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

The logics of formal inconsistency (LFIs) are paraconsistent logics which permit us to internalize the concepts of consistency or inconsistency inside our object language, introducing new operators to talk about them, and allowing us, in principle, to logically separate the notions of contradictoriness and of inconsistency. We present the formal definitions of these logics in the context of General Abstract Logics, argue that they in fact represent the majority of all paraconsistent logics existing up to this point, if not the most exceptional ones, and we single out a subclass of them called C-systems, as the LFIs that are built over the positive basis of some given consistent logic. Given precise characterizations of some received logical principles, we point out that the gist of paraconsistent logic lies in the Principle of Explosion, rather than in the Principle of Non-Contradiction, and we also sharply distinguish these two from the Principle of Non-Triviality, considering the next various weaker formulations of explosion, and investigating their interrelations. Subsequently, we present the syntactical formulations of some of the main C-systems based on classical logic, showing how several well-known logics in the literature can be recast as such a kind of C-systems, and carefully study their properties and shortcomings, showing for instance how they can be used to faithfully

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

This paper discusses how to define logics as deductive limits of sequences of other logics. The case of da Costa's hierarchy of increasingly weaker paraconsistent calculi, known as $C_n$, $\leq n\leq\omega$, is carefully studied. The calculus $C_\omega$, in particular, constitutes no more than a lower deductive bound to this hierarchy, and differs considerably from its companions. A long standing problem in the literature (open for more than 35 years) is to define the deductive limit to this hierarchy, that is its greatest lower deductive bound. The calculus $C_{min}$, stronger than $C_\omega$, is first presented as a step towards this limit. As an alternative to the bivaluation semantics of $C_{min}$ presented thereupon, possible-translations semantics are then introduced and suggested as the standard technique both to give this calculus a more reasonable semantics and to derive some interesting properties about it. Possible-translations semantics are then used to provide both a semantics and a decision procedure for $C_{Lim}$, the real deductive limit of da Costa’s hierarchy. Possible-translations semantics also make it possible to characterize a precise sense of duality: as an example, $D_{min}$ is proposed as the dual to $C_{min}$.

This paper obtains an effective method which assigns two-valued semantics to every finite-valued truth-functional logic (in the direction of the so-called “Suszko’s Thesis”), provided that its truth-values can be individualized by means of its linguistic resources. Such two-valued semantics permit us to obtain new tableau proof systems for a wide class of finitevalued logics, including the main many-valued paraconsistent logics. 1

This is an initial systematic study of the properties of negation from the point of view of abstract deductive systems. A unifying framework of multiple-conclusion consequence relations is adopted so as to allow us to explore symmetry in exposing and matching a great number of positive contextual sub-classical rules involving this logical constant ---among others, well-known forms of proof by cases, consequentia mirabilis and reductio ad absurdum. Finer definitions of paraconsistency and the dual paracompleteness can thus be formulated, allowing for pseudo-scotus and ex contradictione to be di#erentiated and for a comprehensive version of the Principle of Non-Triviality to be presented. A final proposal is made to the e#ect that ---pure positive rules involving negation being often fallible--- a characterization of what most negations in the literature have in common should rather involve, in fact, a reduced set of negative rules.

The logics of formal inconsistency (LFIs) are logics that allow to explicitly formalize the concepts of consistency and inconsistency by means of formulas of their language. Contradictoriness, on the other hand, can always be expressed in any logic, provided its la nguage includes a symbol for negation. Besides being able to represent the distinction between contradiction and inconsistency, LFIs are non-explosive logics, in the sense that a contradiction does not entail arbitrary statements, but yet are gently explosive, in the sense that, adjoining the additional requirement of consistency, then contradictoriness does cause explosion. Several logics can be seen as LFIs, among them the great majority of paraconsistent logics developed under the Brazilian tradition, as well as the sytems developed under the Polish tradition. We present here their semantical interpretations by way of possible-translations semantics, stressing their significance and applications to human reasoning and machine reasoning. We also give tableaux systems for some important LFIs: bC, Ci and LFI1.