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In this paper, logics are conceived as two-sorted rst-order structures, and we argue that this broad de nition encompasses a wide class of logics with theoretical interest as well as interest from the point of view of applications. The language, concepts and methods of model theory can thus be used to describe the relationship between logics through morphisms of structures called transfers. This leads to a formal framework for studying several properties of abstract logics and their attributes such as consequence operator, syntactical structure, and internal transformations.

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

Abstract Using methods of abstract logic and the theory of valuation, we prove that there is no paraconsistent negation obeying the law of double negation and such that ¬(a ∧¬a) is a theorem which can be algebraized by a technique similar to the Tarski-Lindenbaum technique. 135 1What are the features of a paraconsistent negation? Since paraconsistent logic was launched by da Costa in his seminal paper [4], one of the fundamental problems has been to determine what exactly are the theoretical or metatheoretical properties of classical negation that can have a unary operator not obeying the principle of noncontradiction, that is, a paraconsistent operator. What the result presented here shows is that some of these properties are not compatible with each other, so that in constructing a paraconsistent negation as close as possible to classical negation, we have to make a choice among classical properties compatible with the idea of paraconsistency. In particular, there is no paraconsistent negation more classical than all the others. The incompatibility appearing here is between theoretical properties (double negation and ¬(a ∧¬a) as a theorem) and a metatheoretical property (replacement theorem). One who chooses the theoretical properties will not be able to algebraize his system with the usual Tarski-Lindenbaum method and should use some alternative treatments such as that in da Costa [5]. On the other hand, one who chooses the metatheoretical property will have to sacrifice at least one fundamental theoretical property of negation, risking the possibility of dealing with an operator that is a modality rather than a negation. The result presented here is of the same kind as some previous results concerning the incompatibility between the replacement theorem and the paraconsistent logic C1 of [4]. It was soon realized that the replacement theorem is not valid in C1. Mortensen [14] proved that, in fact, it was impossible to define a nontrivial congruence in C1. Urbas [19] proved that the addition of the replacement theorem to

Abstract Algebraic Logic is a general theory of the algebraization of deductive systems arising as an abstraction of the well-known Lindenbaum-Tarski process. The notions of logical matrix and of Leibniz congruence are among its main building blocks. Its most successful part has been developed mainly by BLOK, PIGOZZI and CZELAKOWSKI, and obtains a deep theory and very nice and powerful results for the so-called protoalgebraic logics. I will show how the idea (already explored by WÓJCKICI and NOWAK) ofdeÞning logics using a scheme of “preservation of degrees of truth ” (as opposed to the more usual one of “preservation of truth”) characterizes a wide class of logics which are not necessarily protoalgebraic and provide another fairly general framework where recent methods in Abstract Algebraic Logic (developed mainly by JANSANA and myself) can give some interesting results. After the general theory is explained, I apply it to an inÞnite family of logics deÞned in this way from subalgebras of the real unit interval taken as an MV-algebra. The general theory determines the algebraic counterpart of each of these logics without having to perform any computations for each particular case, and proves some interesting properties common to all of them. Moreover, in the Þnite case the logics so obtained are protoalgebraic, which implies they have a “strong version ” deÞned from their Leibniz Þlters; again, the general theory helps in showing that it is the logic deÞned from the same subalgebra by the truth-preserving scheme, that is, the corresponding Þnite-valued logic in the most usual sense. However, for inÞnite subalgebras the obtained logic turns out to be the same for all such subalgebras and is not protoalgebraic, thus the ordinary methods do not apply. After introducing some (new) more general abstract notions for non-protoalgebraic logics I can Þnally show that this logic too has a strong version, and that it coincides with the ordinary inÞnite-valued logic of Łukasiewicz. 1 1

Fuzzy reasoning can provide techniques for representing and management the imprecision inherent in commonsense reasoning. But, like human reasoning, it conduces to inconsistences (inherent in the imprecise or incomplete knowledge) that might be solved in the frame of fuzzy logic, simulating the human behavior. In this paper we analyze this kind of conflicts and propose a non-monotonic fuzzy logic in order to solve it. Moreover, we show that many (non-monotonic) human reasoning patterns can be modeled by means of this "non-monotonic fuzzy reasoning". 1 Introduction Fuzzy Reasoning Systems have been shown to be an important tool for problems where, due to the complexity or the imprecision, classical tools are unsuccessful. The knowledge is represented as a set of fuzzy propositions expressing in a compact way the domain knowledge. Fuzzy consequences are then obtained using fuzzy inference rules. Nevertheless, what is going on when contradictory fuzzy consequences are obtained? In a firs...

. According to Frege's principle the denotation of a sentence coincides with its truth-value. The principle is investigated within the context of abstract algebraic logic, and it is shown that taken together with the deduction theorem it characterizes intuitionistic logic in a certain strong sense. A deductive system is an algebraic closed-set system over the set Fm of formulas over a language type . The Frege relation over a theory T of S, also called the interderivability relation over T , is dened by e S T = f h'; i 2 Fm 2 : T; ' `S and T; `S ' g. In the context of abstract algebraic logic e S T is interpreted as representing the identity-of-truthvalue relation for S relative to T . The Frege principle asserts that e S T also represents the identity-of-denotation relation relative to T and hence that e S T is compositional, i.e., e S T is a congruence relation on the formula algebra Fm . A deductive system is Fregean if the Frege principle holds. Fregean lo...

Abstract: A Quantified Autoepistemic Logic is axiomatized in a monotonic Modal Quantificational Logic whose modal laws are slightly stronger than S5. This Quantified Autoepistemic Logic obeys all the laws of First Order Logic and its L predicate obeys the laws of S5 Modal Logic in every fixed-point. It is proven that this Logic has a kernel not containing L such that L holds for a sentence if and only if that sentence is in the kernel. This result is important because it shows that L is superfluous thereby allowing the ori ginal equivalence to be simplified by eliminating L from it. It is also shown that the Kernel of Quantified Autoepistemic Logic is a generalization of Quantified Reflective Logic, which coincides with it in the propositional case.

In accordance with Tarski point of view, in this chapter the theory of closure operators is proposed as a unifying tool for fuzzy logics. Indeed, let F be the set of formulas of a given language. Then an abstract fuzzy logic is defined by a fuzzy semantics (i.e. a class of valuations of the formulas in F) and by a closure operator in the lattice of the fuzzy subsets of F (we call deduction operator). One proves that Pavelka’s logic, similarity logic and graded consequence theory can be represented in this way.

invaluable support.