Description of Fuzzy First-Order Modal Logic Based on Constant Domain Semantics

We generalize two well-known model-theoretic characterization theorems from propositional modal logic to first-order modal logic (FML, for short). We first study FML-definable frames and give a version of the Goldblatt–Thomason theorem for this logic. The advantage of this result, compared with the original Goldblatt–Thomason theorem, is that it does not need the condition of ultrafilter reflection and uses only closure under bounded morphic images, generated subframes and disjoint unions. We then investigate Lindstrom type theorems for first-order modal logic. We show that FML has the maximal expressive power among the logics extending FML which satisfy compactness, bisimulation invariance and the Tarski union property.

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.

Many authors have noted that a number of English modal sentences cannot be formalized into standard first-order modal logic. Some widely discussed examples include “There could have been things other than there actually are” and “Everyone who’s actually rich could have been poor.” In response, many authors have introduced an “actually” operator @ into the language of first-order modal logic. It is occasionally noted that some of the example sentences still cannot be formalized with @ if one allows only actualist quantifiers, and embedded versions of these example sentences cannot be formalized even with possibilist quantifiers and @. The typical justification for these claims is to observe that none of the most plausible candidate formalizations succeed. In this paper, we prove these inexpressibility results by using a modular notion of bisimulation for first-order modal logic with “actually” and other operators. In doing so, we will explain in what ways these results do or do not generalize to more expressive modal languages.

This handbook provides a systematic survey of the central areas of philosophical logic. It is divided into four volumes, each devoted to a major sub-field within the disciplines. The articles should be of value to general philosophers, linguists, logicians, mathematicians and computer scientists. Volume 1, Elements of Classical Logic, deals with the background to what has come to be considered the standard formulation of predicate logic - both as far as its semantics and proof theory are concerned. The central chapter on predicate logic is followed by chapters outlining various alternative, but essentially equivalent ways of constructing the semantics for first-order logic as well as its proof theory. In addition, this volume contains a discussion of higher-order extensions of first-order logic and a compendium of the algorithmic and decision-theoretic prerequisites in the study of logical systems. Volume 2, Extensions of Classical Logic, surveys the most significant extensions of predicate logic and their applications to various philosophical fields of inquiry. The 12 chapters in this volume together provide a succinct introduction to a variety of intensional frameworks, a discussion of the most well-known logical systems, as well as an overview of major applications and of the open problems in the respective fields. Volume 3, Alternatives to Classical Logic, consists of a series of surveys of some of the alternatives to the basic assumptions of classical logic. These include many-valued logic, partial logic, free logic, relevance and entailment logics, dialogue logic, quantum logic and intuitionism. Volume 4, Topics in the Philosophy of Language, presents a panorama of the applications of logical tools and methods in the formal analysis of natural language. Since a number of developments in philosophical logic were originally stimulated by concern arising in the semantic analysis of natural language discourse, the chapters in this volume provide some criteria of evaluation of the applications of work in philosophical logic. In revealing both the adequacies and inadequacies of logical investigations in the semantic structures of natural discourse, these chapters also point the way to future developments in philosophical logic in general and thus close again the circle of inquiry relating logic and language.

A well established technique toward developing the proof theory of a Hilbert-style modal logic is to introduce a Gentzen-style equivalent (a Gentzenisation), then develop the proof theory of the latter, and finally transfer the metatheoretical results to the original logic (e.g., [1, 6, 8, 18, 10, 12]). In the first-order modal case, on one hand we know that the Gentzenisation of the straightforward first-order extension of GL, the logic QGL, admits no cut elimination (if the rule is included as primitive; or, if not included, then the rule is not admissible [1]). On the other hand the (cut-free) Gentzenisations of the first-order modal logics M3 and ML3 of [10, 12] do have cut as an admissible rule. The syntactic cut admissibility proof given in [18] for the Gentzenisation of the propositional provability logic GL is extremely complex, and it was the basis of the proofs of cut admissibility of the Gentzenisations of M3 and ML3, where the presence of quantifiers and quantifier rules added to the complexity and length of the proof. A recent proof of cut admissibility in a cut-free Gentzenisation of GL is given in [5] and is quite short and easy to read. We adapt it here to revisit the proofs for the cases of M3 and ML3, resulting to similarly short and easy to read proofs, only slightly complicated by the presence of quantification and its relevant rules.

In the last two decades, modal and description logics have provided a theoretical framework for important applications in many areas of computer science. For this reason, the problem of automated reasoning in modal and description logics has been thoroughly investigated. In this chapter we show how efficient Boolean reasoning techniques have been imported, used and integrated into reasoning tools for modal and description logics. To this extent, we focus on modal logics, and in particular mainly on K(m). In particular, we provide some background in modal logics; we describe a basic theoretical framework and we present and analyze the basic tableau-based and DPLL-based techniques; we describe optimizations and extensions of the DPLL-based procedures; we introduce the automata-theoretic/OBDD-based approach; finally, we present the eager approach.

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.

First-order modal logic, in the usual formulations, is not sufficiently expressive, and as a consequence problems like Frege's morning star/evening star puzzle arise. The introduction of predicate abstraction machinery provides a natural extension in which such difficulties can be addressed. But this machinery can also be thought of as part of a move to a full higher-order modal logic. In this paper we present a sketch of just such a higher-order modal logic: its formal semantics, and a proof procedure using tableaus. Naturally the tableau rules are not complete, but they are with respect to a Henkinization of the true semantics. We demonstrate the use of the tableau rules by proving one of the theorems involved in Godel's ontological argument, one of the rare instances in the literature where higher-order modal constructs have appeared. A fuller treatment of the material presented here is in preparation

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.

Density elimination by substitutions is introduced as a uniform method for removing applications of the Takeuti-Titani density rule from proofs in first-order hypersequent calculi. For a large class of calculi, density elimination by this method is guaranteed by known sufficient conditions for cut-elimination. Moreover, adding the density rule to any axiomatic extension of a simple first-order logic gives a logic that is rational complete; i.e., complete with respect to linearly and densely ordered algebras: a precursor to showing that it is a fuzzy logic (complete for algebras with a real unit interval lattice reduct). Hence the sufficient conditions for cut-elimination guarantee rational completeness for a large class of first-order substructural logics.

Concrete domains have been introduced in description logic (DL) to enable reference to concrete objects (such as numbers) and predefined predicates on these objects (such as numerical comparisons) when defining concepts. The primary research goal in this context was to find restrictions on the concrete domain such that its integration into certain DLs preserves decidability or tractability. In this article, we complement these investigations by studying the abstract expressive power of both first-order and description logics extended with concrete domains, i.e., we analyze which classes of first-order interpretations can be expressed using these logics, compared to what their counterparts without concrete domains can express. We demonstrate that, under natural conditions on the concrete domain D (which also play a role for decidability), extensions of first-order logic (FOL) or the well-known DL ALC with D share important formal characteristics with FOL, such as the compactness and the Löwenheim-Skolem properties. Nevertheless, these conditions do not ensure that the abstract expressive power of the extensions we consider is contained in that of FOL, though in some cases it is. To be more precise, we show, on the one hand, that unary concrete domains leave the abstract expressive power within FOL if we are allowed to introduce auxiliary predicates. As a by-product, we obtain (semi-)decidability results for some fragments of FOL extended with the concrete domains considered in this article. On the other hand, we show that the ability to express equality between elements of D , another condition employed in the context of showing decidability of ALC ( D ), is sufficient to push the abstract expressive power of most first-order fragments with concrete domains beyond that of FOL. While for such concrete domains D decidability is retained for ALC ( D ), we show that the availability of equality in D causes undecidability of the two-variable fragment of FOL( D ), although the two-variable fragment of FOL is decidable.

The Craig interpolation property (CIP) states that an interpolant for an implication exists iff it is valid. The projective Beth definability property (PBDP) states that an explicit definition exists iff a formula stating implicit definability is valid. Thus, the CIP and PBDP reduce potentially hard existence problems to entailment in the underlying logic. Description (and modal) logics with nominals and/or role inclusions do not enjoy the CIP nor the PBDP, but interpolants and explicit definitions have many applications, in particular in concept learning, ontology engineering, and ontology-based data management. In this article, we show that, even without Beth and Craig, the existence of interpolants and explicit definitions is decidable in description logics with nominals and/or role inclusions such as 𝒜ℒ𝒞𝒪, 𝒜ℒ𝒞ℋ, and 𝒜ℒ𝒞ℋ𝒪ℐ and corresponding hybrid modal logics. However, living without Beth and Craig makes these problems harder than entailment: the existence problems become 2ExpTime -complete in the presence of an ontology or the universal modality, and coNExpTime -complete otherwise. We also analyze explicit definition existence if all symbols (except the one that is defined) are admitted in the definition. In this case, the complexity depends on whether one considers individual or concept names. Finally, we consider the problem of computing interpolants and explicit definitions if they exist and turn the complexity upper bound proof into an algorithm computing them, at least for description logics with role inclusions.

Controlled Query Evaluation (CQE) protects confidential information, stored in an information system. It prevents harmful inferences due to a user's knowledge and reasoning. In this article we extend CQE to incomplete first-order databases, a data model which suits a broader range of applications than a previously studied propositional incomplete data model. Because of the complexity of the underlying implication problem, which describes the user's reasoning, the representation of the user's knowledge is the main obstacle to effective inference control. For knowledge representation, we introduce first-order modal logic to CQE. Especially, we deal with knowledge about a restricted data model in first-order logic. The restricted data model considered gives rise to a new problem: if the user is aware of the data model, his reasoning must be modeled appropriately. In the analysis of this “reasoning” model we consider both confidentiality and availability. Finally we show, how the considered data model can be reduced to the propositional case and analyze confidentiality properties of the resulting implementation.

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.

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.