The paper considers the fundamental notions of many- valued logic together with some of the main trends of the recent development of infinite valued systems, often called mathematical fuzzy logics. Besides this logical approach also a more algebraic approach is discussed. And the paper ends with some hints toward applications which are based upon actual theoretical considerations about infinite valued logics. Key words: mathematical fuzzy logic, algebraic semantics, continuous t-norms, left-continuous t-norms, Pavelka-style fuzzy logic, fuzzy set theory, non-monotonic fuzzy reasoning 1 Basic ideas 1.1 From classical to many-valued logic Logical systems in general are based on some formalized language which includes a notion of well formed formula, and then are determined either semantically or syntactically. That a logical system is semantically determined means that one has a notion of interpretation or model 1 in the sense that w.r.t. each such interpretation every well formed formula has some (truth) value or represents a function into

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this article self-contained.

The paper considers some of the main trends of the recent development of mathematical fuzzy logic as an important tool in the toolbox of approximate reasoning techniques. Particularly the focus is on fuzzy logics as systems of formal logic constituted by a formalized language, by a semantics, and by a calculus for the derivation of formulas. Besides this logical approach also a more algebraic approach is discussed. And the paper ends with some hints toward applications which are based upon these theoretical considerations. Key words: mathematical fuzzy logic, algebraic semantics, continuous t-norms, left-continuous t-norms, Pavelka-style fuzzy logic, fuzzy set theory, non-monotonic fuzzy reasoning 1

In [6], C. C. Chang introduced a natural generalization of Lukasiewicz infinite valued propositional logic L. In this logic the truth values are extended from the interval [0,1] to the interval [-1,1]. We will call L # the logic whose designated values are those greater or equal than 0. (Chang calls this logic p # [0].) In this semantics, for a truth assignment v the value of the negation is v(#) = -v(#) . This implies that there are sentences for which v(#) = v(#) = 0 , that is, both sentences are tautologies. Moreover, the sentence # #) is not a tautology so # L # is paraconsistent. Two are the main results of this paper. First we axiomatize the system # L # 0 , the logic whose only designated truth value is 0, that is, the paraconsistent sentences of L # . Then, we prove that the categories # , whose objects are MV --algebras and MV # -- # Funding for the first author has been provided by FONDECYT grant 199--0433 and FOMEC. 1 algebras respectively, with their corresponding morphisms, are equivalent. These categories are associated with Lukasiewicz' infinite valued calculus and with Chang's logic L # , respectively. 1

. # Lukasiewicz's four-valued modal logic is surveyed and analyzed, together with # Lukasiewicz's motivations to develop it. A faithful interpretation of it into classical (non-modal) two-valued logic is presented, and some consequences are drawn concerning its classification and its algebraic behaviour. Some counter-intuitive aspects of this logic are discussed under the light of the presented results, # Lukasiewicz's own texts, and related literature. 1 Introduction The Polish philosopher and logician Jan # Lukasiewicz (Lwow, 1878 -- Dublin, 1956) is one of the fathers of modern many-valued logic, and some of the systems he introduced are presently a topic of deep investigation. In particular his infinitely-valued logic belongs to the core systems of mathematical fuzzy logic as a logic of comparative truth, see [5, 15, 14, 16]. However, it must be stressed here that # Lukasiewicz's logical work bears also a special relationship to modal logic. Actually, modal notions were part of #...

Abstract: We introduce two chains of unary operations in the MVn algebra of Revaz Grigolia; they will be used in establishing many connections between these algebras and n-valued Lukasiewicz-Moisil algebras (LMn algebras for short). The study has four parts. It is by and large self-contained. The main result of the rst part is that MV4 algebras coincide with LM4 algebras. The larger class of \relaxed"-MVn algebras is also introduced and studied. This class is related to the class of generalized LMn pre-algebras. The main results of the second part are that, for n 5, any MVn algebra is an LMn algebra and that the canonical MVn algebra can be identi ed with the canonical LMn algebra. In the third part, the class of good LMn algebras is introduced and studied and it is proved that MVn algebras coincide with good LMn algebras. In the present fourth part, the class of-proper LMn algebras is introduced and studied.-proper LMn algebras coincide (can be identi ed) with Cignoli's proper nvalued Lukasiewicz algebras. MVn algebras coincide with-proper LMn algebras (n 2). We also give the construction of an LM3 (LM4) algebra from the odd (respectively even)-valued LMn algebra (n 5), which proves that LM4 algebras are as much important than LM3 algebras; MVn algebras help to see this point.

It is a well-known fact that MV-algebras, the algebraic counterpart of ̷Lukasiewicz logic, correspond to a certain type of partial algebras: lattice-ordered effect algebras fulfilling the Riesz decomposition property. The latter are based on a partial, but cancellative addition, and we may construct from them the representing ℓ-groups in a straightforward manner. In this paper, we consider several logics differing from ̷Lukasiewicz logics in that they contain further connectives: the P̷L-, P̷L ′-, P̷L ′ △-, and ̷LΠ-logics. For all their algebraic counterparts, we characterise the corresponding type of partial algebras. We moreover consider the representing f-rings. All in all, we get three-fold correspondences: the total algebras- the partial algebras- the representing rings. 1

In this paper we study the logic ̷L ∗ , introduced by C. C.Chang as a natural extension of ̷Lukasiewicz ’ logic ̷L. This logic has positive and negative truth values in the real number interval [−1, 1]. We study deductive filters, we prove a deduction theorem and give detailed proofs of the soundness and completeness theorems. In the last section, we prove that the tautology problem for the logic ̷L ∗ is co–NP. This paper is to be considered a continuation of the paper MV ∗ –Algebras, by the same authors and appearing in this same volume. In that paper we study a class of algebras introduced by Chang as what is now known as an equivalent algebraic semantics for the logic ̷L. For most definitions and other algebraic concepts, the reader is referred to that paper.

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