Recovering a Logic from Its Fragments by Meta-Fibring

Abstract

In this paper we address the question of recovering a logic system by combining two or more fragments of it. We show that, in general, by fibring two or more fragments of a given logic the resulting logic is weaker than the original one, because some meta-properties of the connectives are lost after the combination process. In order to overcome this problem, the categories Mcon and Seq of multiple-conclusion consequence relations and sequent calculi, respectively, are introduced. The main feature of these categories is the preservation, by morphisms, of meta-properties of the consequence relations, which allows, in several cases, to recover a logic by fibring of its fragments. The fibring in this categories is called meta-fibring. Several examples of well-known logics which can be recovered by meta-fibring its fragments (in opposition to fibring in the usual categories) are given. Finally, a general semantics for objects in Seq (and, in particular, for objects in Mcon) is proposed, obtaining a category of logic systems called Log. A general theorem of preservation of completeness by fibring in Log is also obtained. Mathematics Subject Classification (2000). Primary 99Z99; Secondary 00A00.

Combining fragments of classical logic: When are interaction principles needed?

“…In closing, it is worth noting that the essential role played by the saturation requirement in order to explain the semantics of the combined logics seems to suggest that the emergence of interaction principles is connected with a lack of expressiveness of the standard Tarskian framework for the study of logics (as hinted also in [11]), and that the outcome of the present investigation would be entirely different if we were to adopt multiple-conclusion logics, after [26].…”

Juxtaposition: A New Way to Combine Logics

“…51 SeeConiglio (2007) for a variant of fibring designed to preserve metarules. 51 SeeConiglio (2007) for a variant of fibring designed to preserve metarules.…”

Bridge principles and combined reasoning

“…Sometimes, the categorial fibring of two logics computed in categories defined which such a morphisms produces too weak logics; some examples will be given in Section 3. In [9] a category of sequent calculi is proposed in which morphisms preserve sequent rules instead of merely consequence relations. Such rules are a kind of meta-theorems of the logics, and so fibring of logics in this category produces stronger logics than those obtained by fibring using logic translations.…”

“…A second case of spontaneous emergence of a bridge principle, now related to (full) classical propositional logic, is the metafibring of the logic of classical negation and the logic of classical disjunction: as shown in [9], the law of excluded-middle p ∨ ¬p emerges unavoidably in the combined logic.…”