On Wed, Aug 14, 2002 at 04:38:45PM +0200, Bruno Marchal wrote: > Your general question was "Why using modal logic when > quantifying on worlds is enough". My basic answer was > that Kripke's possible world semantics works only on a > subset of the possible modal logics.
Let me generalize my question then. Is it true that for any modal logic that has a semantics, any sentence in that logic has a corresponding sentence in non-modal quantificational logic with the same meaning? In other words, are there any modal sentences whose meaning cannot be expressed by quantifying directly on the appropriate objects? > You can do modal logics without semantics. In fact modal > logic appeared because of apparent existence of modalities. > The main one is "possible" and "necessary". But others > occurred like "permitted" and "obligatory"; "provable" and > consistent", "believable" and "imaginable", etc. > The fundamental motivation of a logician is to give purely > syntactical formula and rules for manipulating formula so that > we can reason and communicating reasoning *without* any > meaning. The traditional joke is that a logician does not > want understand what he talk about! Before the invention of possible world semantics, people had to reason about modalities on a purely syntactical basis. Are there still modal logics for which no semantics is known? We know that in general syntactical formulas and rules are not powerful enough to always let us reason without meaning, because the set of mathematical truths that are derivable syntactically from a fixed set of axioms is just a subset of all mathematical truths. The rest can only be obtained by considering the semantic consequences of the axioms. I think the point of syntax is just to give us a way to obtain at least some of the truths through syntactical manipulation - a way to grab the low-hanging fruit.
