Fuzzy Bisimulation for Gödel Modal Logic

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.

The propositional modal logic is obtained by adding the necessity operator □ to the propositional logic. Each formula in the propositional logic is equivalent to a formula in the disjunctive normal form. In order to obtain the correspondence between the propositional modal logic and the propositional logic, we add the axiom \(\Box\varphi \leftrightarrow\Diamond\varphi \) to K and get a new system K + . Each formula in such a logic is equivalent to a formula in the disjunctive normal form, where □ k (k ≥ 0) only occurs before an atomic formula p, and \(\lnot\) only occurs before a pseudo-atomic formula of form □ k p. Maximally consistent sets of K + have a property holding in the propositional logic: a set of pseudo-atom-complete formulas uniquely determines a maximally consistent set. When a pseudo-atomic formula □ k p i (k,i ≥ 0) is corresponding to a propositional variable q ki , each formula in K + then can be corresponding to a formula in the propositional logic P + . We can also get the correspondence of models between K + and P + . Then we get correspondences of theorems and valid formulas between them. So, the soundness theorem and the completeness theorem of K + follow directly from those of P + .

For a novice this book is a mathematically-oriented introduction to modal logic, the discipline within mathematical logic studying mathematical models of reasoning which involve various kinds of modal operators - `like it is necessary' in philosophy, `it is believed' in cognitive science, `it is provable' in mathematics and `it is true after executing a program' in computer science. It is an advanced text which starts with very fundamental concepts and gradually proceeds to the front line of current research, introducing in full details the modern semantical and algebraic apparatus and covering practically all classical results in the field. It contains both numerous exercises and open problems, and presupposes only minimal knowledge in mathematics. A specialist can use the book as a source of references. For the first time results and methods of many directions in propositional modal logic - from completeness and duality to algorithmic problems - are collected and systematically presented in one volume. Unlike other books, modal logic is treated here as a uniform theory rather than a collection of a few particular systems. It is the only book presenting the theory of superintuitionistic logics.

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.

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.

"This book describes the aspects of mathematical logic related to computer sciences. The materials adopted in this book are intended to attend to both the peculiarities of logical systems and the requirements of computer science."

In multi-agent systems, an agent generally forms her belief based on information from multiple sources, such as messages from other agents or perception of the external environment. While modal epistemic logic has been a standard formalism for reasoning about agent's belief, it lacks the expressive power for tracking information sources. Justification logic (JL) provides the missing expressivity by using justification terms to keep track of the belief formation process. However, because JL does not make a clear distinction between potential and actual evidence, the interpretation of justification formulas in JL turns out to be ambiguous. In this paper, we present a justification-based multi-source reasoning formalism built upon second-order propositional modal logic. Our framework not only inherits the source-tracking advantage of JL but also allows the distinction between the actual observation and simply potential admissibility of evidence.

We consider the relationship between the algorithmic properties of the validity problem for a modal or superintuitionistic propositional logic and the size of the smallest Kripke countermodels for non-theorems of the logic. We establish the existence, for every degree of unsolvability, of a propositional logic whose validity problem belongs to the degree and whose every non-theorem is refuted on a Kripke frame that validates the logic and has the size linear in the length of the non-theorem. Such logics are obtained among the normal extensions of the propositional modal logics $\textbf {KTB}$, $\textbf {GL}$ and $\textbf {Grz}$ as well as in the lattice of superintuitionistic propositional logics. This shows that the computational complexity of a modal or superintuitionistic propositional logic is, in general, not related to the size of the countermodels for its non-theorems.

Propositional modal provability logics like G and Grz have arithmetical interpretations where □φ can be read as ‘formula φ is provable in Peano Arithmetic’. These logics are decidable but are characterized by classes of Kripke frames which are not first‐order definable. By abstracting the aspects common to their characteristic axioms we define the notion of a formula generation map F(P) in one propositional variable. We then focus our attention on the properly displayable subset of all (first‐order definable) Sahlqvist modal logics. For any logic L from this subset, we consider the (provability) logic LF obtained by the addition of an axiom based upon a formula generation map F(P) so that LF = L + F(P). The class of such logics includes G and Grz. By appropriately modifying the right introduction rules for □, we give (not necessarily cut‐free) display calculi for every such logic. We define the pseudo‐displayable subset of these logics as those whose display calculi enjoy cut‐elimination for sequents of the form ⊤ ⊢ φ for any formula φ. We then show that for any provability logic LF having a conservative tense extension, there is a map f on formulae such that LF is pseudo‐displayable if and only if f maps theorems of LF to theorems of the underlying logic L and vice versa. By using a standard renaming technique we can guarantee that there is a polynomial‐time translation from LF into L. All proofs are purely syntactic and show the versatility of display calculi since similar results using traditional Gentzen calculi are not possible for as broad a range of logics and require further conditions. Our maps generalize previously known maps from G into K4. An application of our results gives an O(n.log n)3) translation from the (‘second order’) provability logic Grz into a decidable subset of first‐order logic. Since each of our logics L is a Sahlqvist logic, it is first‐order definable, and hence each L has a translation into first‐order logic. Our results therefore show that all pseudo‐displayable logics LF are ‘essentially first‐order’ even though their characteristic axiom may not be first‐order definable.

It is well known that the propositional modal logic $\mathbf{GL}$ of provability satisfies the de Jongh-Sambin fixed-point property. On the other hand, Montagna showed that the predicate modal system $\mathbf{QGL}$, which is the natural variant of $\mathbf{GL}$, loses the fixed-point property. In this paper, we discuss some versions of the fixed-point property for predicate modal logics. First, we prove that several extensions of $\mathbf{QGL}$ including $\mathbf{NQGL}$ do not have the fixed-point property. Secondly, we prove the fixed-point theorem for the logic $\mathbf{QK} + \Box^{n+1} \bot$. As a consequence, we obtain that the class $\mathsf{FH}$ of Kripke frames which are transitive and finite height satisfies the fixed-point property locally. We also show the failure of the Craig interpolation property for $\mathbf{NQGL}$. Finally, we give a sufficient condition for formulas to have a fixed-point in $\mathbf{QGL}$.

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 introduce a propositional many-valued modal logic which is an extension of the Continuous Propositional Logic to a modal system. Otherwise said, we extend the minimal modal logic to a Continuous Logic system. After introducing semantics, axioms and deduction rules, we establish some preliminary results. Then we prove the equivalence between consistency and satisfiability. As straightforward consequences, we get compactness, an approximated completeness theorem, in the vein of Continuous Logic, and a Pavelka-style completeness theorem.

Modal logics and behavioral equivalences play an important role in the specification and verification of concurrent systems. In this paper, we first present a new notion of bisimulation for nondeterministic fuzzy transition systems, which is distribution based and coarser than state-based bisimulation appeared in the literature. Then, we define a distribution-based bisimilarity metric as the least fixed point of a suitable monotonic function on a complete lattice, which is a behavioral distance and is a more robust way of formalizing behavioral similarity between states than bisimulations. We also propose an on-the-fly algorithm for computing the bisimilarity metric. Moreover, we present a fuzzy modal logic and provide a sound and complete characterization of the bisimilarity metric. Interestingly, this characterization holds for a class of fuzzy modal logics. In addition, we show the nonexpansiveness of a typical parallel composition operator with respect to the bisimilarity metric, which makes compositional verification possible.

This book, written by one of the most distinguished of contemporary philosophers of mathematics, is a fully rewritten and updated successor to the author's earlier The Unprovability of Consistency (1979). Its subject is the relation between provability and modal logic, a branch of logic invented by Aristotle but much disparaged by philosophers and virtually ignored by mathematicians. Here it receives its first scientific application since its invention. Modal logic is concerned with the notions of necessity and possibility. What George Boolos does is to show how the concepts, techniques, and methods of modal logic shed brilliant light on the most important logical discovery of the twentieth century: the incompleteness theorems of Kurt Godel and the 'self-referential' sentences constructed in their proof. The book explores the effects of reinterpreting the notions of necessity and possibility to mean provability and consistency.

Kripke transformed both the model theory of modal logic and the understanding of metaphysical modality. In particular, he analysed the model-theoretic conditions for the Barcan formula and its converse to hold. However, the relation between the model theory of modal logic and the correctness of formulas on their metaphysical readings is not straightforward. To mediate between them, a notion of metaphysical universality is defined such that an intended model structure (on the metaphysical reading) should validate all and only metaphysically universal formulas. Technical problems for defining intended model structures arise in both propositional and first-order modal logic; there may be ‘too many’ worlds and ‘too many’ individuals. Given that these problems can be overcome, a special reason is explained why contingentism but not necessitism forbids an intended model structure for quantified modal logic. But even granted necessitism, it is controversial which first-order modal formulas are metaphysically universal.