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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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