Wednesday, March 19, 2008

Nemo and Outis Revisited

One of the most famous and influential logical puzzles of the past hundred and fifty years is Lewis Carroll's Barber-Shop Paradox. The puzzle is one about hypothetical propositions. The most general form of the puzzle is this:

Suppose these to be true:
(1) If C is true, then, if A is true, B is not true.
(2) If A is true, Bis true.
Can C be true?

This is often called the Barber-Shop Paradox because Carroll imagines Uncle Joe on the way to the barber-shop, where there are three barbers, named, Allen, Brown, and Carr, and where instead of 'true' we say 'out' and instead of "in" we say "out":

(1) If Carr is out, then, if Allen is out, Brown is in.
(2) If Allen is out, Brown is out.
Is Carr out?

Carroll argued this out with John Cook Wilson for nearly two years; Cook Wilson arguing that Carr can't be out (C can't be true), while Carroll claimed that he could be (C can be true). Carroll published some things on it in Mind; Venn wrote about it in Symbolic Logic, both Alfred Sidgwick and W.E. Johnson wrote articles about it in Mind, Bertrand Russell discussed it in The Principles of Mathematics. The standard view, which was Carroll's view (although he worried about certain aspects of it) and Russell's view, and the one everyone has taught to them these days, is that hypothetical propositions are material implications, and from this it follows fairly easily that Carr can be out of the shop. Cook Wilson eventually did come around to Carroll's view; after Carroll's death he wrote an article for Mind arguing that Carr could leave the shop.

In a number of Carroll's works he presents the dispute as a dispute between Nemo (who holds Cook Wilson's view) and Outis (who holds Carroll's view).

Nemo's argument is that (1) amounts to "If C is true then (2) is not true"; but (2) is true ex hypothesi. By modus tollens, C is not true.

Not so, says Outis. The two propositions, "If A is true, B is true," and "If A is true, B is not true" are compatible. What (1) and (2) together require is not that C not be true but that A and C not be true together because "B is true and "B is not true" are incompatible.

On the contrary, says Nemo, Outis is dividing the proposition incorrectly. The absurdity arises not from "B is not true" but from "If A is true, B is not true," and only the assumption that C is true creates the absurdity. Outis is illicitly interpreting (1) as "If C is true [and if A is true], then if A is true, B is not true."

One fun thing to do, if you like logical games, is to extend the debate out as far as you can -- keep the back-and-forth between Nemo and Outis going on as long as you can, making the arguments as plausible as you can.

Of course, on any decent interpretation of hypothetical or conditional propositions, Outis is right. The reason often put forward for this, as noted above, is that "If p, q" is to be interpreted as a material conditional (i.e., as just saying "it's not the case that both p and not-q are true"). But it turns out that Outis's position does not require that interpretation. It's the right position for strict implication, as well, for instance. I have played a lot of Nemo-and-Outis, toying with different interpretations of conditionals, and in every single variation that I have considered Outis wins. The reason appears to be this: Nemo's position requires two different interpretations of the conditional, one for the larger conditionals (1) and (2), and a different one for the embedded conditional ("if A is true, B is not true" in (1)). So if you have a single consistent interpretation for indicative conditionals, Outis is always right. Even for some double interpretations (where embedded conditionals are interpreted in a completely different way than non-embedded conditionals) Nemo's position collapses. But the real question, one to which I don't have an answer, is: are there any double interpretations of conditionals that are not ad hoc (i.e., that could be applied plausibly to at least a small collection of real examples other than Barber Shop problems) on which Nemo's position would beat Outis's?

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