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Beyond Two: Theory and applications of multiplevalued logic
"... Abstract Algebraic Logic is a general theory of the algebraization of deductive systems arising as an abstraction of the wellknown LindenbaumTarski process. The notions of logical matrix and of Leibniz congruence are among its main building blocks. Its most successful part has been developed mainl ..."
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Abstract Algebraic Logic is a general theory of the algebraization of deductive systems arising as an abstraction of the wellknown LindenbaumTarski 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 socalled 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 MValgebra. 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 truthpreserving scheme, that is, the corresponding Þnitevalued 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 nonprotoalgebraic logics I can Þnally show that this logic too has a strong version, and that it coincides with the ordinary inÞnitevalued logic of Łukasiewicz. 1 1
Maximal and premaximal paraconsistency in the framework of threevalued semantics
 STUDIA LOGICA,
, 2011
"... Maximality is a desirable property of paraconsistent logics, motivated by the aspiration to tolerate inconsistencies, but at the same time retain from classical logic as much as possible. In this paper we introduce the strongest possible notion of maximal paraconsistency, and investigate it in the c ..."
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Maximality is a desirable property of paraconsistent logics, motivated by the aspiration to tolerate inconsistencies, but at the same time retain from classical logic as much as possible. In this paper we introduce the strongest possible notion of maximal paraconsistency, and investigate it in the context of logics that are based on deterministic or nondeterministic threevalued matrices. We show that all reasonable paraconsistent logics based on threevalued deterministic matrices are maximal in our strong sense. This applies to practically all threevalued paraconsistent logics that have been considered in the literature, including a large family of logics which were developed by da Costa’s school. Then we show that in contrast, paraconsistent logics based on threevalued properly nondeterministic matrices are not maximal, except for a few special cases (which are fully characterized). However, these nondeterministic matrices are useful for representing in a clear and concise way the vast variety of the (deterministic) threevalued maximally paraconsistent matrices. The corresponding weaker notion of maximality, called premaximal paraconsistency, captures the “core ” of maximal paraconsistency of all possible paraconsistent determinizations of a nondeterministic matrix, thus representing what is really essential for their maximal paraconsistency.
Monadic Fragments of Gödel Logics: Decidability and Undecidability Results
"... Abstract. The monadic fragments of firstorder Gödel logics are investigated. It is shown that all finitevalued monadic Gödel logics are decidable; whereas, with the possible exception of one (G↑), all infinitevalued monadic Gödel logics are undecidable. For the missing case G↑ the decidability of ..."
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Abstract. The monadic fragments of firstorder Gödel logics are investigated. It is shown that all finitevalued monadic Gödel logics are decidable; whereas, with the possible exception of one (G↑), all infinitevalued monadic Gödel logics are undecidable. For the missing case G↑ the decidability of an important subcase, that is well motivated also from an application oriented point of view, is proven. A tight bound for the cardinality of finite models that have to be checked to guarantee validity is extracted from the proof. Moreover, monadic G↑, like all other infinitevalued logics, is shown to be undecidable if the projection operator △ is added, while all finitevalued monadic Gödel logics remain decidable with △. 1
Fuzzy logic as a methodology for the treatment of vagueness
 The Logica Yearbook 2004
, 2005
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Mathematical fuzzy control. A survey of some recent results (submitted
"... The core point of fuzzy control approaches are finite lists of linguistic control rules. For computerbased automatic control these lists have to be transformed into control algorithms which can be realized on a computer. The main general idea of this fuzzy control approach is that such an algorithm ..."
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The core point of fuzzy control approaches are finite lists of linguistic control rules. For computerbased automatic control these lists have to be transformed into control algorithms which can be realized on a computer. The main general idea of this fuzzy control approach is that such an algorithm should yield a fuzzy subset of the output space of the control problem if confronted with a fuzzy subset of the input space. This paper surveys mathematical problems which are connected with, and arose out of these basic ideas. The main formal tools used in these mathematical considerations are fuzzy sets and fuzzy relations together with some generalized, viz. manyvalued logic which underlies these considerations. And the essential way of understanding the mathematical context of fuzzy control is to look at it as an interpolation problem: one has to determine a fuzzy control function out of a finite list of interpolation nodes.
On triangular norm based axiomatic extensions of the Weak Nilpotent Minimum logic
"... In this paper we carry out an algebraic investigation of the Weak Nilpotent Minimum logic (WNM) and its tnorm based axiomatic extensions. We consider the algebraic counterpart of this logic, the variety of WNMalgebras (WNM) and we prove that it is locally finite, so all its subvarieties are gener ..."
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In this paper we carry out an algebraic investigation of the Weak Nilpotent Minimum logic (WNM) and its tnorm based axiomatic extensions. We consider the algebraic counterpart of this logic, the variety of WNMalgebras (WNM) and we prove that it is locally finite, so all its subvarieties are generated by finite chains. We give criteria to compare varieties generated by finite families of WNMchains, in particular varieties generated by standard WNMchains, or equivalently tnorm based axiomatic extensions of WNM, and study their standard completeness properties. We also characterize the generic WNMchains, i.e. those that generate the variety WNM, and we give finite axiomatizations for some tnorm based extensions of WNM.
Xorimplications and eimplications: classes of fuzzy implications based on fuzzy xor
 Electronic Notes in Theoretical Computer Science (ENTCS
"... The main contribution of this paper is to introduce an autonomous definition of the connective “fuzzy exclusive or ” (fuzzy Xor, for short), which is independent from others connectives. Also, two canonical definitions of the connective Xor are obtained from the composition of fuzzy connectives, and ..."
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The main contribution of this paper is to introduce an autonomous definition of the connective “fuzzy exclusive or ” (fuzzy Xor, for short), which is independent from others connectives. Also, two canonical definitions of the connective Xor are obtained from the composition of fuzzy connectives, and based on the commutative and associative properties related to the notions of triangular norms, triangular conorms and fuzzy negations. We show that the main properties of the classical connective Xor are preserved by the connective fuzzy Xor, and, therefore, this new definition of the connective fuzzy Xor extends the related classical approach. The definitions of fuzzy Xorimplications and fuzzy Eimplications, induced by the fuzzy Xor connective, are also studied, and their main properties are analyzed. The relationships between the fuzzy Xorimplications and the fuzzy Eimplications with automorphisms are explored.
Basics of a formal theory of fuzzy partitions
 EUSFLAT LFA 2005
, 2005
"... A theory of fuzzy partitions is an important part of any theory meant to provide a formal framework for fuzzy mathematics. In [3], Henkinstyle higherorder fuzzy logic is introduced and proposed as a foundational theory for fuzzy mathematics. Here we investigate the properties of fuzzy partitions w ..."
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A theory of fuzzy partitions is an important part of any theory meant to provide a formal framework for fuzzy mathematics. In [3], Henkinstyle higherorder fuzzy logic is introduced and proposed as a foundational theory for fuzzy mathematics. Here we investigate the properties of fuzzy partitions within its formal framework. We follow closely the methodology of [2]. Therefore the notions introduced here are inspired by (and deduced from) the corresponding notions of classical mathematics. Sometimes they coincide with already known notions in fuzzy literature. However, we are usually more general (we work in arbitrary fuzzy logic), more expressive (we deal with the graded properties of fuzzy relations, as in [7]), and the proofs are more elegant (resembling the classical proofs).