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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...

To give a similar but somewhat different answer, one might think the problem with the line of reasoning in the question comes here: "But it seems bizarre to say that Russell's invention of the theory of types entails that Bush is the 43rd president...". We were talking about the statement, "If Russell invented the theory of types, then Bush was the 43d president", and now we're talking about entailment? Why? What do these have to do with each other? The move from talking about the truth of conditionals to talking about entailment is what lies, in many ways, behind the invention of (formal) modal logic, by Lewis and Langford in the 1920s. One of the central ambitions of early modal logic was to formalize the notion of entailment. It was with reference to this that Quine spoke of modal logic's being "conceived in sin, the sin of confusing use and mention"---of confusing "if p then q" with "`p' entails `q'". Now, that said, it is undoubtedly a serious question whether the English indicative...

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