Conditionals and the Candy Bar inference

At a conference at Notre Dame in 1987 Paul Teller “made an issue” as he wrote later of “the fallacious form of argument I called the ‘Candy Bar Principle’:

from ‘If I were hungry I would eat some candy bar’ conclude ‘There is some candy bar which I would eat if I were hungry’.”  

And Henry Stapp, whom Teller had criticized mentioned this in his presentation: “Paul Teller has suggested that any proposed proof of the kind I am setting forth must contain such a logical error ….”

I do not want to enter into this controversy, if only because there were so many arguments swirling around Stapp’s proposed proofs.  Instead I want to examine the question:  

is the Candy Bar inference a fallacy?

Let’s formulate it for just a finite case:  there are three candy bars, A, B, and N.  The first two are in this room and the third is next door.  I shall refer to the following form of argument as a Candy Bar inference:

If I choose a candy bar it will be either A or B

therefore,

If I choose a candy bar it will be A, or, if I choose a candy bar it will be B

and I will symbolize this as follows:

C –> (A v B), therefore (C –> A) v (C –> B)

This has a bit of a history of course: it was submitted as valid in Robert Stalnaker’s original theory of conditionals and was rejected by David Lewis in his theory. Lewis showed that Stalnaker’s theory was inadequate, and blamed this principle. But we should quickly add that the problems Lewis raised also disappeared if this principle were kept while another one, shared by Stalnaker and Lewis, was rejected. This is just by the way, for now I will leave all of this aside.

How shall we go about testing the Candy Bar inference?

I imagine that the first intuitive reaction is something like this:

Imagine that I decide to choose a candy bar in this room.  Then it will definitely be either A or B that I choose. But there is nothing definite about which one it will be.  

I could close my eyes and choose at random.

Very fine!  But unfortunately that is not an argument against the Candy Bar inference, but rather against the following different inference:

It is certain that if I choose, then I will choose either A or B,

therefore

Either it is certain that if I choose I will choose A, or, it is certain that if I choose I will choose B

That is not at all the same, for we cannot equate ‘It is certain that if X then Y’ with ‘if X then Y’.  As an example, contrast the confident assertion “If the temperature drops it will rain tomorrow” with “It is certain that if the temperature drops it will rain tomorrow”.  The former will be borne out, the prediction will be verified, if in fact the temperature drops and it rains the next day — but this is not enough to show that the latter assertion was true.

So the intuitive reaction does not settle the matter.  How else can we test the Candy Bar inference?

Can we test it empirically?  Suppose two people, Bob and Alice of course, are asked to predict what I will do, and write on pieces of paper, respectively, “if Bas chooses a candy bar in this room, he will choose A” and  “if Bas chooses a candy bar in this room, he will choose B”.  Surely we will say:  

we know that if Bas chooses a candy bar in this room, he will choose A or B.  

So if he does, either Bob or Alice will turn out to have been right.

And then, if Bas chooses A, we will say “Bob was right”.

That is also an intuitive reaction, which appears to favor the Candy Bar inference.  But again, it does not really establish much.  For it says nothing at all about which of these conditionals, if any, would be true if Bas does not choose a candy bar.  That is the problem with any sort of empirical test: it deals only with facts and does not have access to what would have happened instead of what did happen.

Well there is another empirical approach, not directly to any facts about the choice and the candy bars, but to how reasonable, practical people would let this situation figure in their decision making.

So now we present Alice and Bob with this situation and we ask them to make bets.  These are conditional bets, they will be Gentlemen’s Wagers, which means that they get their money back if Bas does not choose.

Alice first asks herself:  how likely is he to choose a bar from this room, as opposed to from next door (where, you remember, there is bar N)  Suppose she takes that to have probability 3/4.  She accepts a bet that Bas will choose A or B, if he chooses at all, with payoff 1 and price 0.75.  Her expectation value is 0, it is just a fair bet.

Meanwhile Bob agrees with her probability judgment, but is placing two bets, one that if Bas chooses he will choose A, and one that if Bas chooses he will choose B.  These he thinks equally probable, so for a payoff of 1 he agrees to price 3/8 for each.  His expectation value is 1/4(0) + 3/8(1) + 3/8(1) minus what he paid, hence 0:  this too is just a fair bet.  

Thus Alice and Bob pay the same to be in a fair betting situation, where the payoff prices are the same, though one was, in effect, addressing the premise and the other the conclusion of the Candy Bar inference.  So, as far as rational betting behavior is concerned then, again, there is no difference between the two statements.

Betting, however, as we well now by now, is also only a crude measuring instrument for what matters.  The fact that these are Gentlemen’s Wagers, as they pretty well have to be, once again means that we are really only dealing with the scenario in which the antecedent is true.  The counterfactual aspect is beyond our ken.

To be clear:  counterfactual conditionals are metaphysical statements, if they are statements about what is the case, at all.  They are not empirical statements, and this makes the question about the validity of the Candy Bar inference a metaphysical question.

There is quite a lot of every-day metaphysics entrenched at the surface of our ordinary discourse. Think for instance of what Nancy Cartwright calls this-worldly causality, with examples like the rock breaking the window and the cat lapping up the milk. 

Traditional principles about conditionals, just as much as traditional principles about causality, may guide our model building.  And then nature may or may not fit our models …

So the question is not closed, the relation to what is empirically accessible may be more subtle than I managed to get to here.   To be continued …. 

REFERENCES

The Notre Dame Conference in question had its proceedings published as Philosophical Consequences of Quantum Theory:  Reflections on Bell’s Theorem (ed. J. T. Cushing and E. McMullin; University of Notre Dame Press 1989).   

My quotes from Teller and Stapp are from pages 210 and 166 respectively.