Fuzzy logics with modalities

We are going to study, from the logical point of view, expressions as “usually ϕ”, “ϕ is probable”, “for many x, ϕ” and similar. We shall handle them both as generalized quantifiers and as modalities. This should not look unnatural since in general modalities can be viewed as hidden quantifiers. We shall try to show how the theory of generalized quantifiers and modal logic can be applied to the above expressions (stressed by Zadeh as items of the specific agenda of fuzzy logic in the narrow sense) and that they admit “classical” logical analysis. We shall offer a logical apparatus, precise definitions, some results and various problems: here much study and development is still necessary. To be able to use and generalize the results and approaches of Boolean logic, we shall have to review and summarize several notions and facts on generalized quantifiers, some modal logics, and also on measures of uncertainty (that will be used to define various quantifiers and modalities). Thus we shall have two preliminary sections in this chapter: Section 1 on generalized quantifiers (with an appendix on uncertainty measures) and Section 2 on modal logics, both sections in the frame of Boolean (two-valued) logic. Generalized quantifiers in fuzzy logic and fuzzy modal logics are studied in Sections 3, 4.

Consistency degrees of finite theories in Łukasiewicz propositional fuzzy logic

A many‐valued sentential logic with truth values in an injective MV‐algebra is introduced and the axiomatizability of this logic is proved. The paper develops some ideas of Goguen and generalizes the results of Pavelka on the unit interval. The proof for completeness is purely algebraic. A corollary of the Completeness Theorem is that fuzzy logic on the unit interval is semantically complete if and only if the algebra of the truth values is a complete MV‐algebra. In the well‐defined fuzzy sentential logic holds the Compactness Theorem, while the Deduction Theorem and the Finiteness Theorem in general do not hold. Because of its generality and good‐behaviour, MV‐valued logic can be regarded as a mathematical basis of fuzzy reasoning.

Bisimulation is a central concept in the model theory of modal logic with extensive computational applications. It is a relation between two models in which related states have identical atomic properties and matching transition possibilities. Bisimulation captures the expressive power of propositional modal logic in the sense that bisimilar states are not distinguishable by any propositional modal formula. In recent years, fuzzy modal logic has received much attention because of the connection between uncertainty measures and fuzzy modalities. In this paper, we define the notion of fuzzy bisimulation and prove that, in a fuzzified version, it bears the same relationship to fuzzy modal logic that bisimulation bears to propositional modal logic.

A formal fuzzy reasoning system and reasoning mechanism based on propositional modal logic

Pan pointed out that there are three kinds of different negations: contradictory negation, opposite negation and medium negation. He also proposed the novel fuzzy set FScom with the above three kinds of negations, followed by one improved fuzzy set IFScom that is established based on FScom. In this paper, a novel fuzzy propositional logic calculus system that equipped with three kinds of negations is set up to correspond to the FScom and IFScom. The soundness theorem and completeness theorem of the system is proved.

The present paper studies two approaches to the expressiveness of propositional modal logics based on first-degree entailment logic, FDE. We first consider the basic FDE-based modal logic BK and certain systems in its vicinity, and then turn to some FDE-based modal logics in a richer vocabulary, including modal bilattice logic, MBL. On the one hand, model-theoretic proofs of the definability of connectives along the lines of [McCullough, “Logical connectives for intuitionistic propositional logic”, Journal of Symbolic Logic 36, 1 (1971): 15–20. DOI: 10.2307/2271511] and [[17] Wansing, “Logical connectives for constructive modal logic”, Synthese 150, 3 (2006): 459–482. DOI: 10.1007/s11229-005-5518-5] are given for various FDE-based modal logics. On the other hand, building on [Odintsov and Wansing, “Disentangling FDE-based paraconsistent modal logics, Studia Logica 105, 6 (2017): 1221–1254. DOI: 10.1007/s11225-017-9753-9], expressibility is considered in terms of mutual faithful embeddability of one logic into another logic. A distinction is drawn between definitional equivalence, which is defined with respect to a pair of structural translations between two languages, and weak definitional equivalence, which is defined with respect to a weaker notion of translations. Moreover, the definitional equivalence of some FDE-based modal logics is proven, especially the definitional equivalence of MBL and a conservative extension of the logic BK□×BK□, which underlines the central role played by BK among FDE-based modal logics.

The first results in proof theory were obtained using axiomatic systems of logical deduction. The chapter develops propositional and predicate logic in such an axiomatic system. It shows how reasoning under assumption can be regained in an axiomatic system by proving the so-called “deduction theorem” propositional logic (and later, predicate logic as well). The deduction theorem provides the opportunity to introduce one of the main tools needed for proving properties of formal systems, namely, proof by induction on the natural numbers. The differences between minimal, intuitionistic, and classical systems of logic according to the properties of negation characterizing each of the systems are outlined. Independence of theorems such as the law of excluded middle are here given model-theoretically (in later chapters, these are also presented proof-theoretically). After presenting classical predicate logic, the systems of classical and intuitionistic arithmetic are presented. Gentzen’s first substantial result, the so-called Gödel-Gentzen translation, is discussed in the final section where it is proved that if intuitionistic arithmetic is consistent, so is classical arithmetic.

In this paper, we present a generic format for adaptive vague logics. Logics based on this format are able to (1) identify sentences as vague or non-vague in light of a given set of premises, and to (2) dynamically adjust the possible set of inferences in accordance with these identifications, i.e. sentences that are identified as vague allow only for the application of vague inference rules and sentences that are identified as non-vague also allow for the application of some extra set of classical logic rules. The generic format consists of a set of minimal criteria that must be satisfied by the vague logic in casu in order to be usable as a basis for an adaptive vague logic. The criteria focus on the way in which the logic deals with a special ⊡-operator. Depending on the kind of logic for vagueness that is used as a basis for the adaptive vague logic, this operator can be interpreted as completely true, definitely true, clearly true, etc. It is proven that a wide range of famous logics for vagueness satisfies these criteria when extended with a specific ⊡-operator, e.g. fuzzy basic logic and its well known extensions, cf. [7], super- and subvaluationist logics, cf. [6], [9], and clarity logic, cf. [13]. Also a fuzzy logic is presented that can be used for an adaptive vague logic that can deal with higher-order vagueness. To illustrate the theory, some toy-examples of adaptive vague proofs are provided.

For two propositional fuzzy logics, we present analytic proof calculi, based on relational hypersequents. The logic considered first, called MŁ, is based on the finite ordinal sums of Łukasiewicz t-norms. In addition to the usual connectives—the conjunction ⊙, the implication → and the constant 0—we use a further unary connective interpreted by the function associating with each truth value a the greatest ⊙-idempotent below a. MŁ is a conservative extension of Basic Logic. The second logic, called MΠ, is based on the finite ordinal sums of the product t-norm on (0, 1]. Our connectives are in this case just the conjunction and the implication.

Fuzzy normal forms can be generated with an application of "Normal Form Generation Algorithm" on fuzzy truth tables. This takes place at the third level of knowledge representation, i.e., propositional level. It is shown that at least three distinct sets of normal forms can be generated depending on the axioms one is willing to impose on the propositional fuzzy set and logic theories. All are conjunctive-disjunctive and complement based De Morgan logics with the following three classes of axioms that identify each general class of fuzzy normal forms in order of least to most restrictive set of axioms in the following sense: 1) boundary and monotonicity; 2) boundary, monotonicity, associativity and commutativity; and 3) boundary, monotonicity, associativity, commutativity and idempotency.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

In the paper we introduce formal calculi which are a generalization of propositional modal logics. These calculi are called fuzzy modal logics. We introduce the concept of a fuzzy Kripke model and consider a semantics of these calculi in the class of fuzzy Kripke models. The main result of the paper is the completeness theorem of a minimal fuzzy modal logic in the class of fuzzy Kripke models.

Normal forms for fuzzy logic functions and their approximation ability

As an extension of the traditional modal logic, the fuzzy first-order modal logic is discussed in this paper. A description of fuzzy first-order modal logic based on constant domain semantics is given, and a formal system of fuzzy reasoning based on the semantic information of models of first-order modal logic is established. It is also introduced in this paper the notion of the satisfiability of the reasoning system and some properties associated with the satisfiability are proved.

We give a linear nested sequent calculus for the basic normal tense logic \(\mathsf {Kt}\). We show that the calculus enables backwards proof-search, counter-model construction and syntactic cut-elimination. Linear nested sequents thus provide the minimal amount of nesting necessary to provide an adequate proof-theory for modal logics containing converse. As a bonus, this yields a cut-free calculus for symmetric modal logic \(\mathsf {KB}\).