Fregean logics (original) (raw)

Fregean logics with the multiterm deduction theorem and their algebraization

Studia Logica, 2004

A deductive system S (in the sense of Tarski) is Fregean if the relation of interderivability, relative t o a n y g i v en theory T , i.e., the binary relation between formulas f h i : T `S and T `S g is a congruence relation on the formula algebra. The multiterm deduction-detachment theorem is a natural generalization of the deductive theorem of the classical and intuitionistic propositional calculi (IPC) in which a nite system of possibly compound formulas collectively plays the role of the implication connective of IPC. We investigate the deductive structure of Fregean deductive systems with the multiterm deduction-detachment theorem within the framework of abstract algebraic logic. It is shown that each deductive system of this kind has a deductive structure very close to that of the implicational fragment of IPC. Moreover, it is algebraizable and the algebraic structure of its equivalent quasivariety i s v ery close to that of the variety of Hilbert algebras. The equivalent quasivariety i s h o wever not in general a variety. This gives an example of a relatively pointregular, congruence-orderable, and congruence-distributive quasivariety that fails to be a variety, and provides what apparently is the rst evidence of a signi cant di erence between the multiterm deductiondetachment theorem and the more familiar form of the theorem where there is a single implication connective.

Metatheories of deductive systems

2004

A deductive system (Hilbert-style) is an algebraic closure system over the set of formulas of given propositional language. Similarly, a Gentzen system is an algebraic closure system over the set of all sequents, i.e., finite sequences of formulas, of this language. The main feature of this work is a technique that allows us to adapt the methods, previously developed in the area of algebraic logic for work with Hilbert-style deductive systems, to the case of Gentzen systems. Using the properties of the Tarski congruence, a generalization of the Leibnitz congruence, we develop an algebraic hierarchy for Gentzen systems that closely parallels the well-known algebraic hierarchy of deductive systems. This approach allows us to unify in a single framework several previously known results about algebraizable and equivalential Gentzen systems. We also obtain a characterization of weakly algebraizable Gentzen systems. The significance of Gentzen systems and related axiomatizations by Gentzen rules is due in large part to the fact that various metatheoretical properties of deductive systems can be formulated in their terms. It was observed that a number of important non-protoalgebraic deductive system that have a natural algebraic semantics also have a so-called fully adequate Gentzen system associated with them, the conjunction-disjunction fragment of the classical propositional logic being a paradigmatic example. In this work, a general criterion for the existence of a fully adequate Gentzen system for nonprotoalgebraic deductive systems is obtained, and it is shown that many of the known partial results can be explained based on this general criterion. This includes such cases as the existence of fully adequate Gentzen systems for self-extensional logics with conjunction or implication, and the criterions for the existence of a fully adequate Gentzen system for protoalgebraic and weakly algebraizable logics. In another vein, it is shown that the existence of a multiterm deduction-detachment theorem in a deductive system is equivalent to the fact that, so called, axiomatic closure relations for the deductive system form a Gentzen system.

On weakening the Deduction Theorem and strengthening Modus Ponens

Mathematical Logic Quarterly, 2004

This paper studies, with techniques of Abstract Algebraic Logic, the effects of putting a bound on the cardinality of the set of side formulas in the Deduction Theorem, viewed as a Gentzen-style rule, and of adding additional assumptions inside the formulas present in Modus Ponens, viewed as a Hilbert-style rule. As a result, a denumerable collection of new Gentzen systems and two new sentential logics have been isolated. These logics are weaker than the positive implicative logic. We have determined their algebraic models and the relationships between them, and have classified them according to several standard criteria of Abstract Algebraic Logic. One of the logics is protoalgebraic but neither equivalential nor weakly algebraizable, a rare situation where very few natural examples were hitherto known. In passing we have found new, alternative presentations of positive implicative logic, both in Hilbert style and in Gentzen style, and have characterized it in terms of the restricted Deduction Theorem: it is the weakest logic satisfying Modus Ponens and the Deduction Theorem restricted to at most 2 side formulas. The algebraic part of the work has lead to the class of quasi-Hilbert algebras, a quasi-variety of implicative algebras introduced by Pla and Verdú in 1980, which is larger than the variety of Hilbert algebras. Its algebraic properties reflect those of the corresponding logics and Gentzen systems.

Deduction in Non-Fregean Propositional Logic SCI

Axioms, 2019

We study deduction systems for the weakest, extensional and two-valued non-Fregean propositional logic SCI . The language of SCI is obtained by expanding the language of classical propositional logic with a new binary connective ≡ that expresses the identity of two statements; that is, it connects two statements and forms a new one, which is true whenever the semantic correlates of the arguments are the same. On the formal side, SCI is an extension of classical propositional logic with axioms characterizing the identity connective, postulating that identity must be an equivalence and obey an extensionality principle. First, we present and discuss two types of systems for SCI known from the literature, namely sequent calculus and a dual tableau-like system. Then, we present a new dual tableau system for SCI and prove its soundness and completeness. Finally, we discuss and compare the systems presented in the paper.

Implicational formulas in intuitionistic logic

The Journal of Symbolic Logic, Vol. 39, 661-664, 1974

In [1] Diego showed that there are only finitely many nonequivalent formulas in n variables in the positive implicational propositional calculus P. He also gave a recursive construction of the corresponding algebra of formulas, the free Hilbert algebra In, on n free generators. In the present paper we give an alternative proof of the finiteness of I,,, and another construction of free Hilbert algebras, yielding a normal form for implicational formulas. The main new result is that In, is built up from n copies of a finite Boolean algebra. The proofs use Kripke models [2] rather than the algebraic techniques of [1].

A General Algebraic Semantics for Sentential Logics

2017

Introduction 1 1 Generalities on abstract logics and sentential logics 13 2 Abstract logics as models of sentential logics 29 2.1 Models and full models 29 2.2 5-algebras 34 2.3 The lattice of full models over an algebra 38 2.4 Full models and metalogical properties 42 3 Applications to protoalgebraic and algebraizable logics 55 4 Abstract logics as models of Gentzen systems 69 4.1 Gentzen systems and their models 70 4.2 Selfextensional logics with Conjunction 80 4.3 Selfextensional logics having the Deduction Theorem \ • • • 89 5 Applications to particular sentential logics 97 5.1 Some non-protoalgebraic logics 99 5.1.1 CPC AV , the {A, V}-fragment of Classical Logic 99 5.1.2 The logic of lattices 101 5.1.3 Belnap's four-valued logic, and other related logics 102 5.1.4 The implication-less fragment of IPC and its extensions .... 104 5.2 Some Fregean algebraizable logics 105 5.2.1 Alternative Gentzen systems adequate for IPC_ not having the full Deduction Theorem 107 5.3 Some modal logics 108 5.3.1 A logic without a strongly adequate Gentzen system Ill vi Contents 5.4 Other miscellaneous examples. .. Ill 5.4.1 Two relevance logics 112 5.4.2 Sette's paraconsistent logic 113 5.4.3 Tetravalent modal logic 114 5.4.4 Logics related to cardinality restrictions in the Deduction Theorem 115 Bibliography 119

An Intuitionistic Characterization of Classical Logic

By introducing the intensional mappings and their properties, we establish a semantical approach of characterizing intermediate logics. First prove that this new approach provides a general method of characterizing and comparing logics without changing the semantical interpretation of implication connective. Then show that it is adequate to characterize all Kripke_ complete intermediate logics by showing that each of these logics is sound and complete with respect to its (unique) ' weakest characterization property' of intensional mappings. In particular, we show that classical logic has the weakest characterization property cl, which is the strongest among all possible weakest characterization properties of intermediate logics. Finally, it follows from this result that a translation is an embedding of classical logic into intuitionistic logic, iff. its semantical counterpart has the property cl.