The last decade has seen an enormous development in infinite-valued systems and in particular in such systems which have become known as mathematical fuzzy logics. The paper discusses the mathematical background for the interest in such systems of mathematical fuzzy logics, as well as the most important ones of them. It concentrates on the propositional cases, and mentions the first-order systems more superficially. The main ideas, however, become clear already in this restricted setting.

Some aspects of vagueness as presented in Shapiro’s book Vagueness in Context [23] are analyzed from the point of fuzzy logic. Presented are some generalizations of Shapiro’s formal apparatus.

This paper deals with one kind of Kripke-style semantics, which we shall call algebraic Kripke-style semantics, for relevance logics. We first recall the logic R of relevant implication and some closely related systems, their corresponding algebraic structures, and algebraic completeness results. We provide simpler algebraic completeness proofs. We then introduce various types of algebraic Kripke-style semantics for these systems and connect them with algebraic semantics.

In the context of implicational tonoid logics, this paper investigates analogues of Birkhoff’s two theorems, the so-called subdirect representation and varieties theorems, and of Mal’cev’s quasi-varieties theorem. More precisely, we first recall the class of implicational tonoid logics. Next, we establish the subdirect product representation theorem for those logics and then consider some more related results such as completeness. Thirdly, we consider the varieties theorem for them. Finally, we introduce an analogue of Mal’cev’s quasi-varieties theorem for algebras.

A logic for classical conditional events was investigated by Dubois and Prade. In their approach, the truth value of a conditional event may be undetermined. In this paper we extend the treatment to many-valued events. Then we support the thesis that probability over partially undetermined events is a conditional probability, and we interpret it in terms of bets in the style of de Finetti. Finally, we show that the whole investigation can be carried out in a logical and algebraic setting, (...) and we find a logical characterization of coherence for assessments of partially undetermined events. (shrink)

This paper presents several results on the non-commutative fuzzy logic HpsUL, a Hilbert system whose corresponding algebraic semantics is the variety of bounded representable residuated lattices. In particular, we prove that HpsUL is not complete with respect to algebras based on the real unit interval, which answers the question posed by Metcalfe, Olivetti and Gabbay and shows that HpsUL is not the logic of pseudo-uninorms and their residua.

Several investigations in probability theory and the theory of expert systems show that it is important to search for some reasonable generalizations of fuzzy logics (e.g. Łukasiewicz, Gödel or product logic) having a non-associative conjunction. In the present paper, we offer a non-associative fuzzy logic L CBA having as an equivalent algebraic semantics lattices with section antitone involutions satisfying the contraposition law, so-called commutative basic algebras. The class (variety) CBA of commutative basic algebras was intensively studied in several recent papers (...) and includes the class of MV-algebras. We show that the logic L CBA is very close to the Łukasiewicz one, both having the same finite models, and can be understood as its non-associative generalization. (shrink)

An algebraic characterisation is given of the Mares-Goldblatt semantics for quantified extensions of relevant and modal logics. Some features of this more general semantic framework are investigated, and the relations to some recent work in algebraic semantics for quantified extensions of non-classical logics are considered.

This paper is a contribution to Mathematical fuzzy logic, in particular to the algebraic study of t-norm based fuzzy logics. In the general framework of propositional core and Δ-core fuzzy logics we consider three properties of completeness with respect to any semantics of linearly ordered algebras. Useful algebraic characterizations of these completeness properties are obtained and their relations are studied. Moreover, we concentrate on five kinds of distinguished semantics for these logics–namely the class of algebras defined over the real unit (...) interval, the rational unit interval, the hyperreals , the strict hyperreals and finite chains, respectively–and we survey the known completeness methods and results for prominent logics. We also obtain new interesting relations between the real, rational and hyperreal semantics, and good characterizations for the completeness with respect to the semantics of finite chains. Finally, all completeness properties and distinguished semantics are also considered for the first-order versions of the logics where a number of new results are proved. (shrink)

This paper provides a finer analysis of the well-known form of the Local Deduction Theorem in contraction-free logics . An infinite hierarchy of its natural strengthenings is introduced and studied. The main results are the separation of its initial four members and the subsequent collapse of the hierarchy.

The logic FBCI given by linearly ordered BCI-matrices is known not to be an axiomatic extension of the well-known BCI logic. In this paper we axiomatize FBCI by adding a recursively enumerable set of schemes of inference rules to BCI and show that there is no finite axiomatization for FBCI.

This is the continuation of the paper :417–446, 2010). We continue the abstract study of non-classical logics based on the kind of generalized implication connectives they possess and we focus on semilinear logics, i.e. those that are complete with respect to the class of models where the implication defines a linear order. We obtain general characterizations of semilinearity in terms of the intersection-prime extension property, the syntactical semilinearity metarule and the class of finitely subdirectly irreducible models. Moreover, we consider extensions (...) of the language with lattice connectives and generalized disjunctions, study their interplay with implication and obtain axiomatizations and further descriptions of semilinear logics in terms of disjunctions and the proof by cases property. (shrink)

Formal systems of fuzzy logic are well-established logical systems and respected members of the broad family of the so-called substructural logics closely related to the famous logic BCK. The study of fragments of logical systems is an important issue of research in any class of non-classical logics. Here we study the fragments of nine prominent fuzzy logics to all sublanguages containing implication. However, the results achieved in the paper for those nine logics are usually corollaries of theorems with much wider (...) scope of applicability. In particular, we show how many of these fragments are really distinct and we find axiomatic systems for most of them. In fact, we construct strongly separable axiomatic systems for eight of our nine logics. We also fully answer the question for which of the studied fragments the corresponding class of algebras forms a variety. Finally, we solve the problem how to axiomatize predicate versions of logics without the lattice disjunction. (shrink)

Uniform infinite bases are defined for the single-conclusion and multiple-conclusion admissible rules of the implication–negation fragments of intuitionistic logic and its consistent axiomatic extensions . A Kripke semantics characterization is given for the structurally complete implication–negation fragments of intermediate logics, and it is shown that the admissible rules of this fragment of form a PSPACE-complete set and have no finite basis.

In the last few decades many formal systems of fuzzy logics have been developed. Since the main differences between fuzzy and classical logics lie at the propositional level, the fuzzy predicate logics have developed more slowly (compared to the propositional ones). In this text we aim to promote interest in fuzzy predicate logics by contributing to the model theory of fuzzy predicate logics. First, we generalize the completeness theorem, then we use it to get results on conservative extensions of theories (...) and on witnessed models. (shrink)

In abstract algebraic logic, the general study of propositional non-classical logics has been traditionally based on the abstraction of the Lindenbaum-Tarski process. In this process one considers the Leibniz relation of indiscernible formulae. Such approach has resulted in a classification of logics partly based on generalizations of equivalence connectives: the Leibniz hierarchy. This paper performs an analogous abstract study of non-classical logics based on the kind of generalized implication connectives they possess. It yields a new classification of logics expanding Leibniz (...) hierarchy: the hierarchy of implicational logics. In this framework the notion of implicational semilinear logic can be naturally introduced as a property of the implication, namely a logic L is an implicational semilinear logic iff it has an implication such that L is complete w.r.t. the matrices where the implication induces a linear order, a property which is typically satisfied by well-known systems of fuzzy logic. The hierarchy of implicational logics is then restricted to the semilinear case obtaining a classification of implicational semilinear logics that encompasses almost all the known examples of fuzzy logics and suggests new directions for research in the field. (shrink)

In this article we investigate infinitary propositional logics from the perspective of their completeness properties in abstract algebraic logic. It is well-known that every finitary logic is complete with respect to its relatively subdirectly irreducible models. We identify two syntactical notions formulated in terms of intersection-prime theories that follow from finitarity and are sufficient conditions for the aforementioned completeness properties. We construct all the necessary counterexamples to show that all these properties define pairwise different classes of logics. Consequently, we obtain (...) a new hierarchy of logics going beyond the scope of finitarity. (shrink)

This paper proposes a new topic in substructural logic for use in research joining the fields of relevance and fuzzy logics. For this, we consider old and new relevance principles. We first introduce fuzzy systems satisfying an old relevance principle, that is, Dunn’s weak relevance principle. We present ways to obtain relevant companions of the weakening-free uninorm systems introduced by Metcalfe and Montagna and fuzzy companions of the system R of relevant implication and its neighbors. The algebraic structures corresponding to (...) the systems are then defined, and completeness results are provided. We next consider fuzzy systems satisfying new relevance principles introduced by Yang. We show that the weakening-free uninorm systems and some extensions and neighbors of R satisfy the new relevance principles. (shrink)

This paper combines two classes of generalized logics, one of which is the class of weakly implicative logics introduced by Cintula and the other of which is the class of gaggle logics introduced by Dunn. For this purpose we introduce implicational tonoid logics. More precisely, we first define implicational tonoid logics in general and examine their relation to weakly implicative logics. We then provide algebraic semantics for implicational tonoid logics. Finally, we consider relational semantics, called Routley–Meyer–style semantics, for finitary those (...) logics. (shrink)

A many-valued modal logic, called linear abelian modal logic \(\rm {\mathbf{LK(A)}}\) is introduced as an extension of the abelian modal logic \(\rm \mathbf{K(A)}\). Abelian modal logic \(\rm \mathbf{K(A)}\) is the minimal modal extension of the logic of lattice-ordered abelian groups. The logic \(\rm \mathbf{LK(A)}\) is axiomatized by extending \(\rm \mathbf{K(A)}\) with the modal axiom schemas \(\Box(\varphi\vee\psi)\rightarrow(\Box\varphi\vee\Box\psi)\) and \((\Box\varphi\wedge\Box\psi)\rightarrow\Box(\varphi\wedge\psi)\). Completeness theorem with respect to algebraic semantics and a hypersequent calculus admitting cut-elimination are established. Finally, the correspondence between hypersequent calculi and axiomatization (...) is investigated. (shrink)

In this paper we investigate the connections between quantifier elimination, decidability and Uniform Craig Interpolation in Δ-core fuzzy logics added with propositional quantifiers. As a consequence, we are able to prove that several propositional fuzzy logics have a conservative extension which is a Δ-core fuzzy logic and has Uniform Craig Interpolation.

This paper deals with substructural nuclear (image-based) logics and their algebraic and Kripke-style semantics. More precisely, we first introduce a class of substructural logics with connective _N_ satisfying nucleus property, called here substructural _nuclear_ logics, and its subclass, called here substructural _nuclear image-based_ logics, where _N_ further satisfies homomorphic image property. We then consider their algebraic semantics together with algebraic characterizations of those logics. Finally, we introduce _operational Kripke-style_ semantics for those logics and provide two sorts of completeness results for (...) them, one of which is based on algebraic completeness for all the logics and the other of which is based on set-theoretic one for the nuclear image-based logics. (shrink)

In this paper we investigate fuzzy propositional and first order logics which are complete or strongly complete with respect to a single chain, and we relate this properties with the existence of a universal chain for the logic.

Axiomatizations are presented for fuzzy logics characterized by uninorms continuous on the half-open real unit interval [0,1), generalizing the continuous t-norm based approach of Hájek. Basic uninorm logic BUL is defined and completeness is established with respect to algebras with lattice reduct [0,1] whose monoid operations are uninorms continuous on [0,1). Several extensions of BUL are also introduced. In particular, Cross ratio logic CRL, is shown to be complete with respect to one special uninorm. A Gentzen-style hypersequent calculus is provided (...) for CRL and used to establish co-NP completeness results for these logics. (shrink)