Nilpotent Minimum Research Articles

Prelinearity in (quasi-)Nelson logic

The algebraic theory of quasi-Nelson logic, a non-involutive generalization of Nelson's constructive logic with strong negation, has been shown to be surprisingly rich in a series of recent papers. In the present paper we bring quasi-Nelson logic into the fuzzy setting by adding the prelinearity axiom to it. We observe that the resulting system is an extension of the well-known Weak Nilpotent Minimum logic, as well as a rotation logic in the sense of recent work by P. Aglianò and S. Ugolini. We characterize the algebraic models of prelinear quasi-Nelson logic as twist-structures over Gödel algebras endowed with a nucleus operator and use the insight thus gained to look at subvarieties corresponding to extensions of well-known fuzzy systems. Our study of the quasi-Nelson negation in a prelinear setting also allows us to show that the variety of prelinear quasi-Nelson algebras is generated by a single standard algebra, thus obtaining a single chain completeness theorem for the logic.

Monadic NM-algebras: an algebraic approach to monadic predicate nilpotent minimum logic

AbstractIn this paper, we further study the variety of monadic nilpotent minimum (NM)-algebras and their corresponding logic. In order to solve the drawback of monadic NM-algebras, we review some well-known classes of monadic t-norm-based fuzzy logical algebras and then revise the axiomatic system of monadic NM-algebras. Then we show that the variety of monadic NM-algebras is the equivalent algebraic semantics of monadic predicate fuzzy logic $\textbf {mNM}_{\forall }$, which is equivalent to the modal fuzzy logic $\textbf {S5(NM)}$. Moreover, we show that the propositional case of the modal fuzzy logic $\textbf {S5(NM)}$, which is $\textbf{S5}^{\prime}\textbf{(NM),}$ is also complete with respect to the variety of monadic NM-algebras in the sense of Blok and Pigozzi and obtain a necessary and sufficient condition for this logic to be semilinear. Finally, we give some representations of monadic NM-algebras. In particular, we give some characterizations of representable and directly indecomposable monadic NM-algebras.

Logics of formal inconsistency based on distributive involutive residuated lattices

Abstract The aim of this paper is to develop an algebraic and logical study of certain paraconsistent systems, from the family of the logics of formal inconsistency (LFIs), which are definable from the degree-preserving companions of logics of distributive involutive residuated lattices ($\textrm {dIRL}$s) with a consistency operator, the latter including as particular cases, Nelson logic ($\textsf {NL}$), involutive monoidal t-norm based logic ($\textsf {IMTL}$) or nilpotent minimum ($\textsf {NM}$) logic. To this end, we first algebraically study enriched dIRLs with suitable consistency operators. In fact, we consider three classes of consistency operators, leading respectively to three subquasivarieties of such expanded residuated lattices. We characterize the simple and subdirectly irreducible members of these quasivarieties, and we extend Sendlewski’s representation results for the case of Nelson lattices with consistency operators. Finally, we define and axiomatize the logics of three quasivarieties of $ \textrm {dIRL}$s and their corresponding degree-preserving companions that belong to the family of LFIs.

Nilpotent Minimum Logic NM and Pretabularity

This paper deals with pretabularity of fuzzy logics. For this, we first introduce two systems NMnfp and NM½, which are expansions of the fuzzy system NM (Nilpotent minimum logic), and examine the relationships between NMnfp and the another known extended system NM—. Next, we show that NMnfp and NM½ are pretabular, whereas NM is not. We also discuss their algebraic completeness.

Finitary Extensions of the Nilpotent Minimum Logic and (Almost) Structural Completeness

In this paper we study finitary extensions of the nilpotent minimum logic (NML) or equivalently quasivarieties of NM-algebras. We first study structural completeness of NML, we prove that NML is hereditarily almost structurally complete and moreover NM $$^{-}$$ , the axiomatic extension of NML given by the axiom $$\lnot (\lnot \varphi ^{2})^{2}\leftrightarrow (\lnot (\lnot \varphi )^{2})^{2}$$ , is hereditarily structurally complete. We use those results to obtain the full description of the lattice of all quasivarieties of NM-algebras which allow us to characterize and axiomatize all finitary extensions of NML.

Linguistic classification: T-norms, fuzzy distances and fuzzy distinguishabilities

Back in 1967 the linguist Ž. Muljačić used an additive distance between ill-defined linguistic features which is a forerunner of the fuzzy Hamming distance between strings of truth values in standard fuzzy logic. Here we show that if the logical frame is changed one obtains additive distances which are either sorely inadequate, as in the Łukasiewicz or probabilistic case, or coincide with the distance originally envisaged by Muljačić, as happens with a whole class of T-norms (abstract logical conjunctions) which includes the nilpotent minimum. All this strengthens the role of Muljačić distances in linguistic clustering and of Muljačić distinguishabilities (a notion subtly different from distances, but quite inalienable) in linguistic evolution. As a preliminary example we re-take and re-examine Muljačić original data.

Schematic Extensions of MTL by Adding Weak Divisibility Axiom

MTL is a Monoidal t-norm based logic introduced by Esteva and Godo by omitting divisibility axiom from H´ajek's Basic logic (BL). Many logics can be obtained by adding axioms to MTL logic. Logic system WBL is obtained by adding weak divisibility axiom to logic system MTL. Logic system WMV is obtained by adding involution axiom to logic system WBL. WBL-algebra corresponding to logic system WBL and WMV-algebra to logic system WMV are defined respectively. It is proved that the both of logic system Luk and logic system Nilpotent minimum (NM) are the schematic extensions of logic system WMV. Weak Wajsberg algebra and the simplified form of logic system WMV are obtained.

On Deductive Interpolation for the Weak Nilpotent Minimum logic

The Weak Nilpotent Minimum logic WNM was introduced by Esteva and Godo in [6] and shown to be the logic of the class of weak nilpotent minimum triangular norms. In this article, we prove that WNM admits the Deductive Interpolation Property by showing through a model-theoretic argument that the corresponding variety of algebras has the Amalgamation Property.

A temporal semantics for Nilpotent Minimum logic

In [6] a connection among rough sets (in particular, pre-rough algebras) and three-valued Łukasiewicz logic Ł3 is pointed out. In this paper we present a temporal like semantics for Nilpotent Minimum logic NM [22,17], in which the logic of every instant is given by Ł3: a completeness theorem will be shown. This is the prosecution of the work initiated in [5] and [1], in which the authors construct a temporal semantics for the many-valued logics of Gödel [24,14] and Basic Logic [27].

Proof theory for locally finite many-valued logics: Semi-projective logics

We extend the methodology in Baaz and Fermüller (1999) [5] to systematically construct analytic calculi for semi-projective logics—a large family of (propositional) locally finite many-valued logics. Our calculi, defined in the framework of sequents of relations, are proof search oriented and can be used to settle the computational complexity of the formalized logics. As a case study we derive sequent calculi of relations for Nilpotent Minimum logic and for Hajek’s Basic Logic extended with the n-contraction axiom (n≥1). The introduced calculi are used to prove that the decidability problem in these logics is Co-NP complete.

First-order Nilpotent minimum logics: first steps

Inspired by the work done by Baaz et al. (Ann Pure Appl Log 147(1–2): 23–47, 2007; Lecture Notes in Computer Science, vol 4790/2007, pp 77–91, 2007) for first-order Godel logics, we investigate Nilpotent Minimum logic NM. We study decidability and reciprocal inclusion of various sets of first-order tautologies of some subalgebras of the standard Nilpotent Minimum algebra, establishing also a connection between the validity in an NM-chain of certain first-order formulas and its order type. Furthermore, we analyze axiomatizability, undecidability and the monadic fragments.

Algebra and Probability in Many-Valued Reasoning

Many-valued reasoning is a well-established domain of investigation with thriving connections to several areas of logic, mathematics, philosophy, and computer science. In May 2009, an international workshop on Algebra and Probability in Many-Valued Logics was held in Darmstadt, Germany. Established as well as younger researchers in the field gathered together to foster progress on various aspects of many-valued reasoning, with special emphasis on the topics indicated by the meeting's title. Traditionally, algebraic techniques play an important role in the investigation of manyvalued logical systems. More recently, efforts to extend the scope of the theory of probabilities to such logics are attracting much attention. As a follow-up to the workshop, we are very pleased to offer the present special issue of Studia Logica. The papers herein contained were submitted to the journal in response to the call for paper that was issued after the workshop. Some of them are related to presentations that their authors made at the meeting; others were submitted independently of the conference. In all cases, submitted papers have been reviewed, and then selected or rejected, according to the usual standards of the journal. The paper by Aguzzoli and Gerla is concerned with the logic of nilpotent minimum (NM), a schematic extension of the logic of all left-continuous triangular norms introduced by Esteva and Godo. The authors establish for the finitely generated Tarski-Lindenbaum algebras of NM a key result about generalised assignments of probabilities, known as states. Namely, they show that an appropriate axiomatisation in the style of Kolmogorov for such states provides a precise mathematical formulation of the intuitive notion of average degree of truth of a formula, where the average is computed by integrating the function expressed by the formula with respect to an appropriate measure. Results of this type had previously been proved for

Probability Measures in the Logic of Nilpotent Minimum

We axiomatize the notion of state over finitely generated free NM-algebras, the Lindenbaum algebras of pure Nilpotent Minimum logic. We show that states over the free n-generated NM-algebra \({{\fancyscript{NM}_{n}}}\) exactly correspond to integrals of elements of \({{\fancyscript{NM}_{n}}}\) with respect to Borel probability measures.

On expansions of WNM t-norm based logics with truth-constants

This paper focuses on completeness results about generic expansions of propositional weak nilpotent minimum (WNM) logics with truth-constants. Indeed, we consider algebraic semantics for expansions of these logics with a set of truth-constants { r ¯ | r ∈ C } , for a suitable countable C ⊆ [ 0 , 1 ] , and provide a full description of completeness results when: (i) the t-norm is a weak nilpotent minimum satisfying the finite partition property and (ii) the set of truth-constants covers all the unit interval in the sense that each interval of the partition contains values of C in its interior.

Continuity of triple I methods based on several implications

This paper focuses on the continuity problem of fully implicational triple I methods for fuzzy reasoning. Based on the residual implications generated by continuous triangular norms, the residual implication generated by nilpotent minimum with the standard negation and the Zadeh implication, respectively, continuity and uniform continuity properties of triple I methods are examined in Hamming and uniform metrics. We also investigate the continuity of the model of a fuzzy if-then rule corresponding to the triple I methods.

Spectral Duality for Finitely Generated Nilpotent Minimum Algebras, with Applications

We establish a categorical duality for the finitely generated Lindenbaum-Tarski algebras of propositional nilpotent minimum logic. The latter's conjunction is semantically interpreted by a left-continuous (but not continuous) triangular norm; implication is obtained through residuation. Our duality allows one to transfer to nilpotent minimum logic several known results about inutitionistic logic with the prelinearity axiom (also called Godel-Dummett logic), mutatis mutandis. We give several such applications.

NMŁ, a schematic extension of F.Esteva and L.Godo's logic MTL

A schematic extension NMŁ of F.Esteva and L.Godo's Logic MTL is introduced in this paper. Based on a new left-continuous but discontinuous t-norm, which was proposed by S.Jenei and can be regarded as a kind of distorted nilpotent minimum, the semantics of NMŁ is interpreted and the standard completeness theorem of NMŁ is proved. The fact that the maximum and the minimum are definable from the negation and implication in NMŁ and NM is discovered, which also leads to a modification of the NM axiom system.

New family of triangular norms via contrapositive symmetrization of residuated implications

In this paper a certain transformation of triangular norms (called N-annihilation) is studied. This problem is strongly related to the contrapositive symmetry property of residuated implications. We characterize those continuous triangular norms where the annihilated binary operation is a triangular norm. Some surprising properties of nilpotent triangular norms are presented e.g. the nilpotent minimum is described as limit of nilpotent triangular norms. As a consequence, a new family of triangular norms (called nilpotent ordinal sums) owing several attractive properties is discovered. The new family contains the nilpotent triangular norms and the nilpotent minimum as extremal cases and can be admitted into investigations in the theory of Girard monoids as well.

Contrapositive symmetry of fuzzy implications

Contrapositive symmetry of R- and QL-implications defined from t-norms, t-conorms and strong negations is studied. For R-implications, characterizations of contrapositive symmetry are proved when the underlying t-norm satisfies a residuation condition. Contrapositive symmetrization of R-implications not having this property makes it possible to define a conjunction so that the residuation principle is preserved. Cases when this associated conjunction is a t-norm are characterized. As a consequence, a new family of t-norms (called nilpotent minimum) owing several attractive properties is discovered. Concerning QL-implications, contrapositive symmetry is characterized by solving a functional equation. When the underlying t-conorm is continuous and the t-norm is Archimedean, the t-conorm must be isomorphic to the Lukasiewicz one, while the t-norm must be isomorphic to a member from the well-known Frank family of t-norms. Finally, contrapositive symmetry for some new families of fuzzy implications is investigated.