On Combined Connectives

Abstract

Combined connectives arise in combined logics. In fibrings, such combined connectives are known as shared connectives and inherit the logical properties of each component. A new way of combining connectives (and other language constructors of propositional nature) is proposed by inheriting only the common logical properties of the components. A sound and complete calculus is provided for reasoning about the latter. The calculus is shown to be a conservative extension of the original calculus. Examples are provided contributing to a better understanding of what are the common properties of any two constructors, say disjunction and conjunction.Mathematics Subject Classification (2010). Primary 03B62; Secondary 03B05.

A modal logic amalgam of classical and intuitionistic propositional logic

“…In [3], the authors propose cryptofibring semantics as a generalization of fibring semantics and show that this provides a solution to the collapsing problem. Other solutions to the problem are given by graph-theoretic fibring [18,19] and, more recently, by a new method called meet combination of logics [20,21].…”

Categorical Abstract Algebraic Logic: Meet-Combination of Logical Systems

“…To avoid some of the drawbacks and undesired effects involved in the application of fibring, Sernadas et al [11,12] introduced, recently, another way of combining logical systems, called meet-combination, in which the combined connectives, instead of inheriting all properties they enjoy in the component logical systems, inherit only those properties that are common to both connectives. A very illuminating example of the difference that this entails as contrasted to the fibring method consists of the result of combining two logics L ∧ and L ∨ , one including a classical conjunction ∧ and one including a classical disjunction ∨, with the intention of obtaining a combined connective "identifying" these two connectives from the component logics.…”

“…In [11] Sernadas et al start from a given logical system L with a Hilbert style calculus and with a matrix semantics and define a new logic L × that incorporates all meetcombinations of connectives of L of the same arity. Moreover, this system includes in a canonical way the connectives of the original logical system.…”

“…The matrix semantics consists, also roughly speaking, of the direct squares of the matrices in the original matrix semantics. In the main results, [11,Theorems 3.9 and 3.13], it is shown that soundness and a special form of completeness, called concrete completeness, are inherited in L × from L. Moreover, Sernadas et al [11] investigate in some detail the case of classical propositional logic, which constitutes the main motivation and paradigmatic example behind their work. Based on classical propositional calculus, they present several interesting examples, which, in addition, serve as illustrations for various sensitive points of the general theory.…”

“…In the present paper, we adapt the framework of [11] to a categorical level, using notions and techniques of categorical abstract algebraic logic [13,14]. Our main goal is providing a framework in which, starting from a -institution whose closure system is axiomatized by a set of rules of inference, we may construct a new -institution that includes, in a precise technical sense, natural transformations corresponding to meet-combinations of operations available in the originalinstitution.…”