δ(p/φ) and δ(p/ψ) being obtained by substituting φ for p and ψ for p in δ respectively. This prop... more δ(p/φ) and δ(p/ψ) being obtained by substituting φ for p and ψ for p in δ respectively. This property is possessed by classical and intuitionistic propositional logics (CPC and IPC respectively) and can be taken as a formalization of Frege’s principle of compositionality for truth-values, or extensionality principle. Logics satisfying it are called Fregean, after Suszko [11]. Examples of important non-Fregean logics abound: almost all the logics of the modal family are not. Nevertheless non-Fregean logics, such as the local consequence relation of the normal modal logic K (see below), often satisfy the following weaker replacement property: for any formulas φ, ψ, δ and any propositional variable p,
A Leibniz class is a class of logics closed under the formation of term-equivalent logics, compat... more A Leibniz class is a class of logics closed under the formation of term-equivalent logics, compatible expansions, and non-indexed products of sets of logics. We study the complete lattice of all Leibniz classes, called the Leibniz hierarchy. In particular, it is proved that the classes of truth-equational and assertional logics are meet-prime in the Leibniz hierarchy, while the classes of protoalgebraic and equivalential logics are meet-reducible. However, the last two classes are shown to be determined by Leibniz conditions consisting of meet-prime logics only.
In this paper we consider the structure of the class FGModS of full generalized models of a deduc... more In this paper we consider the structure of the class FGModS of full generalized models of a deductive system S from a universal-algebraic point of view, and the structure of the set of all the full generalized models of S on a fixed algebra A from the latticetheoretical point of view; this set is represented by the lattice FACSSA of all algebraic closed-set systems C on A such that 〈A, C〉 ∈ FGModS. We relate some properties of these structures with tipically logical properties of the sentential logic S. The main algebraic properties we consider are the closure of FGModS under substructures and under reduced products, and the property that for any A the lattice FACSSA is a complete sublattice of the lattice of all algebraic closed-set systems over A. The logical properties are the existence of a fully adequate Gentzen system for S, the Local Deduction Theorem and the Deduction Theorem for S. Some of the results are established for arbitrary deductive systems, while some are found to ...
A notion of interpretation between arbitrary logics is introduced, and the poset $\mathsf {Log}$ ... more A notion of interpretation between arbitrary logics is introduced, and the poset $\mathsf {Log}$ of all logics ordered under interpretability is studied. It is shown that in $\mathsf {Log}$ infima of arbitrarily large sets exist, but binary suprema in general do not. On the other hand, the existence of suprema of sets of equivalential logics is established. The relations between $\mathsf {Log}$ and the lattice of interpretability types of varieties are investigated.
We consider the equationally orderable quasivarieties and associate with them deductive systems d... more We consider the equationally orderable quasivarieties and associate with them deductive systems defined using the order. The method of definition of these deductive systems encompasses the definition of logics preserving degrees of truth we find in the research areas of substructural logics and mathematical fuzzy logic. We prove several general results, for example that the deductive systems so defined are finitary and that the ones associated with equationally orderable varieties are congruential.
δ(p/φ) and δ(p/ψ) being obtained by substituting φ for p and ψ for p in δ respectively. This prop... more δ(p/φ) and δ(p/ψ) being obtained by substituting φ for p and ψ for p in δ respectively. This property is possessed by classical and intuitionistic propositional logics (CPC and IPC respectively) and can be taken as a formalization of Frege’s principle of compositionality for truth-values, or extensionality principle. Logics satisfying it are called Fregean, after Suszko [11]. Examples of important non-Fregean logics abound: almost all the logics of the modal family are not. Nevertheless non-Fregean logics, such as the local consequence relation of the normal modal logic K (see below), often satisfy the following weaker replacement property: for any formulas φ, ψ, δ and any propositional variable p,
A Leibniz class is a class of logics closed under the formation of term-equivalent logics, compat... more A Leibniz class is a class of logics closed under the formation of term-equivalent logics, compatible expansions, and non-indexed products of sets of logics. We study the complete lattice of all Leibniz classes, called the Leibniz hierarchy. In particular, it is proved that the classes of truth-equational and assertional logics are meet-prime in the Leibniz hierarchy, while the classes of protoalgebraic and equivalential logics are meet-reducible. However, the last two classes are shown to be determined by Leibniz conditions consisting of meet-prime logics only.
In this paper we consider the structure of the class FGModS of full generalized models of a deduc... more In this paper we consider the structure of the class FGModS of full generalized models of a deductive system S from a universal-algebraic point of view, and the structure of the set of all the full generalized models of S on a fixed algebra A from the latticetheoretical point of view; this set is represented by the lattice FACSSA of all algebraic closed-set systems C on A such that 〈A, C〉 ∈ FGModS. We relate some properties of these structures with tipically logical properties of the sentential logic S. The main algebraic properties we consider are the closure of FGModS under substructures and under reduced products, and the property that for any A the lattice FACSSA is a complete sublattice of the lattice of all algebraic closed-set systems over A. The logical properties are the existence of a fully adequate Gentzen system for S, the Local Deduction Theorem and the Deduction Theorem for S. Some of the results are established for arbitrary deductive systems, while some are found to ...
A notion of interpretation between arbitrary logics is introduced, and the poset $\mathsf {Log}$ ... more A notion of interpretation between arbitrary logics is introduced, and the poset $\mathsf {Log}$ of all logics ordered under interpretability is studied. It is shown that in $\mathsf {Log}$ infima of arbitrarily large sets exist, but binary suprema in general do not. On the other hand, the existence of suprema of sets of equivalential logics is established. The relations between $\mathsf {Log}$ and the lattice of interpretability types of varieties are investigated.
We consider the equationally orderable quasivarieties and associate with them deductive systems d... more We consider the equationally orderable quasivarieties and associate with them deductive systems defined using the order. The method of definition of these deductive systems encompasses the definition of logics preserving degrees of truth we find in the research areas of substructural logics and mathematical fuzzy logic. We prove several general results, for example that the deductive systems so defined are finitary and that the ones associated with equationally orderable varieties are congruential.
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