Abstract In this work, we provide constructions, characterizations, geometrical dualities and a McNaughton theorem for non-archimedean MV-algebras, which are the semantics of super-Łukasiewicz logics introduced by Komori.

Abstract We offer a new logic, called Epistemic Łukasiewicz logic of partial knowledge that is represented as multimodal epistemic Łukasiewicz logic $$K\text{\L} _\text {P}(n)$$ K Ł P ( n ) with n knowledge operators $$\Box _i$$ □ i $$(1 \le i \le n)$$ ( 1 ≤ i ≤ n ) interpreted in a non-archimedean monadic MV-algebra. We choose knowledge operators, which can be estimated by some grading (different kinds of knowledge): absolute knowledge or partial knowledge. We consider a very special type of partial knowledge. Actually, we take infinitesimal elements (the radical) of perfect MV-algebras as a range of this estimation. The choice of infinitesimal elements seems suitable for actual situations like a measure of partial information.

Abstract In this paper we aim at completing the study of $\sigma $-complete Riesz MV-algebras that started in Di Nola et al. (2018, J. Logic Comput., 28, 1275–1292). To do so, we discuss polynomials, algebraic geometry and dualities in the infinitary variety of such algebras. In particular, we characterize the free objects as algebras of Baire-measurable functions and we generalize two dualities, namely the Marra–Spada duality and the Gelfand duality, obtaining a duality with basically disconnected compact Hausdorff spaces and an equivalence with Rickart $C^*$-algebras.

Abstract It is introduced an immune dynamicn-valued Łukasiewicz logic $$ID{\L }_n$$ IDŁn on the base ofn-valued Łukasiewicz logic $${\L }_n$$ Łn and corresponding to it immune dynamic $$MV_n$$ MVn -algebra ( $$IDL_n$$ IDLn -algebra), $$1< n < \omega $$ 1<n<ω , which are algebraic counterparts of the logic, that in turn represent two-sorted algebras $$(\mathcal {M}, \mathcal {R}, \Diamond )$$ (M,R,◊) that combine the varieties of $$MV_n$$ MVn -algebras $$\mathcal {M} = (M, \oplus , \odot , \sim , 0,1)$$ M=(M,⊕,⊙,∼,0,1) and regular algebras $$\mathcal {R} = (R,\cup , ;, ^*)$$ R=(R,∪,;,∗) into a single finitely axiomatized variety resemblingR-module with “scalar” multiplication $$\Diamond $$ ◊ . Kripke semantics is developed for immune dynamic Łukasiewicz logic $$ID{\L }_n$$ IDŁn with application in immune system.

We extend Łukasiewicz logic obtaining the infinitary logic Infinitary Riesz Logic (⁠|$\mathcal{IR}$|Ł) whose models are algebras C(X, [0, 1]), where X is a basically disconnected compact Hausdorff space. Equivalently, our models are unit intervals in Dedekind |$\sigma $|-complete Riesz spaces with strong unit. The Lindenbaum–Tarski algebra of |$\mathcal{IR}$|Ł is, up to isomorphism, an algebra of [0, 1]-valued Borel functions. Finally, our system enjoys standard completeness with respect to the real interval [0, 1].

We study commutative idempotent semirings in general, and some examples in particular. We show that the class Red of semiring reducts of MV-algebras, although axiomatized by a first order theory, is not axiomatized by a geometric theory (in the topos-theoretic sense) or a universal-existential first order theory. Then we perform comparisons between the class Red, the class of all semirings, and some so-called exotic semirings.

Formal Concept Analysis (FCA) and its fuzzy extension have been widely used to arrange data into a lattice that is an effective data structure useful to address several aims, such as: data mining, ontology learning and merging, and so on. In literature it is possible to distinguish two main approaches to address fuzzy FCA implementation: the one-sided threshold and the fuzzy closure one. This work focuses on a specific definition of one-sided threshold algorithm and fuzzy closure one. Specifically, it shows that these methods can be unified, since the one-sided threshold approach can be seen as a specialization of the fuzzy closure. The lattice generated using one-sided fuzzy threshold approach is a substructure of the lattice generated using the fuzzy closure approach. In addition, an experimentation has been performed on both implementations of the fuzzy FCA, one-sided threshold and fuzzy closure. In particular, the results are compared in terms of running time and number of extracted fuzzy concepts by varying the t-norm function Łukasiewicz, Gödel, and Product.

We focus on Riesz MV-algebras, which are MV-algebras equipped with a multiplication by numbers in the real interval [0,1]. In analogy with a work in preparation for MV-algebras by the same authors, we consider for every integer n the Riesz MV-algebra of all continuous functions from the n-th power of [0,1] to [0,1] and the Riesz MV-subalgebras thereof. In particular we study the Riesz MV-subalgebras isomorphic to free Riesz MV-algebras with finitely many generators, possibly different from the usual linear models given by what we call Riesz-McNaughton functions (and which generalize McNaughton functions used in the case of MV-algebras). In doing this we characterise zerosets of Riesz-McNaughton functions by means of polyhedra, and we extend to Riesz MV-algebras a duality for MV-algebras exposed in a paper by Marra and Spada.

In this article, first we generalize from the MV algebra [0,1] to an arbitrary MV algebra A the well-known Galois connection (V,I) between the powerset of each power of [0,1] and the powerset of the corresponding free MV algebra. Then, in analogy with the Nullstellensatz of classical algebraic geometry, we study the closure operators obtained by composing the functors V and I.

Abstract In this paper we try to apply universal algebraic geometry to MV algebras, that is, we study “MV algebraic sets” given by zeros of MV polynomials, and their “coordinate MV algebras”. We also relate algebraic and geometric objects with theories and models taken in Łukasiewicz many valued logic with constants. In particular we focus on the structure of MV polynomials and MV polynomial functions on a given MV algebra.

Classical logic, as is well known, can be analyzed in a great part by algebraic methods using the Lindenbaum algebra obtained from the formal system. For example the completeness theorem for this logic becomes equivalent to the semisimplicity of the obtained Lindenbaum algebra. Since Chang [4, 5], Łukasiewicz logic has also been analyzed algebraically through the associated Lindenbaum type algebra, that is the algebra of equivalence classes obtained from the relation of provable equivalence. In this case this algebra is an MV-algebra [4]. Once again logical notions have an algebraic counterpart, for example, completeness relates strongly to semisimplicity [4, 5]. However, unlike the classical case where the algebras in question are Boolean and always semisimple, not all MV-algebras are semisimple. This fact, in a sense, enriches the theory of MV-algebras. Now every MV-algebra can be considered a Lindenbaum type algebra, namely an algebra associated to Łukasiewicz logic with additional axioms. Thus we can carry over to any MV-algebra various logical notions such as (in) completeness, consistency, satisfiability, etc. Two important logical notions are those of “formal consequence” and “semantical consequence”. The former just says that a wff α is deducible from a set of wff via the axioms and rules of inference, while the latter just says that every evaluation that “satisfies” all the members of also “satisfies” α. Informally call these relations F, S respectively; consider them as binary relations, Fα and Sα. Now the completeness theorem just states F = S. Thus we can talk about an MV-algebra being “complete” provided the associated relations F, S are equal.

We define the Lukasiewicz Transform by mean of the partition of unity defined in [4] and semimodules over the semiring reducts of an MV-algebra. Then we describe the "Lukasiewicz Transform Based" (LTB) algorithm for image processing, showing some results of its application.

Conditioning plays a central role, both from a theoretical and practical point of view, in domains such as logic and probability, or rule–based expert systems. In classical approaches to probability, there is the notion of "conditional probability" P(E|H), but usually there is no meaning given to E|H itself. In 1935 de Finetti 5 was the first to mention "conditional events" outside the function P. We shall refer to a concept of conditional event extensively discussed in 4, where the idea of de Finetti of looking at E|H, with H≠∅ (the impossible event), as a three–valued logical entity (true when both E and H are true, false when H is true and E is false, "undetermined" when H is false) is generalized (or better, in a sense, is given up) by letting the third "value" t(E, H)suitably depend on the given ordered pair(E, H) and not being just an undetermined common value for all pairs. Here an axiomatic definition is given of Partial Algebraic Conditional Spaces (PACS), that is a set of conditional events endowed with two partial operations (denoted by ⊕ and ⊙): we then show that the structure discussed through a betting scheme in 4 (i.e., a class of particular random variables with suitable partial sum and product) is a "natural" model of a PCAS. Moreover, it turns out that the map t(E, H) can be looked on – with this choice of the two operations ⊕ and ⊙ – as a conditional probability (in its most general sense related to the concept of coherence) satisfying the classic de Finetti – Popper axioms.

In fuzzy logic, every word or phrase describing uncertainty is represented by a real number from the interval [0, 1]. There are only denumerable many words and phrases and continuum many real numbers; thus, not every real number corresponds to some common sense degree of uncertainty. In this article, for several fuzzy logics, we describe which numbers are describing such degrees, i.e., in mathematical terms, which real numbers are definable in the corresponding fuzzy logic. © 2003 Wiley Periodicals, Inc.

We enrich the three valued Lukasiewicz logic, L/sub 3/, endowing it with an additional unary connective /spl diams/. We obtain a logic, D/sub 2/-logic, in which the /spl diams/ operator can be interpreted as an asymmetric negation. We show the completeness of D/sub 2/-logic and study its algebraic models.

We propose a genetic-level clustering methodology able to cluster objects represented by R/sup p/ spaces. The unsupervised cluster algorithm is based on a fuzzy clustering c-means method that searches the best fuzzy partition of the universe assuming that the evaluation of each object respect to some features is unknown, but knowing that it belongs to circular region of R/sup 2/ space.

In this paper we define internal cut-free sequent calculi for any n-valued Lukasiewicz logic Ln. These calculi are based on a representation of formulas of Ln, by n - 1 many {0, 1}-valued formulas of Ln. They enjoy the usual properties of sequent systems like symmetry, subformula property and invertibility of the rules. Upon dualizing our calculi one obtains Hähnle's tableau systems. Then they provide a reformulation of Hähnle's approach to theorem proving that makes no use of nonlogical elements.

This paper presents selected aspects of information processing in frames. The slots of frames are considered as taking on linguistic values represented as fuzzy sets. Matching procedures are developed and algorithms resulting within this context are proposed. The question of modeling cases involving many exemplar frames associated with a single prototype frame is posed. Subsequently the resulting model is effectively applied to fill the values missing in an exemplar frame. The optimization task involving entropy criterion enables to minimize ambiguity associated with the reconstruction problem. The resulting interval-valued fuzzy constructs are interpreted in terms of relevancy of the reconstructed information.>