In this paper we introduce canonical extensions of partially ordered sets and monotone maps and a corresponding discrete duality. We then use these to give a uniform treatment of completeness of relational semantics for various substructural logics with implication as the residual(s) of fusion.

In this paper we consider distributive modal logic, a setting in which we may add modalities, such as classical types of modalities as well as weak forms of negation, to the fragment of classical propositional logic given by conjunction, disjunction, true, and false. For these logics we define both algebraic semantics, in the form of distributive modal algebras, and relational semantics, in the form of ordered Kripke structures. The main contributions of this paper lie in extending the notion of Sahlqvist (...) axioms to our generalized setting and proving both a correspondence and a canonicity result for distributive modal logics axiomatized by Sahlqvist axioms. Our proof of the correspondence result relies on a reduction to the classical case, but our canonicity proof departs from the traditional style and uses the newly extended algebraic theory of canonical extensions. (shrink)

This paper is a study of duality in the absence of canonicity. Specifically it concerns double quasioperator algebras, a class of distributive lattice expansions in which, coordinatewise, each operation either preserves both join and meet or reverses them. A variety of DQAs need not be canonical, but as has been shown in a companion paper, it is canonical in a generalized sense and an algebraic correspondence theorem is available. For very many varieties, canonicity (as traditionally defined) and correspondence lead on (...) to topological dualities in which the topological and correspondence components are quite separate. It is shown that, for DQAs, generalized canonicity is sufficient to yield, in a uniform way, topological dualities in the same style as those for canonical varieties. However topology and correspondence are no longer separable in the same way. (shrink)

A logic is selfextensional if its interderivability (or mutual consequence) relation is a congruence relation on the algebra of formulas. In the paper we characterize the selfextensional logics with a conjunction as the logics that can be defined using the semilattice order induced by the interpretation of the conjunction in the algebras of their algebraic counterpart. Using the charactrization we provide simpler proofs of several results on selfextensional logics with a conjunction obtained in [13] using Gentzen systems. We also obtain (...) some results on Fregean logics with conjunction. (shrink)

Provability, Computability and Reflection.

In this paper we introduce canonical extensions of partially ordered sets and monotone maps and a corresponding discrete duality. We then use these to give a uniform treatment of completeness of relational semantics for various substructural logics with implication as the residual(s) of fusion.

The purpose of this note is to expose a new way of viewing the canonical extension of posets and bounded lattices. Specifically, we seek to expose categorical features of this completion and to reveal its relationship to other completion processes.

Frege's Principle asserts that the denotation of a propositional sentence coincides with its truth value. In the context of algebraizable logics the principle can be interpreted as the compositionality of interderivability relation $\Fr{S}$, defined formally by $\Fr{S}T=\{\langle \phi, \psi\rangle\in\Fml^2\mid T,\phi \dashv\vdash_{\mathcal S}T,\psi \}$, for given deductive system $\mathcal S$ and any $\mathcal S$-theory $T$. Of special interest are the deductive systems for which the property of being Fregean is inherited by all full 2nd-order models, so called, \it{fully Fregean} deductive systems. (...) The main result of this paper is a characterization of fully Fregean deductive systems over countable languages using properties of the strong Frege operator on the formula algebra. The example of a Fregean, but not fully Fregean deductive system $\mathcal S$ is provided. $\mathcal S$ also turns out to be selfextensional, but not fully selfextensional, and, in addition, the three principal algebraic semantics for $\mathcal S$ are different, i.e., $\Alg^\ast\mathcal S\subsetneq\Alg\mathcal S\subsetneq\Var$. (shrink)