## Thursday, November 05, 2009

### Considering All the Possibilities

There's a cute logic puzzle up at "Cosmic Variance" making a point about dysrationalia:

Jack is looking at Anne, but Anne is looking at George. Jack is married, but George is not. Is a married person looking at an unmarried person?

A) Yes.

B) No.

C) Cannot be determined.

Apparently over 80% of respondents pick C, when the answer is supposed to be A; and this is supposed to be an example of dysrationalia -- intelligent people picking the wrong answer through a failure to think through the possibilities.

Unfortunately, as some of the commenters note, there are problems here, because (C) is not demonstrably irrational. The basic problem is this. We have a nontransitive relation, R, and three individuals, j, a, and g. The following relations obtain:

jRa
aRg

in addition, j is classified as married and g is classified as unmarried. The question is, given these facts, is there a person who is married who stands in relation R to a person who is unmarried?

And the answer, in a strict logical sense, is (C) -- it can't be determined. In order to restrict the answer to (A) you have to make the following assumptions:

(1) j, a, and g are all persons (if any of them is a horse, for instance, (A) is incorrect, because the question asks about married and unmarried persons)

(2) j, a, and g all must be either married or unmarried (if it's a category mistake to apply these labels to one of them, (A) is incorrect -- for instance, there are circumstances in which we would say that someone, like a baby, is not the sort of thing that can be either married or unmarried)

In many cases these would be entirely reasonable assumptions to make. But whether they are, in fact, reasonable assumptions depends entirely on which universe of discourse we are considering. Sometimes (A) will be the most rational answer. Sometimes it will not. The problem is not precise enough to rule out possibilities in which it is not, because it failed to specify any universe of discourse.

Consider the following analogy:

A borders on B, and B borders on C. A has a democratic government. C has an undemocratic government. Therefore there is country with a democratic government that borders on a country with an undemocratic one. For B must have either a democratic government or an undemocratic government. If it has a democratic government, then it borders on C, which has an undemocratic government. If it has an undemocratic government, then A, which has a democratic government, borders on it.

Beautifully reasoned. But B has no government at all; it is an ocean or an unclaimed wasteland, not a country. (If you want a philosophical example of the same underlying idea, you can look to Kant, because his resolution of the antinomies, e.g., about whether the world had a beginning, makes use of this very same feature.)

Thus the only way you could think (C) definitely wrong is if you are not considering all possibilities -- that is, if you are making assumptions that restrict the possibilities in play. But this really has nothing to do with rationality one way or another: there will be times when it will be rational to consider the possibility that Anne is not a person and times when it will be rational not to do so. There will be times when it will be rational to get hung up on the question of whether there is a tertium quid between 'married' and 'unmarried', and times when you should obviously be dichotomizing. It will depend entirely on the domain of discourse (and is an example of why domain of discourse is important for logical analysis).