## Journal of Symbolic Logic

### Fibred Semantics and the Weaving of Logics Part 1: Modal and Intuitionistic Logics

D. M. Gabbay

#### Abstract

This is Part 1 of a paper on fibred semantics and combination of logics. It aims to present a methodology for combining arbitrary logical systems $\mathbf{L}_i, i \in I$, to form a new system $\mathbf{L}_I$. The methodology fibres' the semantics $\mathscr{K}_i$ of $\mathbf{L}_i$ into a semantics for $\mathbf{L}_I$, and weaves' the proof theory (axiomatics) of $\mathbf{L}_i$ into a proof system of $\mathbf{L}_I$. There are various ways of doing this, we distinguish by different names such as fibring', dovetailing' etc, yielding different systems, denoted by $\mathbf{L}^F_I, \mathbf{L}^D_I$ etc. Once the logics are weaved', further interaction' axioms can be geometrically motivated and added, and then systematically studied. The methodology is general and is applied to modal and intuitionistic logics as well as to general algebraic logics. We obtain general results on bulk, in the sense that we develop standard combining techniques and refinements which can be applied to any family of initial logics to obtain further combined logics. The main results of this paper is a construction for combining arbitrary, (possibly not normal) modal or intermediate logics, each complete for a class of (not necessarily frame) Kripke models. We show transfer of recursive axiomatisability, decidability and finite model property. Some results on combining logics (normal modal extensions of $\mathbf{K}$) have recently been introduced by Kracht and Wolter, Goranko and Passy and by Fine and Schurz as well as a multitude of special combined systems existing in the literature of the past 20-30 years. We hope our methodology will help organise the field systematically.

#### Article information

Source
J. Symbolic Logic, Volume 61, Issue 4 (1996), 1057-1120.

Dates
First available in Project Euclid: 6 July 2007

https://projecteuclid.org/euclid.jsl/1183745126

Mathematical Reviews number (MathSciNet)
MR1456098

Zentralblatt MATH identifier
0872.03007

JSTOR