# Tag Archives: Relevant Logic

## 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.

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.

## What Is Relevance Anyhow?

But What Does ‘Relevance’ Mean? If (at the time of this writing) one googles for a definition of the word ‘relevance’, the gist of what they will get will be something like:  a state of affairs1 p is relevant to a state of affairs q when p is connected to q in some way and that connection is important to us in some way.  The connection matters.

Any given state of affairs will of course bear a very large (perhaps indefinitely large) number of connections to any other state of affairs.  I am trivially connected for example to all people in the world whose last name begins with ‘W’ (I bear a W connection to each of them); and I am trivially connected to everyone else in the world whose last name does not begin with ‘W’ (I bear a non-W connection to each of them).

But some connections matter to us, perhaps in relation to some particular goal, or in relation to some highly pervasive desire.  The importance of the connection selects out those cases in which p is relevant to q.

The Ice Example:  Warning — I Intend To Use This As A Metaphor For Implication:  For example, the thickness/thinness (or even complete absence) of the ice covering a river (state of affairs p)  is connected to my reaching the other bank of the river (state of affairs q) by way of enabling/hindering/rendering-impossible my reaching that other bank.  This connection matters to me when I have the goal of reaching the other side alive, or at least in some reasonable approximation thereto.  (And I have this goal because of something else that matters to me.  I need, say, to evade the secret police on this side, or the only food there is exists only on the other side.)  The importance of this connection, the place it has in the web of my goals, renders p relevant to q.

So when the Relevant Logician insists that p be relevant to q in propositions of the form IF p THEN q, they can plausibly be construed as asserting that there is some connection between p and q, and this connection is important to us.  What this connection is and why it is important to us may be suggested by the following examples.  The first example to follow (Madame Olensky) does not quite get us to this connection, but it is suggestive enough to put us on the right track leading to it (The Doorbell).

The Matter Regarding Madame Olensky And Professor Plum:  When Madame Olensky is caught standing over the body of Professor Plum with a smoking gun in her hand, this state of affairs (p) bears a definite connection to another (quite) possible state of affairs, namely, that Madame Olensky murdered Professor Plum (q). This connection consists in the fact that p‘s obtaining/being true increases the probability (in this case drastically) that q obtains/is true.  That probability is now somewhere greater than 0 but equal to or less than 1.  The connection matters to us whenever we are concerned enough to ask (say, out of a desire for justice, I should hope, or at least out of a general desire to get things right):  Did Madame Olensky murder Professor Plum?  Because this increases-the-probability connection matters to us, it renders Madame Olensky’s standing over the body of Professor Plum (whose last twitches ceased just one second ago) with a smoking gun relevant to the possible state of affairs comprising Madame Olensky’s just having murdered Professor Plum.

But the Relevant Logician will want something a bit stronger for the connection between p and q that will make p relevant to q in propositions of the form IF p THEN q.  For in propositions of that form, the obtaining/being true of q is guaranteed should p obtain/be-true.  In other words, the probability of q, given p, needs to be 1.  Not 0.86, not 0.9999, but 1.  Implication needs to be completely reliable.

In other words, the ice needs to be so solid that the chances of falling through, of losing one’s footing and plunging into deep cold water while trying to cross to the consequent q are zero.

Although Madame Olensky’s standing over the body of Professor Plum with a smoking gun definitely increases the probability that she is the murderer of Professor Plum beyond 0, that probability is doubtlessly not 1.  For a sufficiently competent writer of mystery novels can invent a scenario just barely within the realm of possibility in which, despite the bald fact that Madame Olensky is standing over the body of Professor Plum with a smoking gun in her hand, she is in fact not the actual murderer of Professor Plum.  The probability is, say, a mere 0.99999999999.

In the matter regarding Madame Olensky and Professor Plum, there is a minuscule, but real chance that one might fall through the ice, lose their footing, plunge into the deep cold swift water while crossing to the other bank of the river.

So the statement

1) IF Madame Olensky is standing over the body of Professor Plum with a smoking gun, THEN Madame Olensky is the murderer of Professor Plum

is false.  It is false because, although the state of affairs comprising Madame Olensky’s standing over the body of Professor Plum with a smoking gun is definitely relevant to the possible state of affairs comprising Madame Olensky’s being the murderer of Professor Plum, the connection which generates this relevance is not the right relevance-making connection.

The Doorbell (In Perfect Working Order):  The right relevance connection does exist, I think, taking a cue from Fred Dretske, in the case of a doorbell whose wiring is in perfect condition.  Given the condition of the wiring, the probability, when the doorbell is ringing (p), that someone outside is pushing the doorbell button, or that, at least, something is depressing that button (q), is 1.  The constraint created by the perfect condition of the wiring makes p a completely reliable indicator of q.  So this IF THEN statement:

2) IF the doorbell is ringing THEN someone or something is depressing the button outside

is true.  That someone or something outside is depressing the doorbell button is guaranteed by the doorbell’s ringing inside.

This particular increases-the-probability (to 1) connection between the doorbell’s ringing and someone-or-something’s depressing the button outside matters to (most of) us because there is, I should think, a pervasive desire to get things right, to know how things actually stand outside the room, to know what is actually the case among the things that are not immediately present to us, to be able to tell what is happening.  This mattering selects out this particular connection as a relevance-making connection between p and q.  Because of this relevance of p to q, 2) above is true.

The doorbell’s ringing (when the condition of the wiring is perfect) is, of course, the classic example of Information That, of informational content.  The ringing (r, for reception) is information that the button outside is getting depressed (s, for source), if we follow 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

I will dwell on the knowledge k part of this definition in some detail later.

That the conditional probability of the button outside’s getting depressed increases to 1 when the doorbell rings is both what makes the ringing a signal, information that the button outside is getting depressed and what makes p relevant to q in 2) above.  Therefore, it is tempting to identity the relevance-making relation between p an q with the information-that relation.   Implication, it is tempting to say, is always information that.  The following:

3) IF Cliff lives in Houston THEN the earth has just one moon

fails to be a true implication because Cliff’s living in Houston is not information that the earth has just one moon.  I will be returning to this point later.2

To revert back to the river ice metaphor, the antecedent in 3) is ice that never formed in the first place.  There is no chance one can cross to the consequent q on the basis of p.  One cannot even lose their footing here, because there was only ever swift cold water to plunge into.

However, there are of course a number of rather severe challenges to the notion that implication is always information.  I will consider some of these in the snippets that follow.

At the time of this writing, I am suffering under the delusion that once all the challenges that I have considered so far have been dealt with, one ends up with the concept of relevant implication as always to be made sense of in terms of the concept of information — sometimes as full-blooded information, sometimes as degenerate or denatured information, and sometimes as the radical absence of information.  Whichever is the case, there is always the reference to the concept of information.  We will see if I end up having to eat crow on this point.

Some Housekeeping:  First, however, I want to do some housekeeping.  The careful reader will notice that I keep shifting back and forth between talking about p and q as states of affairs and as propositions.  I will continue to shift back and forth because I will be following Roderick Chisholm in treating propositions as a subspecies of states of affairs.3  The state of affairs comprising this cat, Munti sitting on this Persian mat can obtain or not obtain at different times.  The state of affairs comprising ‘Munti is sitting on on this Persian mat on October 31 at 12:00 am’ either always obtains or never obtains according as it was true or not true October 31 at 12:00 am that Munti was sitting on the Persian mat.  The latter is a state of affairs (obtaining or not obtaining) that is also a proposition (true or false); the former is a state of affairs (obtaining or not obtaining) that is not also a proposition.

Propositions are true or false; a proposition can follow from another or fail to follow from it.  Implication, therefore, is a relation between sets of states of affairs obtaining/failing to obtain being true/failing to be true at particular times (the doorbell is ringing at times t0, t1, t2, t… tn) and the button outside is getting pushed at times t0, t1, t2, t… tn).

One Final Point:  I have defined relevance in terms of mattering.  Since in Relevant Logic p has to be relevant to q in implication propositions in order for those implications to be true, does this mean that no implication statement was true before any sentient creature existed to whom anything could matter?  (I don’t think so, but this still needs to be shown, of course.)  If so, is this a weirdness that is off-putting enough to make one prefer Classical Logic to Relevant Logic?

1 I will leave ‘state of affairs’ as an undefined primitive.
2 One reason p is not information that q here is, of course, that the earth has just one moon is “old information” and therefore not information at all. But the more important reason is that even if this were not “old information”, Cliff’s living in Houston would still not be information that the earth has just one moon because the former, by itself, leaves the probability of the latter at 0. This ‘even if’ is pertinent to my claim that implication is to be understood in terms of information even if a particular example of an implication proposition is not an instance of information that.
3 Roderick Chisholm, THEORY OF KNOWLEDGE SECOND EDITION, Englewood Cliffs, Prentice-Hall, Inc., 1977, pp. 87-88.

Edit Log:  June 04, 2017:  Made some fairly minor edits in an always-ongoing and never-fully-accomplished effort to avoid complete and total embarrassment.

## 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.

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.

## Re-Igniting An Old Flame

A few weeks ago my interest in the French Philosopher Maurice Merleau-Ponty (1908-1961) suddenly got re-ignited upon finding out that a paper I published in a previous life (THE CONCEPT OF THE ECSTASIS, Journal Of The British Society For Phenomenology, 14(1):  79-90, 1983) actually got listed in the bibliography of Stephen Priest’s MERLEAU-PONTY:  THE ARGUMENTS OF THE PHILOSOPHERS.

The sudden explosion of this renewed interest is a bit like the result of throwing a lighted match on a bunch of rags soaked in gasoline.  In its heat, I’ve decided to start a new category of blog posts comprising an attempt to gain a deeper, fuller understanding of the topic of that paper.  What positions stated in the paper do I still hold?  What positions must I mark to market?  (<yes I am being ironic>Doubtlessly none — surely my paper is sacred text.</yes I am being ironic>) What can be stated more clearly, argued for more carefully?  Doing this kind of thing is what blogs are ideal for:

…you can work around the edges of an idea over days and weeks and months [and years] and really   come to understand it. It’s this process that blogging does better than pretty much any other medium.

Anil Dash On Blogging

The topic of my paper is, essentially:

The question concerning corporeity connects also with Merleau-Ponty’s reflections on space (l’espace) and the primacy of the dimension of depth (la profondeur) as implied in the notion of being in the world (être au monde; to echo Heidegger’s In-der-Welt-sein) and of one’s own body (le corps propre).

Wikipedia Article On Maurice Merleau-Ponty

So in the months and years to come I will be re-reading, working through, and blogging on Merleau-Ponty (THE PHENOMENOLOGY OF PERCEPTION, THE VISIBLE AND INVISIBLE, and other works) in order to really come to understand, truly get my head around, get a maximal grasp of this notion of ‘the primacy of the dimension of depth as implied in the notion of being in the world and of one’s own body.’  As part of this effort, I will be re-reading and blogging on George Berkeley’s works as well, which, partly as foil, partly in a kind of concurrence, shed light in an interesting way on Merleau-Ponty.

These efforts will fall under the category ‘Primacy Of The Dimension Of Depth.’

Of course, I am far from having finished the other two main categories I have been working on in this blog, to wit: ‘The Argument That Tagalog Lacks A Subject’ (a thread inspired largely by Paz Buenaventura Naylor’s article), and ‘Material Implication And Information Theory’ (inspired largely by Fred Dretske’s KNOWLEDGE AND THE FLOW OF INFORMATION and by Edwin D. Mares’ RELEVANT LOGIC).  I intend to continue working on these threads at the same time that I am re-igniting an old flame, my crush on Merleau-Ponty.

If I bore anyone, tough.  You don’t have to read these incoherent/semi-incoherent ramblings.  I am writing largely in order to learn, to get as much clarity as I can in my own head regarding these topics.

Of course, it would be nice if someone else were interested in them, and, even better yet, had something useful and interesting to say about them, whether in disagreement or agreement with me.

It would also be nice if Ashton Kutcher gave me a call.

(No post even touching on philosophy would be completed without an homage to Plato’s SYMPOSIUM.)  I wonder if Alkibiades was as gorgeous.

## Some Boring MetaBlogging

Number 14 of this pretty much describes what I am trying to do here.  In particular:

…you can work around the edges of an idea over days and weeks and months [and years] and really come to understand it. It’s this process that blogging does better than pretty much any other medium.

This is what I am trying to do with the Relevant Logic/Material Implication/Information Theory viewed through the eyes of Fred Dretske stuff (repeated endlessly).  Who knows, I might even do some endless blogging someday to gain a ‘maximal grasp’ (Merleau-Ponty) on the Roderick Chisholm stuff.

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

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.

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.

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

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.

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.