Fibring is a meta-logical constructor that applied to two logics produces a new logic whose formulas allow the mixing of symbols. Homogeneous fibring assumes that the original logics are presented in the same way (e.g via Hilbert calculi). Heterogeneous fibring, allowing the original logics to have different presentations (e.g. one presented by a Hilbert calculus and the other by a sequent calculus), has been an open problem. Herein, consequence systems are shown to be a good solution for heterogeneous fibring when one of the logics is presented in a semantic way and the other by a calculus and also a solution for the heterogeneous fibring of calculi. The new notion of abstract proof system is shown to provide a better solution to heterogeneous fibring of calculi namely because derivations in the fibring keep the constructive nature of derivations in the original logics. Preservation of compactness and semi-decidability is investigated.

abstract. This paper is a guided tour through the theory of fibring as a general mechanism for combining logics. We present the main ideas, constructions and difficulties of fibring, from both a model and a proof-theoretic perspective, and give an outline of soundness, completeness and interpolation preservation results. Along the way, we show how the current algebraic semantics of fibring relates with the original ideas of Dov Gabbay. We also analyze the collapsing problem, the challenges it raises, and discuss a number of future research directions. 1

ness is preserved by a combination mechanism . and it is known that logic system is given by . L ## , then the completeness of follows from the completeness of ## . No wonder that much theoretical e#ort has been dedicated to establishing preservation results and/or finding preservation counterexamples about di#erent combination mechanisms. For an early overview of the practical and theoretical issues see also [4]. Several forms of combination have been studied, like product [30, 21, 22, 23], fusion [38, 28, 29, 40, 19], temporalization [12, 13, 41, 14], parameterization [6], synchronization [33] and fibring [15, 16, 3, 17, 34, 42]. Fusion is the best understood combination mechanism. In short, the fusion of two modal systems leads to a bimodal system including the two original modal operators and common propositional connectives. Several interesting properties of logic systems (like soundness, weak completeness, Craig interpolation property and decidability) were shown

Combining logics in an important topic in applied logics [7, 1] that raises interesting theoretical problems related to transference results. The objective is to produce a new logic from two (or more) given logics by using a meta operator – the combination mechanism. Of special interest is to investigate whether the mechanism preserves logical properties of the original logics. In general, sufficient conditions can be given for preservation. Fibring, proposed by Gabbay in [5], is one of the most challenging mechanisms for combining logics, which includes fusion of modal logics [10] as a particular case. Fibring can be and has been investigated from a deductive point of view (mainly using Hilbert calculi [11], labelled deductive systems [8] and tableau systems [2]) and also from a model-theoretic perspective (using either an algebraic approach or a modal-like semantics [6]). Several transference results have been obtained for these constructions, namely for soundness and completeness [11], several guises of interpolation and semi-decidability. Up to now, work on fibring sequent calculi has not been considered. A possibility