Sunday, February 17, 2008

More on correspondence

Dunn finally gave an indication of the sense of correspondence he has been using. He says that the sense of correspondence is the same as the correspondence between axioms of modal logic and frame conditions, namely accessibility relation conditions. He cites van Benthem in this regard as originating the term "correspondence." As was said before, the conditions on º correspond to various logical systems for implication. This is because there is a tight connection between º and implications, as mentioned before. There clearly needs to be a bridge here as we have logical systems where want something like frames. The bridge is an accessibility relation of sorts. Dunn uses a ternary relation R, which acts like an accessibility relation, to define fusion and various implication operations. We are concerned with frames, just frames of the basic form (U,R), with U a set of points and R a ternary relation on them.

Given a basic frame as above, we can give a canonical definition of R for →.For a,b,c, subsets of U, Rabc iff ∀ x,y, if x∈ a and x→y ∈ b, then y∈ c. I think this is the entry point for semantics for relevance logic. From the tiny bit I know about it, I think it is Rabc iff a+b=c, for a binary operation +.

To return to the first paragraph, there is a theorem about what frame properties are first-order definable. It is called the Sahlqvist theorem. It is in terms of the standard modal language with the standard connectives, box, top and bottom. It syntactically picks out a class of conditional sentences, whose general form is somewhat arcane, and shows that those give rise to frame conditions through a translation into first-order sentences. (My description seems to be fairly inadequate so please look at the Wikipedia page. I should do a post on the Sahlqvist theorem by itself...) The Sahlqvist theorem gives a sufficient, but not necessary, condition for a modal formula to pick out a first-order definable class of frames. A question that popped up while reading through the Dunn book was what sorts of ternary relations were first-order definable. A related question is are there any forms of equations that pick out those frames, in the same way that Sahlqvist formulae do for first-order definable modal frames. An answer is not forthcoming in Dunn. I'm also not really sure what the relation is between first-order logic and the algebraic logic stuff in Dunn's book. It doesn't seem like we've used anything beyond first order logic lately, but I'm not sure if that is an accident of the chapters or a feature of algebraic logic. There isn't an overriding concern to explain which structures and which of their properties are first-order definable. From the opening chapters, it didn't seem like algebraic logic was in any essential way restricted to first-order logic or less. Maybe the later chapters will shed light on this.

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