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Suppose P is true and Q is true, then it follows logically that P --> Q, that Q --> P and therefore that P Q. Now, suppose that P is 'George W. Bush is the 43rd President of the US' and Q is 'Bertrand Russell invented the ramified theory of types', both propositions are true, and therefore the truth of both guarantees the truth the aforementioned propositions. But it seems bizarre to say that Russell's invention of the theory of types entails that Bush is the 43rd president, as well as the other logical consequences. After all we can conceive of a scenario where Russell invents the ramified theory of types, but Bush becomes a plumber (say), if that is a possible scenario, it would seem that the proposition "If Russell invents the ramified theory of types then Bush is the 43rd President of the US" is false given the definition of 'if then'. But after all, does it make sense to say that a proposition entails another only in the actual world? (That doesn't seem to have as much generality as we...

Briefly, yeah. I think I see what you're getting at. When P and Q are true (which I think is what you mean by 'P&Q'), then P->Q, Q->P, and P is materially equivalent to Q. But note that this was the case for your earlier puzzle, too. Keep in mind that in standard first-order propositional logic, P->Q is a matter of only "material" implication, and P equivalent to Q is a matter only of "material" equivalence. That is, all 'P->Q' says is that when P is true Q is also true. All the equivalence says is that they have the same truth values. It doesn't say that there's some reason for the truth values being as they are, that there's any other connection between the statements, or that P and Q will be true in every possible or imaginary world. In our world P and Q happen both to be true, and that's enough. The issue you're pointing to is addressed by what's come to be called "Relevance Logic" and it is sometimes used to mark a difference in the use of the terms "implication" and "entailment." In relevance...

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