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Fibred semantics and the weaving of logics. I: Modal and intuitionistic logics. (English) Zbl 0872.03007

Let \({\mathbf L}\) be a logic with some connectives and let \({\mathbf L}_i\) \((i\in I)\) be a family of logics which are conservative extensions of \({\mathbf L}\). Assume that \({\mathbf L}_i\cap{\mathbf L}_j={\mathbf L}\), for all \(i\neq j\) \((i,j\in I)\) and that each \({\mathbf L}_i\) has, among others, the set \({\mathcal C}_i\) of additional connectives to those of \({\mathbf L}\). Then the general weaving of logics problem is to characterize the set of all logics \(\{{\mathbf L}^\alpha_I\mid\alpha\) is a name of combining \({\mathbf L}_i\), \(i\in I\}\) which are built on the connectives of \({\mathbf L}\) and \(\bigcup^n_{i=1}{\mathcal C}_i\), and which are conservative extensions of each \({\mathbf L}_i\) \((i\in I)\). In this paper, the author develops a methodological framework for the mathematical presentation of, and possible solutions to, the general weaving of logics problem. More accurately, the various ways as ‘fibring’, ‘dovetailing’ etc., yielding different systems, are considered and proposed. This general methodology is applied to modal and intuitionistic logics as well as to general algebraic logics. The standard combining techniques and refinements which can be applied to any family of initial logics to obtain further combined logics are presented.
The central concept introduced in the paper is the notion of fibred semantics arising naturally in response to the needs of several independent areas of logic and its applications. The fibred semantics can be considered as a new kind of possible world semantics.
As the crucial result of the paper we point out a construction for combining arbitrary modal or superintuitionistic logics, each complete with respect to a class of Kripke-type semantics. The problem of transfer of recursive axiomatizability, decidability and finite model property is also considered and solved.
The proposed general methodology will help to organize this field systematically and gives many new interesting ideas for further research.

MSC:

03B22 Abstract deductive systems
03B45 Modal logic (including the logic of norms)
03G99 Algebraic logic
03B20 Subsystems of classical logic (including intuitionistic logic)
03B55 Intermediate logics
68T27 Logic in artificial intelligence
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