# Varieties of Logic

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• Varieties of Logic!

Access provided by: anon Sign Out. The substructural logic usually associated with K is an algebraizable logic that has K as its equivalent algebraic semantics, and is a logic that preserves truth, i.

In this article, we introduce another logic associated with K, namely the logic that preserves degrees of truth, in the sense that it preserves lower bounds of truth values in inferences. We study this second logic mainly from the point of view of abstract algebraic logic. We determine its algebraic models and we classify it in the Leibniz and the Frege hierarchies: we show that it is always fully selfextensional, that for most varieties K it is non-protoalgebraic, and that it is algebraizable if and only K is a variety of generalized Heyting algebras, in which case it coincides with the logic that preserves truth.

We also characterize the new logic in three ways: by a Hilbert style axiomatic system, by a Gentzen style sequent calculus and by a set of conditions on its closure operator. Concerning the relation between the two logics, we prove that the truth-preserving logic is the extension of the one that preserves degrees of truth with either the rule of Modus Ponens or the rule of Adjunction for the fusion connective.

So looked at down at the infinitesimal level, is linear, and gives its slope at — so is the derivative of. Now that is indeed interesting. The usual complaint by the intuitionist is that adding the law of excluded middle unjustifiably collapses important distinctions in particular the distinction between and.

What are we to make of this? I guess there is quite a lot to be said here. It is a nice question, for example, how much sense we can make of all this outside the topos-theoretic context where the Kock-Lawvere theory of smooth analysis had its original home.