Bruno Marchal wrote:

All right. But modal logic are (traditionaly) extension of classical logic, so that causal implication, or >natural language entailment, when study mathematically are generally defined through modalities + >"material implication".

So in a sense, you confuse yourself by premature anticipation.

Well, I guess "in every possible world where X is true, Y is true also" can only be false if there's a possible world where the classical logical statement "X -> Y" is false (because in that possible world, X is true but Y is false). So perhaps the possible-world statement would be equivalent to the modal-logic statement "it is necessarily true that X->Y"--would this be an example of modal logics "extending" classical logic? In any case, in classical logic X -> Y can only be false if X is true in *our* world, whereas the possible-world version of "if X then Y" does not require that X is true in our world, although it must be true in some possible world. And like I said, I think the possible-world statement more accurately captures the meaning of the natural-language statement.

Well, I guess "in every possible world where X is true, Y is true also" can only be false if there's a possible world where the classical logical statement "X -> Y" is false (because in that possible world, X is true but Y is false). So perhaps the possible-world statement would be equivalent to the modal-logic statement "it is necessarily true that X->Y"--would this be an example of modal logics "extending" classical logic? In any case, in classical logic X -> Y can only be false if X is true in *our* world, whereas the possible-world version of "if X then Y" does not require that X is true in our world, although it must be true in some possible world. And like I said, I think the possible-world statement more accurately captures the meaning of the natural-language statement.

Jesse