A Survey of Abstract Algebraic Logic (original) (raw)

Assertional logics, truth-equational logics, and the hierarchies of abstract algebraic logic

Outstanding contributions to logic, 2018

We establish some relations between the class of truth-equational logics, the class of assertional logics, other classes in the Leibniz hierarchy, and the classes in the Frege hierarchy. We argue that the class of assertional logics belongs properly in the Leibniz hierarchy. We give two new characterizations of truth-equational logics in terms of their full generalized models, and use them to obtain further results on the internal structure of the Frege hierarchy and on the relations between the two hierarchies. Some of these results and several counterexamples contribute to answer a few open problems in abstract algebraic logic, and open a new one.

Applying Algebraic Logic; a General Methodology

2000

Connections between Algebraic Logic and (ordinary) Logic. Algebraic co- unterpart of model theoretic semantics, algebraic counterpart of proof theory, and their connections. The class Alg(L) of algebras associated to any logic L. Equivalence theorems stating that L has a certain logical property iff Alg(L) has a certain algebraic property. (E.g. L admits a strongly complete Hilbert- style inference system iff

Algebraic Properties of Propositional Calculus

In this short note we relate some known properties of propositional calculus to purely algebraic considerations of a Boolean algebra. Classes of formulas of propositional calculus are considered as elements of a Boolean algebra. As such they can be represented by uniquely defined elements of this algebra which we call "logical primes". The algebraic notations appear useful because they make it possible to derive well known properties of propositional calculus by simple calculations or to substitute lengthy logical considerations by schematic algebraic manipulations.

On the Interpretation of the Propositional Calculus

The question considered is 'How can formulae of the propositional calculus be brought into a representational relation with the world?'. Four approaches are discussed: (1) the denotational approach, on which formulae are taken to denote objects, (2) the abbreviational approach, on which formulae and connectives are taken to abbreviate natural-language expressions, (3) the truth-conditional approach, on which truth-conditions are stipulated for formulae, and (4) the modelling approach, on which formulae, together with either valuation- or proof-theory, are regarded as an abstract structure capable of bearing (via stipulation) a representational relation to the world. The modelling approach is developed here for the first time. The simple technical apparatus used for this is then applied to two issues in the philosophy of logic. (1) I demonstrate a corollary or converse to Carnap's result that certain 'non-normal' valuation-functions can be added to the set of admissible valuations of formulae without destroying the soundness and completeness of standard proof-theories. This sheds considerable light on a recent thread of the inferentialism debate which involves dialectical use of Carnap's result. (2) I show how the approach can be extended to quantification theory, by defining a model-theoretic notion of validity equivalent to the usual one, but making use of a proof-theoretic apparatus in place of the device of assigning values to formulae. This sheds light on the close relationship between proof- and valuation-theory.

Chapter 1: Generalities on Abstract Logics and Sentential Logics

2009

In this chapter we include the main definitions, notations, and general properties concerning logical matrices, abstract logics and sentential logics. Most of the results reproduced here are not new; however, those concerning abstract logics are not well-known, so it seems useful to recall them in some detail, and to prove some of the ones that are new. Useful references on these topics are Brown and Suszko [1973], Burris and Sankappanavar [1981] and Wójcicki [1988]. Algebras In this monograph (except in Chapter 5, where we deal with examples) we will always work with algebras A = A,. .. of the same, arbitrary, similarity type; thus, when we say "every/any/some algebra" we mean "of the same type". By Hom(A, B) we denote the set of all homomorphisms from the algebra A into the algebra B. The set of congruences of the algebra A will be denoted by ConA. Many of the sets we will consider have the structure of a (often complete, or even algebraic) lattice, but we will not use a different symbol for the lattice and for the underlying set, since no confusion is likely to arise. Given any class K of algebras, the set Con K A = {θ ∈ ConA : A/θ ∈ K} is called the set of K-congruences of A; while this set is ordered under ⊆, in general it is not a lattice. This set will play an important role in this monograph.

Peirce's implicative lattices, Propositional Logic in the Algebraic Tradition

In this paper I will focus my attention on one of the most significant moments of the process through which, inside the Algebraic Tradition, the idea of Propositional Logic emerged, underlining the need for an operation corresponding to the implication (logical connective). The aim of this essay wil be to point out the crucial role played by Peirce who, as I hope to elucidate soon, gave to the notion of implication, in its role of inverse operation of the conjunction in implicative lattices, a more general scope than that of the Filonian Material Implication.

Basic propositional calculus I

1998

We present an axiomatization for Basic Propositional Calculus BPC and give a completeness theorem for the class of transitive Kripke structures. We present several refinements, including a completeness theorem for irreflexive trees. The class of intermediate logics includes two maximal nodes, one being Classical Propositional Calculus CPC, the other being E l , a theory axiomatized by T-+ 1. The intersection C P C n El is axiomatizable by the Principle of the Excluded Middle A V-A. If B is a formula such that (T-+ B)-+ B is not derivable, then the lattice of formulas built from one propositional variable p using only the binary connectives, is isomorphically preserved if B is substituted for p. A formula (T + B)-+ B is derivable exactly when B is provably equivalent to a formula of the form