Splitting Logics

This paper addresses the question of factoring a logic into families of (generally simpler) components, estimating the top– down perspective, splitting, versus the bottom–up, splicing. Three methods are carefully analyzed and compared: possible–translations semantics, nondeterministic semantics and plain fibring (joint with its particularization, direct union of matrices). The possibilities of inter–definability between these methods are also examined. Finally, applications to some well–known logic systems are given and their significance evaluated. 1 Splitting logics, splicing logics and their use One of fundamental questions in the philosophy of logic, “Why there are so many logics instead of just one?” (or even, instead of none), is naturally counterposed by another: If there are indeed many logics, are they excluding alternatives, or are they compatible? Is it possible to combine them into coherent systems, with the purpose of using them in applications and of taking profit of this composionality capacity to better understand logics? And if we can compose, why not decompose logics? One of the first, and one of the most general, approaches for the question of combining logics is the concept of fibring introduced by D. Gabbay in [Gabbay, 1996]. Fibring is able to combine logics creating new and expressive systems, in the direction of what we call splicing logics. The other direction is called splitting logics. Though, as we shall argue, there is no essential distinction between splicing and splitting, there are important differences with respect to the aims one may have in mind. Splitting as a process for investigating logics has been under–appreciated, and we intend to stress here some results and some views that we believe to be of interest for the sake of splitting in the trade of combining logics. 1 1The process tags “splicing” and “splitting” logics were introduced in [Carnielli and Coniglio, 1999]. As a noun, “splitting” is also used in the literature in a completely different sense, viz., to designate a “logic that splits a class”, as e.g. in W.J. Blok, “On the degree of incompleteness of modal logics” (abstract). Bulletin of the Section of Logic of the Polish Academy of Sciences, 7(4):167-175, December 1978. ¡book title¿, 1–26. c © ¡year¿, the author. 2 Walter Carnielli and Marcelo E. Coniglio Possible–translations semantics were proposed in [Carnielli, 1990], and were designed to help solve the problem of assigning semantic interpretations to non–classical logics. The idea behind possible–translations semantics is to build an interpretation for a given logic by taking into account a specific set of translations from its formulas into a class of simpler logics, with known or acceptable semantics. For a certain time it was even called “non– deterministic semantics” (as in [Carnielli and D’Ottaviano, 1997]), due to the apparent ambiguity of having several translations from the same domain. Such semantics comprise a flexible and widely applicable tool for endowing logics with recursive and palatable semantic interpretation: detailed examples will be given in Section 3, but it is worth mentioning that several paraconsistent logics (as fragments of classical logic) which are not characterizable by finite matrices can be characterized by suitable combinations of many–valued logics. The reader is invited to check details for the case of N. da Costa’s hierarchy {Cn}n∈N in [Carnielli, 2000] and [Marcos, 1999]. Examples of possible–translations semantics go in the direction of splitting, illustrating how a complex logic can be analyzed into less complex factors. We also analyze here the nondeterministic semantics (see Section 4) and the direct union of matrices and plain fibring (see Section 5). The traditional notion of matrix semantics, due to J. Lukasiewicz and E. Post, is also briefly reviewed in Section 3. Matrix semantics generalize algebraic semantics, as used in algebraic logic. They constitute a method for assigning semantic meaning for logics, as well as a method for defining logical systems. The fact that possible–translations semantics are a widely applicable tool is witnessed by our results below, which show that both nondeterministic semantics and matrix semantics are particular cases of possible–translations semantics. As the notion of matrix semantics proves to be adequate for any structural deductive system, so are possible–translations semantics. Another application, better suited for many–valued logics, is the concept of society semantics (cf. [Carnielli and Lima-Marques, 1999] and [Fernández and Coniglio, 2003]) that we do not treat here. Possible–translations semantics (and their particular cases) work not only as general tools for assigning semantics for logics, but also as a tool for splitting logics as well. In the same manner they work for the direct unions of matrices and the plain fibring, which are not reducible to possible–translations semantics. The particular cases are not to be discounted by any means: on the contrary, they are significant, specially when regarded from the splitting standpoint, in the measure that they provide operative methods for com-