Topics Logic

Question 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 intutively ascribe to logic). Maybe a possible solution might be to say that the propositions that are at stake are not what they appear. So, what I am in fact saying is that: If P & Q, then P <--> Q and Q --> P and P --> Q. So, saying that "If Russell invents the ramified theory of types then Bush is the 43rd President of the US" is in fact a false conditional, but not so for the proposition "If Russell invents the ramified theory of types and Bush is the 43rd President of the US then If Russell invents the ramified theory of types then Bush is the 4rd President of the US" which is not only true, but the triuth of the antecedent guarantees the truth of the consequent, as it should, given that the conditional is presumed true. Does this work? What can be said about this? Thanks a lot.

Accepted:
January 28, 2010

Comments

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 logics, it's not enough that Q just happens to be true in worlds where P just happens (perhaps for no related reason) to be true. In relevance logic there must be some connection, causal or conceptual, between P and Q that explains or underwrites Q being true when P is true, etc. So, in these terms, while Russell's inventing the Theory of Types might materially imply that W was the 43rd President of the US, it doesn't entail that he was. See Alan R. Anderson and Nuel D. Belnap, Jr., Entailment: The Logic of Relevance and Necessity (1975; vol. 2 1992); J. Michael Dunn, ‘Relevance Logic and Entailment’, in Handbook of Philosophical Logic: Alternatives to Classical Logic, vol. 3, eds. D. Gabbay and F. Guenthner (1986), 117-24; and Stephen Read, Relevant Logic (1988)

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 conditional, "if p, then q", is properly understood as material. Note that this is an empirical question about how English actually works. It could have worked so that "if...then..." was material; if it doesn't, then that is a contingent fact. The great majority of semanticsts these days would suppose that the English conditional was not material, but the literature on conditionals is extremely complicated. There's an old book called If p, Then q, by David Sanford, which is still a good introduction.