1.1 Contradictoriness and inconsistency, consistency and non-contradictoriness In traditional logic, contradictoriness (the presence of contradictions in a theory or in a body of knowledge) and triviality (the fact that such a theory

Fibring is recognized as one of the main mechanisms in combining logics, with great signicance in the theory and applications of mathematical logic. However, an open challenge to bring is posed by the collapsing problem: even when no symbols are shared, certain combinations of logics simply collapse to one of them, indicating that bring imposes unwanted interconnections between the given logics. Modulated bring allows a ner control of the combination, solving the collapsing problem both at the semantic and deductive levels. Main properties like soundness and completeness are shown to be preserved, comparison with bring is discussed, and some important classes of examples are analyzed with respect to the collapsing problem. 1

Fibring is de ned as a mechanism for combining logics with a rstorder base, at both the semantic and deductive levels. A completeness theorem is established for a wide class of such logics, using a variation of the Henkin method that takes advantage of the presence of equality and inequality in the logic. As a corollary, completeness is shown to be preserved when bring logics in that class. A modal rst-order logic is obtained as a bring where neither the Barcan formula nor its converse hold.

Society Semantics, introduced in [5] by W. Carnielli and M. LimaMarques, is a method for obtaining new logics from the combination of agents (valuations) of a given logic. The goal of this paper is to present several generalizations of this method, as well as to show some applications to many-valued logics. After a reformulation of Society Semantics in a wider setting, we develop in detail two examples of application of the new formalism, characterizing a hierarchy of paraconsistent logics called P n (for n 2 N) and a hierarchy of paracomplete logics I n (for n 2 N). We also propose three increasing generalizations, obtaining Society Semantics for several many-valued logics, including a hierarchy of logics called I n P k which are both paraconsistent and paracomplete. Keywords: society semantics, paraconsistent logics, paracomplete logics, manyvalued logics, combinations of logics, agents.

wing the ideas in [4], to cope with possible non{truth{functionality of constructors. In the spirit of the theory of institutions and general logics [8, 9], we consider a logic to consist of an indexing functor to a suitable category of logic systems. In our case, the logic systems of interest are non{truth{functional (ntf) rooms . For simplicity, we shall only work at this level of abstraction. As shown in [3], everything can be smoothly lifted to the fully edged indexed case. In the sequel, AlgSig' denotes the category of algebraic many sorted signatures with a distinguished sort ' (for formulae) and morphisms preserving it. Given one such signature , we denote by Alg() the category of {algebras and {algebra homomorphisms, and by cAlg() the class of all pairs hA; i with A a<

generalizes the common situation when truth-values are ordered, we require a whole Tarskian closure operation as in [2]. In the sequel, AlgSig denotes the category of algebraic many sorted signatures with a distinguished sort (for formulae) and morphisms preserving it. Given such a signature hS; Oi, we denote by Alg(hS; Oi) the category of hS; Oi- algebras, and by cAlg(hS; Oi) the class of all pairs hA; i with A 2 jAlg(hS; Oi)j and a closure operation on A . Denition 1. A layered parchment is a tuple P = hSig; L; Mi where: { Sig is a category (of abstract<F13

A well-known result by W\'ojcicki-Lindenbaum shows that any tarskian logic is many-valued, and another result by Suszko shows how to provide 2-valued semantics to these very same logics. This paper investigates the question of obtaining 2-valued semantics for many-valued logics, including paraconsistent logics, in the lines of the so-called ``Suszko's Thesis". We set up the bases for developing a general algorithmic method to transform any truth-functional finite-valued semantics satisfying reasonable conditions into a computable quasi tabular 2-valued semantics, that we call dyadic. We also discuss how ``Suszko's Thesis" relates to such a method, in the light of truth-functionality, while at the same time we reject an endorsement of Suszko's philosophical views about the misconception of many-valued logics.

Abstract. The semantic collapse problem is perhaps the main difficulty associated to the very powerful mechanism for combining logics known as fibring. In this paper we propose cryptofibred semantics as a generalization of fibred semantics, and show that it provides a solution to the collapsing problem. In particular, given that the collapsing problem is a special case of failure of conservativeness, we formulate and prove a sufficient condition for cryptofibring to yield a conservative extension of the logics being combined. For illustration, we revisit the example of combining intuitionistic and classical propositional logics.

Fibring is a metalogical constructor that permits to combine different logics by operating on their deductive systems under certain natural restrictions, as for example that the two given logics are presented by deductive systems of the same type. Under such circumstances, fibring will produce a new deductive system by means of the free use of inference rules from both deductive systems, provided the rules are schematic, in the sense of using variables that are open for application to formulas with new linguistic symbols (from the point of view of each logic component). Fibring is a generalization of fusion, a less general but wider developed mechanism which permits results of the following kind: if each logic component is decidable (or sound, or complete with respect to a certain semantics) then the resulting logic heirs such a property. The interest for such preservation results for combining logics is evident, and they have been achieved in the more general setting of fibring in several cases. The Craig interpolation property and the Maehara interpolation have a special significance when combining logics, being related to certain problems of complexity theory, some properties of model theory and to the usual (global) metatheorem of deduction. When the peculiarities of the distinction between local and global deduction interfere, justifying what we call careful reasoning, the question of preservation of interpolation becomes more subtle and other forms of interpolation can be distinguished. These questions are investigated and several (global and local) preservation results for interpolation are obtained for fibring logics that fulfill mild requirements. AMS Classification: 03C40, 03B22, 03B45 1

Abstract. In this paper a variation of the fibred semantics of D. Gabbay called plain fibring is proposed, with the aim of combining logics given by matrix semantics. It is proved that the plain fibring of matrix logics is also a matrix logic. Moreover, it is proved that any logic obtained by plain fibring is a conservative extension of the original logics. It is also proposed a simpler version of plain fibring of matrix logics called direct union. This technique is applied to the study of the class of fuzzy logics defined by t-norms.