rwas rwas wrote: >IF: > AB:C > 11 1 > 10 0 > 01 1 > 00 1 > >Can someone explain the "IF" table?
I refer you to my last post to George Levy in the present thread, where I say a little more. I would like to add something, though. Someone asked (I guess jokingly) to Bertrand Russell to explain why (2 = 1) -> Russell is the pope Russell seems to have give the following (I guess jokingly too) answer: I and the pope are two, so if two is equal to one, it means I and the pope are one, which entails I am the pope. This is just funny but, strictly speaking, wrong. The correct answer should have been: (2=1) is false, so (2=1) -> X, is always true, as you can seen in the truth table of "IF". (But then of course the joke disappear). So the explanation of the truth table of the "IF" is that it is a definition. Nevertheless, in this list, perhaps we can still provide a genuine reason to appreciate that definition. Most of us accept some notion of "block universe". Indeed, I take myself a sort of block universe, which is just the set of "arithmetical truth". In such a set, it is nice that we have a purely "non deductival and acausal" weak form of implication, where A->B means really nothing more that A is false OR B is true. I should mention that classical propositional logic *is* difficult, and I'm sure a lot of things will appear in a clearer way with the modal logics. Indeed, when you teach logic, the subtil but very important point is to understand that A --- B and A -> B A are quite different things. --- means that B B can be deduced from A (by a finite number of applications of the inference rule with the axioms). A->B means nothing a priori. It is just a string. That string has the truth values given by some semantics, when we have a semantics. But then we must explain that for classical propositional logic, it can be proved (and indeed it has been proved by Jacques Herbrand in the 1920...) that each time you have A --- B you have A -> B, and each time you have A->B, you have A --- B (by modus ponens actually). This is a form of Herbrand deduction theorem (or the metadeduction theorem as it is called sometimes). In that sense, modal logic will appear more easy, because the metadeduction theorem is just false for modal logic. And this you can understand with what we have seen. Indeed S5 is closed for the rule of inference of necessitation: A --- A But A->A is not valid in Leibniz semantics. A true in a world does not entails A true in all worlds! So by the soundness/completeness result for S5 with respect to Leibniz semantics S5 does not prove A->A. And so, the metadeduction theorem is false for modal logics, and so it is easier to explain the difference between "Russell's material implication" (->) and deduction. And this means that the sequel will be easier! In logic it is the beginning which is difficult, because it is too easy (!) in the sense that a lot of *important nuances* (like implication and deduction) are blurred. Bruno