James G. Raftery

Adding involution to residuated structures

Two constructions for adding an involution operator to residuated ordered monoids are investigated. One preserves integrality and the mingle axiom x2≤x but fails to preserve the contraction property x≤x2. The other has the opposite preservation properties. Both constructions preserve commutativity as well as existent nonempty meets and joins and self-dual order properties. Used in conjunction with either construction, a result of R.T. Brady can be seen to show that the equational theory of commutative distributive residuated lattices (without involution) is decidable, settling a question implicitly posed by P. Jipsen and C. Tsinakis. The corresponding logical result is the (theorem-) decidability of the negation-free axioms and rules of the logic RW, formulated with fusion and the Ackermann constant t. This completes a result of S. Giambrone whose proof relied on the absence of t.

ASSERTIONALLY EQUIVALENT QUASIVARIETIES

A translation in an algebraic signature is a finite conjunction of equations in one variable. On a quasivariety K, a translation τ naturally induces a deductive system, called the τ-assertional logic of K. Two quasivarieties are τ-assertionally equivalent if they have the same τ-assertional logic. This paper is a study of assertional equivalence. It characterizes the quasivarieties equivalent to ones with various desirable properties, such as τ-regularity (a general form of point regularity). Special attention is paid to structural properties of quasivarieties that are assertionally equivalent to their varietal closures under an indicated translation.

Fragments of r-Mingle

The logic RM and its basic fragments (always with implication) are considered here as entire consequence relations, rather than as sets of theorems. A new observation made here is that the disjunction of RM is definable in terms of its other positive propositional connectives, unlike that of R. The basic fragments of RM therefore fall naturally into two classes, according to whether disjunction is or is not definable. In the equivalent quasivariety semantics of these fragments, which consist of subreducts of Sugihara algebras, this corresponds to a distinction between strong and weak congruence properties. The distinction is explored here. A result of Avron is used to provide a local deduction-detachment theorem for the fragments without disjunction. Together with results of Sobociński, Parks and Meyer (which concern theorems only), this leads to axiomatizations of these entire fragments — not merely their theorems. These axiomatizations then form the basis of a proof that all of the basic fragments of RM with implication are finitely axiomatized consequence relations.

Representable idempotent commutative residuated lattices

It is proved that the variety of representable idempotent commutative residuated lattices is locally finite. The n-generated subdirectly irreducible algebras in this variety are shown to have at most 3n+1 elements each. A constructive characterization of the subdirectly irreducible algebras is provided, with some applications. The main result implies that every finitely based extension of positive relevance logic containing the mingle and Godel-Dummett axioms has a solvable deducibility problem.

Order algebraizable logics

Abstract This paper develops an order-theoretic generalization of Blok and Pigozziʼs notion of an algebraizable logic. Unavoidably, the ordered model class of a logic, when it exists, is not unique. For uniqueness, the definition must be relativized, either syntactically or semantically. In sentential systems, for instance, the order algebraization process may be required to respect a given but arbitrary polarity on the signature. With every deductive filter of an algebra of the pertinent type, the polarity associates a reflexive and transitive relation called a Leibniz order, analogous to the Leibniz congruence of abstract algebraic logic (AAL). Some core results of AAL are extended here to sentential systems with a polarity. In particular, such a system is order algebraizable if the Leibniz order operator has the following four independent properties: (i) it is injective, (ii) it is isotonic, (iii) it commutes with the inverse image operator of any algebraic homomorphism, and (iv) it produces anti-symmetric orders when applied to filters that define reduced matrix models. Conversely, if a sentential system is order algebraizable in some way, then the order algebraization process naturally induces a polarity for which the Leibniz order operator has properties (i)–(iv).

A finite model property for RMImin

It is proved that the variety of relevant disjunction lattices has the finite embeddability property. It follows that Avrons relevance logic RMImin has a strong form of the finite model property, so it has a solvable deducibility problem. This strengthens Avrons result that RMImin is decidable. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

A perspective on the algebra of logic

Equations are the most basic formulas of algebra, and the logical rules for manipulating them are so intuitive that they are seldom formalized. Consequently, non-algebraic deductive systems (or ‘logics’) are very often interpreted in equational languages—although this is not always possible. For the optimal transfer of algebraic techniques, we require invertible interpretations that respect the structure of substitution; they should also induce isomorphism between the extension lattice of a system and that of its algebraic counterpart. The successful resolution of concrete logical problems in the presence of such an isomorphism has inspired (1) a robust general notion of equivalence between deductive systems, (2) a precise account of ‘algeb-raizable’ logics (pioneered by Blok and Pigozzi) and (3) a stock of ‘bridge theorems’ between logic and algebra. Moreover, an algebraic invariant in the theory of equivalence—called the Leibniz operator—has given rise to (4) a classification of deductive systems, analogous to the Maltsev classification of varieties in universal algebra. The present paper is a selective exposition of these developments.

Quasivarieties of Logic, Regularity Conditions and Parameterized Algebraization

Relatively congruence regular quasivarieties and quasivarieties of logic have noticeable similarities. The paper provides a unifying framework for them which extends the Blok-Pigozzi theory of elementarily algebraizable (and protoalgebraic) deductive systems. In this extension there are two parameters: a set of terms and a variable. When the former is empty or consists of theorems, the Blok-Pigozzi theory is recovered, and the variable is redundant. On the other hand, a class of ‘membership logics’ is obtained when the variable is the only element of the set of terms. For these systems the appropriate variant of equivalent algebraic semantics encompasses the relatively congruence regular quasivarieties.

Contextual Deduction Theorems

Logics that do not have a deduction-detachment theorem (briefly, a DDT) may still possess a contextual DDT—a syntactic notion introduced here for arbitrary deductive systems, along with a local variant. Substructural logics without sentential constants are natural witnesses to these phenomena. In the presence of a contextual DDT, we can still upgrade many weak completeness results to strong ones, e.g., the finite model property implies the strong finite model property. It turns out that a finitary system has a contextual DDT iff it is protoalgebraic and gives rise to a dually Brouwerian semilattice of compact deductive filters in every finitely generated algebra of the corresponding type. Any such system is filter distributive, although it may lack the filter extension property. More generally, filter distributivity and modularity are characterized for all finitary systems with a local contextual DDT, and several examples are discussed. For algebraizable logics, the well-known correspondence between the DDT and the equational definability of principal congruences is adapted to the contextual case.

Residuated Structures, Concentric Sums and Finiteness Conditions

This article is motivated by a concern with finiteness conditions on varieties of residuated structures—particularly residuated meet semilattice-ordered commutative monoids. A “concentric sum” construction is developed and is used to prove, among other results, a local finiteness theorem for a class that encompasses all n-potent hoops and all idempotent subdirect products of residuated chains. This in turn implies that a range of residuated lattice-based varieties have the finite embeddability property, whence their quasi-equational theories are decidable. Applications to substructural logics are discussed.