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

calong with (in the case of subjective probability) knowledgekmakes the conditional probability ofqequal to 1 givenpis what makesprelevant toq.

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

pThenp

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

*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!!!!!!!)*

**two**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

pThenp

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.