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References in:
A study of truth predicates in matrix semantics
Review of Symbolic Logic 11 (4):780-804 (2018)
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According to Frege's principle the denotation of a sentence coincides with its truth-value. The principle is investigated within the context of abstract algebraic logic, and it is shown that taken together with the deduction theorem it characterizes intuitionistic logic in a certain strong sense.A 2nd-order matrix is an algebra together with an algebraic closed set system on its universe. A deductive system is a second-order matrix over the formula algebra of some fixed but arbitrary language. A second-order matrix A is (...) |
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This paper is a contribution to the study of the rôle of disjunction inAlgebraic Logic. Several kinds of (generalized) disjunctions, usually defined using a suitable variant of the proof by cases property, were introduced and extensively studied in the literature mainly in the context of finitary logics. The goals of this paper are to extend these results to all logics, to systematize the multitude of notions of disjunction (both those already considered in the literature and those introduced in this paper), (...) |
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In the paper we study the class of weakly algebraizable logics, characterized by the monotonicity and injectivity of the Leibniz operator on the theories of the logic. This class forms a new level in the non-linear hierarchy of protoalgebraic logics. |
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There exist important deductive systems, such as the non-normal modal logics, that are not proper subjects of classical algebraic logic in the sense that their metatheory cannot be reduced to the equational metatheory of any particular class of algebras. Nevertheless, most of these systems are amenable to the methods of universal algebra when applied to the matrix models of the system. In the present paper we consider a wide class of deductive systems of this kind called protoalgebraic logics. These include (...) |
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It is shown that the class of reduced matrices of a logic is a 1 st order -class provided the variety associated with has the finite replacement property in the sense of [7]. This applies in particular to all 2-valued logics. For 3-valued logics the class of reduced matrices need not be 1 st order. |
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The notion of local deduction theorem (which generalizes on the known instances of indeterminate deduction theorems, e.g. for the infinitely-valued ukasiewicz logic C ) is defined. It is then shown that a given finitary non-pathological logic C admits the local deduction theorem iff the class Matr(C) of all matrices validating C has the C-filter extension property (Theorem II.1). |
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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. (...) |
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The first known statements of the deduction theorems for the first-order predicate calculus and the classical sentential logic are due to Herbrand [8] and Tarski [14], respectively. The present paper contains an analysis of closure spaces associated with those sentential logics which admit various deduction theorems. For purely algebraic reasons it is convenient to view deduction theorems in a more general form: given a sentential logic C (identified with a structural consequence operation) in a sentential language I, a quite arbitrary (...) |
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A deductive system (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. |
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