin flippings to perform an unpleasant task, the eventual selection of Herman out of 8 possible choices reduced the number of possibilities from 8 to 1.  The selection of Herman generates INFORMATION THAT Herman was selected because of this reduction in possibilities.  But in my situation, already knowing that the peanut is located under shell #4, the number of possibilities regarding where the peanut is located is already just 1.  Turning over shell #3 to prove that it is empty does not reduce the number of possibilities from 2 to 1 — that number was 1 in the first place.  So in my situation shell #3’s proving to be empty does not generate, is not information that, the peanut is located under shell #4.

That the number of possibilities is in my situation just 1, not 2 means of course that the conditional probability that the peanut is located under shell #4 is not 1/2, but 1.  This means that shell #3’s proving to be empty does not make the conditional probability that the peanut is located under shell #4 equal to 1.  For that conditional probability was already equal to 1.  We are supposing that I already know that the peanut is located under shell #4, but I would not know this if the conditional probability were not already 1.  The very strange conditions that would have to obtain to make the conditional probability say, 1 in 2 would rule out this knowledge.  The peanut would have to exist under both shell #3 and shell #4 at the same time while unobserved, then “collapse” to a single location under one of the shells upon turning over the other shell and observing its empty condition.  So to say that I already know the location of the shell is to say that the conditional probability the peanut is at that location equals 1.

Now in the first paragraph of this screed I said (maybe ‘pontificated’ is the better word):

…whatever condition c along with (in the case of subjective probability) knowledge k makes the conditional probability of q equal to 1 given p is what makes p relevant to q.

Here my knowledge k (the peanut is located under shell #4) presupposes certain conditions c (the peanut does not exist in a kind of locational smear when unobserved, only to collapse to a single location when an observation is made).  Here p (shell #3 proves to be empty) along with k and the presupposed c definitely does not make the conditional probability of q equal to 1.  This conditional probability was, given k and its presupposed c, already 1.  So in my situation, p is not relevant to q.

So were I, in my situation of already knowing that the peanut is located under shell #4, to  utter GROUNDHOG-DAY-wise:

If shell #3 is empty, Then the peanut is located under shell #4

My statement would be false for exactly the same reason that the following is false:

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

In both cases, the antecedent is irrelevant to the consequent by failing to make the conditional probability of the consequent 1, rendering the corresponding IF-THEN statement false.  The antecedent “If shell #3 is empty” is in my situation irrelevant to the consequent “the peanut is located under shell #4” in exactly the same way that “Cliff lives in Houston” is irrelevant to “the earth has just one moon.” (In exactly the same way?  Yes, at least according to the perhaps narrow definition of relevance I postulated above.  But does this narrowness weaken my claim?  Might the emptiness of shell #3 be relevant to the peanut’s being located under shell #4 in some ((perhaps)) vague way even given my knowledge k?)

To re-iterate (this is a GROUNDHOG DAY post after all), the shell statement is false in my situation for exactly the same reason that “shell #3 is empty” fails to be information that “the peanut is located under shell #4.”  In this particular case, the relevance-making condition which is lacking is identical with an INFORMATION THAT relation.

If so, however, one is faced with a consequence that may strike some as at least equally unappealing as the paradoxes of Material Implication.  (Warning:  I am about to wallow in more GROUNDHOG DAY iterations.)  For when I utter:

If shell #3 is empty, Then the peanut is located under shell #4

the statement I utter is false, but when you hear:

If shell #3 is empty, Then the peanut is located under shell #4

and your situation is such that you have seen both shells #1 and #2 are empty and you do not know that the peanut is located under shell #4, the statement you hear is true!  The same statement is both true and false at the same time, given different situations.  Put another way, what is true or false (at least for a certain class of IF-THEN statements) is not the statement, but the statement as it shows up in a particular situation.

At least in the case of subjective probability, then, truth is relative in much the same way that Galilean motion is relative.

On a purely autobiographical note, I am not sure this relativity bothers me any more than Galilean relativity (there is the possibility of an ultimate reference frame) or for that matter Einsteinian relativity (there is no ultimate reference frame which would assign a single value to the speed of a moving object) does.  The idea that a person walking inside a flying jet is moving at a speed of 1 mile per hour relative to the reference frame of the jet but at a speed of 501 miles per hour relative to the reference frame of the earth (suppose the jet’s speed is 500 miles per hour) is perfectly intuitive even though it means a contradiction is true (the person is both moving at a speed of 1 mile per hour and is not moving at a speed of 1 mile per hour).

Likewise, the contradiction of claiming that (GROUNDHOG DAY alert):

If shell #3 is empty, Then the peanut is located under shell #4

Is both true and false at the same time seems to me to be intuitive if one casts it as a matter in which a conclusion’s following (not following) from its premise hinges upon what other knowledge or evidence one has (does not have).  But I do suspect that some would prefer to this relativity of truth and the attendant tolerance of contradiction the weirdness of Classical Logic’s Material Implication which arises from treating Implication as purely truth functional.

shell_02

This statement (GROUNDHOG DAY alert):

If shell #3 is empty, Then the peanut is located under shell #4

is variously true or false — even at the same time — depending upon the already-existing knowledge (or lack of it) of the person uttering or hearing the statement.  By contrast, the following statement is true regardless of what anyone knows, and true in any situation:

If the peanut is located under shell #4, Then the peanut is located under shell #4

In other words:

If p Then p

That the peanut is located under shell #4 clearly suffices to make the conditional probability that the peanut is located under shell #4 1.  So according to my account of what makes p relevant to q, p is relevant to p. p is relevant to itself.  p is in a relation to itself.  I am of course beginning to sound very weird (or maybe weirder) and very Hegelian…and I am beginning to wonder if I can get out of this weirdness by talking about 1-place relations, which are perfectly respectable mathematically.  (Not just 1-place relations!  0-place relations are also quite respectable mathematically!  What is more, Chris Date’s Relational Algebra recognizes two 0-place relations, TABLE DEE which is identical with the that weird proposition in logic TRUE, and TABLE DUM, which is identical with the equally weird proposition in logic FALSE!!!!!!!)

In this section of my post, I will decide that I am Relational-Algebra-weird by treating “If p Then p” as a 1-place INFORMATION THAT relation.  This in turn is part of my larger project to go through each example of IF-THEN statements I’ve adduced in previous posts and decide whether the relevance-making RELATION is in that particular case an INFORMATION-THAT relation or not.

Remember that to generate information is to reduce the number of possibilities to one.  When Herman is selected through 3 successive coin flips out of 8 candidates to perform the unpleasant task, the number of possibilities is reduced from 8 to 1.  The probability of Herman’s getting selected was initially 1 in 8, then became 1.  Whenever any event occurs, some states of affairs comes to obtain, some thing acquiring some property, the probability of that occurrence goes from 1 in (some usually gargantuan number) to just 1.  So any occurrence of p (Herman’s getting selected, shell #3 proving to be empty, a ruby having formed through whatever geological processes exactly one mile underneath where I happen to be sitting now typing this disreputable screed into a WordPress blog, the doorbell’s ringing) generates information.  Sometimes the occurrence of p generates information that q (that the peanut is under shell #4…that someone or something is depressing the button outside….).  But whatever else the occurrence of p generates information about, it generates at the very least the information that p.  Herman’s selection generated the information that Herman was selected, whether or not this information gets transmitted from the source situation in which the selection occurred (the room where the employees performed 3 coin flips) to the situation which is waiting for the information (the room where the boss is sitting).  When the information does get transmitted from source to receiver, the INFORMATION THAT relation is a 2-place relation comprising two situations, source and receiver.  When the information does not get transmitted, but stays where it is in the source, the INFORMATION THAT relation is a 1-place relation, comprising simply the source situation.

When the relevance-making relation that makes If p Then q true is an INFORMATION THAT relation, the occurrence (obtaining, existence) of p generates the information that q.  We have just seen that the occurrence (obtaining, existence) of p generates the information that p. So we get:

If p Then p

as a 1-place INFORMATION THAT relation.  Rather than saying, rather weirdly and rather Hegelianishly, that p is related to itself by virtue of being relevant to itself, we simply say that there exists a 1-place relation comprising the source at which the information that p was generated, and only that source.  This remains an INFORMATION THAT relation even though nothing ever tells me, informs me, shows me that, for example, a ruby exists exactly 1 mile beneath where I am now sitting, typing this disreputable screed into WordPress, or that the peanut is in fact underneath shell #4.  It is just a 1-place, not a 2-place relation, and an INFORMATION THAT relation to boot.

So in all of the following,

If a ruby exists exactly 1 mile underneath where I am now sitting, Then a ruby exists exactly 1 mile underneath where I am now sitting

If the peanut is located underneath shell #4, Then the peanut is located underneath shell #4

If Herman was selected to perform the unpleasant task, Then Herman was selected to perform the unpleasant task

the general relevance-making relation, i.e., the occurrence (obtaining, existence) of p making the conditional probability that p equal to 1, is identical with an INFORMATION THAT relation.  (My ((probably non-existent)) reader will remember that the relevance-making relation is not always an INFORMATION THAT relation.)

And this (after having brought in a ruby example and a Herman’s getting selected example) concludes my working through of most of the peanut-under-a-shell examples.  I still have one more peanut and shell example to work through, namely,

If I turn over shell #4, I will see the peanut

which I will work through in a future post.

SmallShell

Today’s homage to Plato’s SYMPOSIUM is Channing Tatum, who has recently appeared in MAGIC MIKE II.

ChanningTatum_02png

Channing Tatum is the very walking, talking, breathing, living definition of the words ‘age 35 and beautiful and sexy.’  One of these days I will get around to contemplating Plato’s Form of Beauty itself.  For now, though, I will rest content just contemplating the form of Channing Tatum.

SmallShell

July 18, 2015:  extensive revisions made in probably futile attempt to hide the vastness of the extent of my confusion.

July 21, 2015:  made one more revision in order to try to hide the lack of control I have over the subject matter.

August 02, 2015:  made yet another revision for the same dubious reasons as listed above.

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Shells, Peanuts, And Doorbells: Subjective Probability And The Relevance-Making Relation

So far then, we have the following:  following Relevant Logic, we can avoid Classical Logic’s paradoxes of Material Implication, according to which the following statements are true…

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

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

…by insisting that the antecedent p be relevant to the consequent q.  The question now is:  what is the relation that makes p relevant to q?  I propose that this relation (henceforth the ‘CONDITIONAL PROBABILITY IS 1 relation) can be stated as follows:  given p, the conditional probability of q, (under conditions c, and possibly given knowledge k) would be, or would become 1.

We will see that this relation involves a dependency on p of the value of the conditional probability of q; this dependency though is different from the dependencies I’ve discussed in the previous posts. This dependency is the relevance-making relation we are looking for in our quest to escape from the evil clutches of the Classical Logician.

             shell_02shell_02shell_02shell_02

There are two items in the way I have just stated the CONDITIONAL PROBABILITY IS 1 relation that cry out for discussion.  The first item is the distinction between subjective and objective probability.  (I am a bit surprised that I have not yet seen so far a discussion of this distinction by Dretske, though perhaps I have run across such a discussion but forgotten about it.) The second item is the phrase ‘given that.’

OBJECTIVE VS. SUBJECTIVE PROBABILITY:  In the doorbell examples given in the post below, the CONDITIONAL PROBABILITY IS 1 relation is in both cases objective. In the non-poltergeist example, were the doorbell ringing, the conditional probability would be 1 that someone or something is depressing the button outside. This probability would be 1 regardless of what anyone thinks, knows, or feels. The probability is objective. Likewise, in the poltergeist example, the conditional probability that the doorbell is ringing inside were I to press the button outside would be 1, regardless of what anyone thinks, knows, or feels. In both the poltergeist and the non-poltergeist examples, the CONDITIONAL PROBABILITY IS 1 relation is objective.

By contrast, when I first come across the four shells (in a situation in which I already know that there is a peanut located underneath one of the shells), the conditional probability that the peanut is underneath shell #4 would become 1 in three were shell #1 to prove to be empty; would then become 1 in 2 were shell #2 prove also to be empty, and finally would become 1 were shell #3 to turn out to be empty.  In each case, starting from the very beginning, the conditional probability hinges upon what I already know about the situation and changes with the alterations in my knowledge.  The CONDITIONAL PROBABILITY IS 1 relation in this case is subjective.

Henceforth I will use the phrase ‘would be’ to suggest that the CONDITIONAL PROBABILITY IS 1 relation is objective, and ‘would become’ to suggest that the relation is subjective.  ‘Would be’ suggests that the conditional probability is set from the very beginning and does not change with changes in a person’s knowledge of the situation; ‘would become’ suggests that the conditional probability is not fixed from the very beginning, and does change with increases (or decreases) in a person’s knowledge.

If we allow both objective and subjective probability and identify the relevance of p to q with the CONDITIONAL PROBABILITY IS 1 relation, we then get the result that IF-THEN statements are relative when the relevance relation is based on subjective probability.  In your situation, when you have first come upon the 4 shells (and you may not even know that there is a peanut is located underneath one of the shells!), the statement:

1)  If shell #3 turns out to be empty, Then a (the) peanut is located under shell #4

is false, because in your situation the Conditional Probability that a peanut is located under shell #4 would clearly not become 1 were shell #3 to turn out to be empty.  But in my situation, given what I know, that statement is true.  The Conditional Probability would definitely, in my situation, become 1 were shell #3 to prove to be empty.  So at least those IF-THEN statements belonging to a certain class — i.e., those whose relevance relation is based on subjective probability — display a relativity similar to the Galilean relativity of motion.

If one wants to avoid this (possibly, for some — at least for me –) counter-intuitive, paradoxical-seeming result, they may want to rule out subjective probability and base IF-THEN statements only on objective probability.  But what would ‘objective probability’ be in the case of the shell game?  I think it makes intuitive sense to claim something like:  ‘given that the peanut was located under shell #4 from the very beginning, chances were always 100% (the conditional probability was always 1) from the very beginning that the peanut was under shell #4.  (In other words, given p, the conditional probability of p is 1.  OMG — If p Then p!)   But let’s take a closer look at the phrase ‘given that’.

GIVEN THAT:  ‘Given that p, the conditional probability of q is 1′ means, I take it, that what the conditional probability of q is hinges upon, depends upon, p.  In the non-poltergeist doorbell example, that conditional probability of the button outside being pushed is 1 hinges upon the doorbell’s ringing.  If there is no ringing, the conditional probability of the button’s being depressed is not 1, but 1/100, or 1/100,000, or whatever.  (Remember that the conditions c of the doorbell’s defective wiring are such that 1% of the time the doorbell does not ring when the button outside is getting pushed.)  No ringing, no conditional probability equaling 1.   In the poltergeist doorbell example, that the conditional probability of the doorbell’s ringing inside is 1 and not 1/2, or 1/10,000, or whatever, hinges upon my pressing the button outside.  (Remember that in this example the conditions c of the doorbell’s defective wiring are such that 1% of the time the doorbell rings even when no one or nothing is depressing the button, creating the impression that a poltergeist must be dwelling inside the doorbell apparatus.)  No pressing of the button, no conditional probability equaling 1.

Note that this is a case of the value of the conditional probability of q hinging upon p.  This is to be distinguished from, for example, the ringing’s causally depending upon the button’s getting depressed, or the fact that I am about to see the peanut causally depends upon my lifting shell #4 (plus other factors).

Now if we do not allow subjective probability, the only GIVEN THAT relation that holds in the case of the shell game example is ‘given that the peanut is under shell #4, the conditional probability of the peanut’s being under shell #4 is 1’.  This is the only case that does not depend upon what a person already knows.  So statements 1 through 3 below would all be false for exactly the reason that 4) is false:  there is no longer any relation that would make p relevant to q by p‘s giving the conditional probability of q the value of 1:

1)  If shell #3 turns out to be empty, Then a (the) peanut is located under shell #4

2) If shell #1 turns out to be empty, Then a (the) peanut is located under shell #4

3) If shell #2 turns out to be empty, Then a (the) peanut is located under shell #4

4)  If Cliff lives in Houston, then a (the) peanut is located under shell #4

But there are situations in which statements 1 through 3 are true — situations in which my knowledge and yours vary.  I submit then that the price of jettisoning subjective probability is one that is too high to pay.  We need to keep subjective probability, and along with it the Galilean-like relativity of those IF-THEN statements whose relevance-making CONDITIONAL PROBABILITY is 1 relation is an instance of subjective probability.

Let me see what I will make of all of this in the morning, when I am sober.

Today’s homage to Plato’s SYMPOSIUM comprises Sal Mineo and the guy he crushed on, James Dean.

James_Dean_SalMineo_4

Beauty so wonderful, so fleeting.


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 toccurs 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.)