NDLI: Fregean logics with the multiterm deduction theorem and their algebraization

Content Provider	Springer Nature Link
Author	Czelakowski, J. Pigozzi, D.
Copyright Year	2004
Abstract	A deductive system $$\mathcal{S}$$ (in the sense of Tarski) is Fregean if the relation of interderivability, relative to any given theory T, i.e., the binary relation between formulas $$\{ \left\langle {\alpha ,\beta } \right\rangle :T,\alpha \vdash s \beta and T,\beta \vdash s \alpha \} ,$$ is a congruence relation on the formula algebra. The multiterm deduction-detachment theorem is a natural generalization of the deduction theorem of the classical and intuitionistic propositional calculi (IPC) in which a finite 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 is very close to that of the variety of Hilbert algebras. The equivalent quasivariety is however not in general a variety. This gives an example of a relatively point-regular, congruence-orderable, and congruence-distributive quasivariety that fails to be a variety, and provides what apparently is the first evidence of a significant difference between the multiterm deduction-detachment theorem and the more familiar form of the theorem where there is a single implication connective.
Starting Page	171
Ending Page	212
Page Count	42
File Format	PDF
ISSN	00393215
Journal	Studia Logica
Volume Number	78
Issue Number	1-2
e-ISSN	15728730
Language	English
Publisher	Kluwer Academic Publishers
Publisher Date	2004-01-01
Publisher Place	Dordrecht
Access Restriction	One Nation One Subscription (ONOS)
Subject Keyword	abstract algebraic logic protoalgebraic logic quasivariety equivalential logic algebraizable logic self-extensional logic Leibniz congruence deduction theorem Logic Mathematical Logic and Foundations Computational Linguistics
Content Type	Text
Resource Type	Article
Subject	Logic History and Philosophy of Science

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