Talk:Barbershop paradox

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[edit] Why would Carroll have considered this a paradox?

I'm a little confused as to why Carroll considered this a paradox. In his notes he says
The paradox is a very real difficulty in the Theory of Hypotheticals....Are two Hypotheticals, of the forms If A then B and If A then not-B, compatible?
Of course they are! Why should (P > Q) and (P > ~Q) be contradictory? The contradiction of (P > Q) is ~(P > Q), which resolves to (P . ~Q), which is not the same as (P > ~Q) at all. So my question is: were the rules of logic different in Carroll's day? Was it not considered a standard law that from a falsity you can prove anything (F > R)? And if so, should we make it clear that he made the paradox under different logical rules? — Asbestos | Talk (RFC) 14:39, 19 May 2006 (UTC)

Asbestos,

My understanding is that when Carroll wrote his essay (1894) the notion of truth-functional material implication was not yet current in English logic; it was introduced mainly through the efforts of Bertrand Russell about a decade later. Truth-functional implication doesn't always line up very well with the intuitive use of "If-then" constructs so Carroll could not take for granted some of the features of material implication that are taken for granted by philosophers today. Note the questions that Carroll poses for logicians in his concluding "Note":

Can a Hypothetical, whose protasis is false, be regarded as legitimate?

Are two Hypotheticals, of the forms "If A then B" and "If A then not-B, compatible"?

What difference in meaning, if any, exists between the following Propositions?

# A, B, C, cannot be all true at once;

# If C and A are true, B is not true;

# If C is true, then, if A is true, B is not true;

# If A is true, then, if C is true, B is not true

If you think that material implication unproblematically expresses anything that you need to express by means of a conditional, then the "paradox" is simply obsolete: each of Carroll's questions has a definite and easy answer -- a hypothetical with a false protasis is regarded not only as legitimate, but as always true; two hypotheticals with the same antecedent but contradictory consequents are compatible so long as the antecedent is not true; and all of the listed propositions, if expressed truth-functionally (~(A . B . C), (C . A) > ~B, C > (A > ~B), A > (C > ~B)) are logically equivalent. And those answers provide a handy solution to the paradox proposed: Carr can be out as long as Allen is not also out, as per the answer to the second question.

If, on the other hand, you think that material implication fails to capture something interesting or important about conditional statements (which a lot of philosophers do think, for independent reasons), then Carroll's questions are likely to remain live ones for whatever notion of a hypothetical you think is uncaptured by material implication.

Hope this helps. Radgeek 06:24, 21 May 2006 (UTC)