Thursday, May 26, 2005

3VL - Modal logic

One of the reasons I liked dealing with logic is that it provides a richer and consistent formal language for discussing the truth values of propositions: certain(true), not necessary, possible, not possible (false), doubtful or equivocal (U), and two-valued or dichotomous.

I had begun with the idea of trying to reproduce a modal logic which would correspond to the similarities, and with the introduction of the strict Lukasiewicz conditional, I could managed to reproduce versions of all the axioms of S5.
Later I found a term, "formal similarity" to describe the relationship, because there are substantial differences between this version and Lewis-type systems.

Lewis used a different notion of possibility than I do. His corresponds to the existence of a possibility, where mine takes it in a somewhat different sense of "not impossible". He associated "possible" with "non-self-contradictory" and would probably not include doubtful or equivocal statements as possible, where I would include them. I believe he would argue that "It is possible that I will eat a turkey on Thanksgiving and it is possible that I will not eat a turkey on Thanksgiving, but it is not possible that I will and won't eat a turkey on thanksgiving." I would inquire about whether he is using "and" in a truth-functional sense or an additive sense and whether a half-eaten turkey would make "will and will not" plausible.

He also bases his logic strictly on two-valued classical logic, and assumes that all its laws hold without restriction (including the law of the excluded middle). I don't; I have to modify some of them to account for the existence and effect of doubtful propositions, which he doesn't acknowledge or account for.

He also used a different notion for his strict conditional. He defines "strictly implies" as "it is necessary that if P then Q", and "it is not possible for P and not Q". While I do find a use for "it is certain that if P then Q", The idea that "it is not possible for P and not Q" is too strict for my purposes. My version is weaker but still adequate.

The effect of these differences is that there is a significant difference in substance between the two systems, but since I can reproduce analogues of all the Lewis axioms as theorems, I can reproduce corresponding analogues of any theorem of S5, and I can decide whether a conjecture is or is not a theorem, and if not, why not, far more easily. "Anything you can do, I can do better?"

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