Showing papers on "Normal modal logic published in 2015"

Contingency and knowing whether

Abstract: A proposition is noncontingent, if it is necessarily true or it is necessarily false. In an epistemic context, ‘a proposition is noncontingent’ means that you know whether the proposition is true. In this paper, we study contingency logic with the noncontingency operator Δ but without the necessity operator □. This logic is not a normal modal logic, because Δ(φ → ψ) → (Δφ → Δψ) is not valid. Contingency logic cannot define many usual frame properties, and its expressive power is weaker than that of basic modal logic over classes of models without reflexivity. These features make axiomatizing contingency logics nontrivial, especially for the axiomatization over symmetric frames. In this paper, we axiomatize contingency logics over various frame classes using a novel method other than the methods provided in the literature, based on the ‘almost-definability’ schema AD proposed in our previous work. We also present extensions of contingency logic with dynamic operators. Finally, we compare our work to the related work in the fields of contingency logic and ignorance logic, where the two research communities have similar results but are apparently unaware of each other’s work. One goal of our paper is to bridge this gap.

Epistemic equilibrium logic

Abstract: We add epistemic modal operators to the language of here-and-there logic and define epistemic here-and-there models. We then successively define epistemic equilibrium models and autoepistemic equilibrium models. The former are obtained from here-and-there models by the standard minimisation of truth of Pearce's equilibrium logic; they provide an epistemic extension of that logic. The latter are obtained from the former by maximising the set of epistemic possibilities; they provide a new semantics for Gelfond's epistemic specifications. For both definitions we characterise strong equivalence by means of logical equivalence in epistemic here-and-there logic.

A Logic of Knowing How

Abstract: In this paper, we propose a single-agent modal logic framework for reasoning about goal-direct “knowing how” based on ideas from linguistics, philosophy, modal logic and automated planning. We first define a modal language to express “I know how to guarantee ϕ given ψ” with a semantics not based on standard epistemic models but labelled transition systems that represent the agent’s knowledge of his own abilities. A sound and complete proof system is given to capture the valid reasoning patterns about “knowing how” where the most important axiom suggests its compositional nature.

Fuzzy Bisimulation for Gödel Modal Logic

Abstract: 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.

Possibility Frames and Forcing for Modal Logic

Abstract: Possibility Frames and Forcing for Modal Logic ∗ Wesley H. Holliday Department of Philosophy & Group in Logic and the Methodology of Science University of California, Berkeley December 30, 2015 Abstract This paper develops the model theory of normal modal logics based on partial “possibilities” instead of total “worlds,” following Humberstone [1981] instead of Kripke [1963]. Possibility semantics can be seen as extending to modal logic the semantics for classical logic used in weak forcing in set theory, or as semanticizing a negative translation of classical modal logic into intuitionistic modal logic. Thus, possibility frames are based on posets with accessibility relations, like intuitionistic modal frames, but with the constraint that the interpretation of every formula is a regular open set in the Alexandrov topology on the poset. The standard world frames for modal logic are the special case of possibility frames wherein the poset is discrete. The analogues of classical Kripke frames, i.e., full world frames, are full possibility frames, in which propositional variables may be interpreted as any regular open sets. We develop the beginnings of duality theory, definability/correspondence theory, and complete- ness theory for possibility frames. The duality theory, relating possibility frames to Boolean algebras with operators (BAOs), shows the way in which full possibility frames are a generalization of Kripke frames. Whereas Thomason [1975a] established a duality between the category of Kripke frames with p-morphisms and the category of complete (C), atomic (A), and completely additive (V) BAOs with com- plete BAO-homomorphisms (these categories being dually equivalent), we establish a duality between the category of full possibility frames with possibility morphisms and the category of CV-BAOs, i.e., allow- ing non-atomic BAOs, with complete BAO-homomorphisms (the latter category being dually equivalent to a reflective subcategory of the former). This parallels the connection between forcing posets and Boolean-valued models in set theory, but now with accessibility relations and modal operators involved. Generalizing further, we introduce a class of principal possibility frames that capture the generality of V-BAOs. If we do not require a full or principal frame, then every BAO has an equivalent possibility frame, whose possibilities are proper filters in the BAO. With this filter representation, which does not require the ultrafilter axiom, we are lead to a notion of filter-descriptive possibility frames. Whereas Goldblatt [1974] showed that the category of BAOs with BAO-homomorphisms is dually equivalent to the category of descriptive world frames with p-morphisms, we show that the category of BAOs with ∗ For helpful comments or discussion, I wish to give special thanks to Johan van Benthem, Guram Bezhanishvili, Nick Bezhanishvili, Matthew Harrison-Trainor, and Tadeusz Litak. I also wish to thank Ivano Ciardelli, Josh Dever, Davide Grossi, Lloyd Humberstone, Thomas Icard, Hans Kamp, Larry Moss, Lawrence Valby, Yanjing Wang, and Dag Westerstahl, as well as the participants in my Fall 2014 or Spring 2015 graduate seminars at UC Berkeley: Russell Buehler, Sophia Dandelet, Matthew Harrison-Trainor, Alex Kocurek, Alex Kruckman, James Moody, James Walsh, and Kentaro Yamamoto. I am also thankful for feedback I received when presenting some of this material at the following venues: the Modal Logic Workshop on Consistency and Structure at Carnegie Mellon University in April 2014; my course at the 3rd East-Asian School on Logic, Language and Computation (EASSLLC) at Tsinghua University in July 2014; the Advances in Modal Logic conference at the University of Groningen in August 2014 (see Holliday 2014); the Workshop on the Future of Logic in honor of Johan van Benthem in Amsterdam in September 2014; the Hans Kamp Seminar in Logic and Language at the University of Texas at Austin in April 2015; the 4th CSLI Workshop on Logic, Rationality and Intelligent Interaction at Stanford University in May 2015; and the New Mexico State University Mathematics Colloquium in November 2015. Finally, I wish to gratefully acknowledge an HRF grant from UC Berkeley that allowed me to complete this work in Fall 2015.

Focused Labeled Proof Systems for Modal Logic

Abstract: Focused proofs are sequent calculus proofs that group inference rules into alternating positive and negative phases. These phases can then be used to define macro-level inference rules Gentzen's original and tiny introduction and structural rules. We show here that the inference rules of labeled proof systems for modal logics can similarly be described as pairs of such phases within the LKF focused proof system for first-order classical logic. We consider the system $${ G3K} $$ of Negri for the modal logic $$K$$ and define a translation from labeled modal formulas into first-order polarized formulas and show a strict correspondence between derivations in the two systems, i.e.,i¾?each rule application in $${ G3K} $$ corresponds to a bipole--a pair of a positive and a negative phases--in $${ LKF} $$. Since geometric axioms when properly polarized induce bipoles, this strong correspondence holds for all modal logics whose Kripke frames are characterized by geometric properties. We extend these results to present a focused labeled proof system for this same class of modal logics and show its soundness and completeness. The resulting proof system allows one to define a rich set of normal forms of modal logic proofs.

On Nested Sequents for Constructive Modal Logics

Abstract: We present deductive systems for various modal logics that can be obtained from the constructive variant of the normal modal logic CK by adding combinations of the axioms d, t, b, 4, and 5. This includes the constructive variants of the standard modal logics K4, S4, and S5. We use for our presentation the formalism of nested sequents and give a syntactic proof of cut elimination.

A Van Benthem Theorem for Modal Team Semantics

Abstract: The famous van Benthem theorem states that modal logic corresponds exactly to the fragment of first-order logic that is invariant under bisimulation. In this article we prove an exact analogue of this theorem in the framework of modal dependence logic (MDL) and team semantics. We show that Modal Team Logic (MTL) extending MDL by classical negation captures exactly the FO-definable bisimulation invariant properties of Kripke structures and teams. We also compare the expressive power of MTL to most of the variants and extensions of MDL recently studied in the area.

Separation logics and modalities: a survey

Abstract: Like modal logic, temporal logic, and description logic, separation logic has become a popular class of logical formalisms in computer science, conceived as assertion languages for Hoare-style proof systems with the goal to perform automatic program analysis. In a broad sense, separation logic is often understood as a programming language, an assertion language and a family of rules involving Hoare triples. In this survey, we present similarities between separation logic as an assertion language and modal and temporal logics. Moreover, we propose a selection of landmark results about decidability, complexity and expressive power.

Neighborhood Contingency Logic

Abstract: A formula is contingent, if it is possibly true and possibly false; a formula is non-contingent, if it is not contingent, i.e., if it is necessarily true or necessarily false. In this paper, we propose a neighborhood semantics for contingency logic, in which the interpretation of the non-contingency operator is consistent with its philosophical intuition. Based on this semantics, we compare the relative expressivity of contingency logic and modal logic on various classes of neighborhood models, and investigate the frame definability of contingency logic. We present a decidable axiomatization for classical contingency logic (the obvious counterpart of classical modal logic), and demonstrate that for contingency logic, neighborhood semantics can be seen as an extension of Kripke semantics.

Modular Sequent Calculi for Classical Modal Logics

Abstract: This paper develops sequent calculi for several classical modal logics. Utilizing a polymodal translation of the standard modal language, we are able to establish a base system for the minimal classical modal logic E from which we generate extensions (to include M, C, and N) in a modular manner. Our systems admit contraction and cut admissibility, and allow a systematic proof-search procedure of formal derivations.

Proof Search in Nested Sequent Calculi

Abstract: We propose a notion of focusing for nested sequent calculi for modal logics which brings down the complexity of proof search to that of the corresponding sequent calculi. The resulting systems are amenable to specifications in linear logic. Examples include modal logic $$\mathsf {K}$$, a simply dependent bimodal logic and the standard non-normal modal logics. As byproduct we obtain the first nested sequent calculi for the considered non-normal modal logics.

Budget-Constrained Knowledge in Multiagent Systems

Abstract: The paper introduces a modal logical system for reasoning about knowledge in which information available to agents might be constrained by the available budget. Although the system lacks an equivalent of the standard Negative Introspection axiom from epistemic logic S5, it is proven to be sound and complete with respect to an S5-like Kripke semantics.

On Nested Sequents for Constructive Modal Logics

Abstract: We present deductive systems for various modal logics that can be obtained from the constructive variant of the normal modal logic CK by adding combinations of the axioms d, t, b, 4, and 5. This includes the constructive variants of the standard modal logics K4, S4, and S5. We use for our presentation the formalism of nested sequents and give a syntactic proof of cut elimination.

Intuitionistic Epistemology and Modal Logics of Verification

Abstract: The language of intuitionistic epistemic logic, IEL [3], captures basic reasoning about intuitionistic knowledge and belief, but its language has expressive limitations. Following Godel’s explication of IPC as a fragment of the more expressive system of classical modal logic S4 we present a faithful embedding of IEL into S4V – S4 extended with a verification modality. The classical modal framework is finer-grained and more flexible, allowing us to make explicit various properties of verification.

On modal expansions of t-norm based logics with rational constants

Abstract: According to Zadeh, the term “fuzzy logic” has two different meanings: wide and narrow. In a narrow sense it is a logical system which aims a formalization of approximate reasoning, and so it can be considered an extension of many-valued logic. However, Zadeh also says that the agenda of fuzzy logic is quite different from that of traditional many-valued logic, as it addresses concepts like linguistic variable, fuzzy if-then rule, linguistic quantifiers etc. Hajek, in the preface of his foundational book Metamathematics of Fuzzy Logic, agrees with Zadeh’s distinction, but stressing that formal calculi of many-valued logics are the kernel of the so-called Basic Fuzzy logic (BL), having continuous triangular norms (t-norm) and their residua as semantics for the conjunction and implication respectively, and of its most prominent extensions, namely Lukasiewicz, Godel and Product fuzzy logics. Taking advantage of the fact that a t-norm has residuum if, and only if, it is left-continuous, the logic of the left-continuous t-norms, called MTL, was soon after introduced. On the other hand, classical modal logic is an active field of mathematical logic, originally introduced at the beginning of the XXth century for philosophical purposes, that more recently has shown to be very successful in many other areas, specially in computer science. That are the most well-known semantics for classical modal logics. Modal expansions of non-classical logics, in particular of many-valued logics, have also been studied in the literature. In this thesis we focus on the study of some modal logics over MTL, using natural generalizations of the classical Kripke relational structures where propositions at possible words can be many-valued, but keeping classical accessibility relations. In more detail, the main goal of this thesis has been to study modal expansions of the logic of a left-continuous t-norm, defined over the language of MTL expanded with rational truth-constants and the Monteiro-Baaz Delta-operator, whose intended (standard) semantics is given by Kripke models with crisp accessibility relations and taking the unit real interval [0, 1] as set of truth-values. To get complete axiomatizations, already known techniques based on the canonical model construction are uses, but this requires to ensure that the underlying (propositional) fuzzy logic is strongly standard complete. This constraint leads us to consider axiomatic systems with infinitary inference rules, already at the propositional level. A second goal of the thesis has been to also develop and automated reasoning software tool to solve satisfiability and logical consequence problems for some of the fuzzy logic modal logics considered. This dissertation is structured in four parts. After a gentle introduction, Part I contains the needed preliminaries for the thesis be as self-contained as possible. Most of the theoretical results are developed in Parts II and III. Part II focuses on solving some problems concerning the strong standard completeness of underlying non-modal expansions. We first present and axiomatic system for the non-nodal propositional logic of a left-continuous t-norm who makes use of a unique infinitary inference rule, the “density rule”, that solves several problems pointed out in the literature. We further expand this axiomatic system in order to also characterize arbitrary operations over [0, 1] satisfying certain regularity conditions. However, since this axiomatic system turn out to be not well-behaved for the modal expansion, we search for alternative axiomatizations with some particular kind of inference rules (that will be called conjunctive). Unfortunately, this kind of axiomatization does not necessarily exist for all left-continuous t-norms (in particular, it does not exist for the Godel logic case), but we identify a wide class of t-norms for which it works. This “well-behaved” t-norms include all ordinal sums of Lukasiewiczand Product t-norms. Part III focuses on the modal expansion of the logics presented before. We propose axiomatic systems (which are, as expected, modal expansions of the ones given in the previous part) respectively strongly complete with respect to local and global Kripke semantics defined over frames with crisp accessibility relations and worlds evaluated over a “well-behaved” left-continuous t-norm. We also study some properties and extensions of these logics and also show how to use it for axiomatizing the possibilistic logic over the very same t-norm. Later on, we characterize the algebraic companion of these modal logics, provide some algebraic completeness results and study the relation between their Kripke and algebraic semantics. Finally, Part IV of the thesis is devoted to a software application, mNiB-LoS, who uses Satisfability Modulo Theories in order to build an automated reasoning system to reason over modal logics evaluated over BL algebras. The acronym of this applications stands for a modal Nice BL-logics Solver. The use of BL logics along this part is motivated by the fact that continuous t-norms can be represented as ordinal sums of three particular t-norms: Godel, Lukasiewicz and Product ones. It is then possible to show that these t-norms have alternative characterizations that, although equivalent from the point of view of the logic, have strong differences for what concerns the design, implementation and efficiency of the application. For practical reasons, the modal structures included in the solver are limited to the finite ones (with no bound on the cardinality).

A Syntactic Proof of Arrow's Theorem in a Modal Logic of Social Choice Functions

Abstract: We show how to formalise Arrow's Theorem on the impossibility of devising a method for preference aggregation that is both independent of irrelevant alternatives and Pareto efficient by using a modal logic of social choice functions. We also provide a syntactic proof of the theorem in that logic. While prior work has been successful in applying tools from logic and automated reasoning to social choice theory, this is the first human-readable formalisation of the Arrovian framework allowing for a direct derivation of the theorem.

Sabotage Modal Logic: Some Model and Proof Theoretic Aspects

Abstract: We investigate some model and proof theoretic aspects of sabotage modal logic. The first contribution is to prove a characterization theorem for sabotage modal logic as the fragment of first-order logic which is invariant with respect to a suitably defined notion of bisimulation (called sabotage bisimulation). The second contribution is to provide a sound and complete tableau method for sabotage modal logic. We also chart a number of open research questions concerning sabotage modal logic, aiming at integrating it within the current landscape of logics of model update.

A Modal-Layered Resolution Calculus for K

Abstract: Resolution-based provers for multimodal normal logics require pruning of the search space for a proof in order to deal with the inherent intractability of the satisfiability problem for such logics. We present a clausal modal-layered hyper-resolution calculus for the basic multimodal logic, which divides the clause set according to the modal depth at which clauses occur. We show that the calculus is complete for the logics being considered. We also show that the calculus can be combined with other strategies. In particular, we discuss the completeness of combining modal layering and negative resolution. In addition, we present an incompleteness result for modal layering together with ordered resolution.

Characterizing Frame Definability in Team Semantics via the Universal Modality

Abstract: Let Open image in new window denote the fragment of modal logic extended with the universal modality in which the universal modality occurs only positively. We characterize the definability of Open image in new window in the spirit of the well-known Goldblatt–Thomason theorem. We show that an elementary class \({\mathbb {F}}\) of Kripke frames is definable in Open image in new window if and only if \({\mathbb {F}}\) is closed under taking generated subframes and bounded morphic images, and reflects ultrafilter extensions and finitely generated subframes. In addition, we initiate the study of modal frame definability in team-based logics. We show that, with respect to frame definability, the logics Open image in new window , modal logic with intuitionistic disjunction, and (extended) modal dependence logic all coincide. Thus we obtain Goldblatt–Thomason -style theorems for each of the logics listed above.

A Strong and Rich 4-Valued Modal Logic Without Łukasiewicz-Type Paradoxes

Abstract: The aim of this paper is to introduce an alternative to Łukasiewicz’s 4-valued modal logic Ł. As it is known, Ł is afflicted by “Łukasiewicz (modal) type paradoxes”. The logic we define, PŁ4, is a strong paraconsistent and paracomplete 4-valued modal logic free from this type of paradoxes. PŁ4 is determined by the degree of truth-preserving consequence relation defined on the ordered set of values of a modification of the matrix MŁ characteristic for the logic Ł. On the other hand, PŁ4 is a rich logic in which a number of connectives can be defined. It also has a simple bivalent semantics of the Belnap–Dunn type.

Semantic pollution and syntactic purity

Abstract: Logical inferentialism claims that the meaning of the logical constants should be given, not model-theoretically, but by the rules of inference of a suitable calculus. It has been claimed that certain proof-theoretical systems, most particularly, labelled deductive systems for modal logic, are unsuitable, on the grounds that they are semantically polluted and suffer from an untoward intrusion of semantics into syntax. The charge is shown to be mistaken. It is argued on inferentialist grounds that labelled deductive systems are as syntactically pure as any formal system in which the rules define the meanings of the logical constants.

The expressive power of modal logic with inclusion atoms

Abstract: Modal inclusion logic is the extension of basic modal logic with inclusion atoms, and its semantics is defined on Kripke models with teams. A team of a Kripke model is just a subset of its domain. In this paper we give a complete characterisation for the expressive power of modal inclusion logic: a class of Kripke models with teams is definable in modal inclusion logic if and only if it is closed under k-bisimulation for some integer k, it is closed under unions, and it has the empty team property. We also prove that the same expressive power can be obtained by adding a single unary nonemptiness operator to modal logic. Furthermore, we establish an exponential lower bound for the size of the translation from modal inclusion logic to modal logic with the nonemptiness operator.

Modal logics of metric spaces

Abstract: It is a classic result (McKinsey & Tarski, 1944; Rasiowa & Sikorski, 1963) that if we interpret modal diamond as topological closure, then the modal logic of any dense-in-itself metric space is the well-known modal system S4. In this paper, as a natural follow-up, we study the modal logic of an arbitrary metric space. Our main result establishes that modal logics arising from metric spaces form the following chain which is order-isomorphic (with respect to the ⊃ relation) to the ordinal ω + 3:It follows that the modal logic of an arbitrary metric space is finitely axiomatizable, has the finite model property, and hence is decidable.

Deciding the First Levels of the Modal mu Alternation Hierarchy by Formula Construction

Abstract: We construct, for any sentence of the modal mu calculus Psi, derived sentences in the modal fragment and the fragment without least fixpoints of the modal mu calculus such that Psi is equivalent to a formula in these fragments if and only if it is equivalent to these formulas. The formula without greatest fixpoints that Psi is equivalent to if and only if it is equivalent to any formula without greatest fixpoint is obtained by duality. This yields a constructive proof of decidability of the first levels of the modal mu alternation hierarchy. The blow-up incurred by turning Psi into the modal formula is shown to be necessary: there are modal formulas that can be expressed sub-exponentially more efficiently with the use of fixpoints. For the fragments with only greatest or least fixpoints however, as long as formulas are in disjunctive form, the transformation into a formula syntactically in these fragments does not increase the size of the formula.

Final Coalgebras from Corecursive Algebras.

Abstract: We give a technique to construct a final coalgebra in which each element is a set of formulas of modal logic. The technique works for both the finite and the countable powerset functors. Starting with an injectively structured, corecursive algebra, we coinductively obtain a suitable subalgebra called the "co-founded part". We see—first with an example, and then in the general setting of modal logic on a dual adjunction—that modal theories form an injectively structured, corecursive algebra, so that this construction may be applied. We also obtain an initial algebra in a similar way. We generalize the framework beyond Set to categories equipped with a suitable factorization system, and look at the examples of Poset and Set-op .

Deontic Logic for Human Reasoning.

Abstract: Deontic logic is shown to be applicable for modelling human reasoning. For this the Wason selection task and the suppression task are discussed in detail. Different versions of modelling norms with deontic logic are introduced and in the case of the Wason selection task it is demonstrated how differences in the performance of humans in the abstract and in the social contract case can be explained. Furthermore, it is shown that an automated theorem prover can be used as a reasoning tool for deontic logic.

Reasoning with Global Assumptions in Arithmetic Modal Logics

Abstract: We establish a generic upper bound \(\textsc {ExpTime} \) for reasoning with global assumptions in coalgebraic modal logics. Unlike earlier results of this kind, we do not require a tractable set of tableau rules for the instance logics, so that the result applies to wider classes of logics. Examples are Presburger modal logic, which extends graded modal logic with linear inequalities over numbers of successors, and probabilistic modal logic with polynomial inequalities over probabilities. We establish the theoretical upper bound using a type elimination algorithm. We also provide a global caching algorithm that offers potential for practical reasoning.

A Modal Logic for Non-deterministic Information Systems

Abstract: In this article, we propose a modal logic for non-deterministic information systems. A deductive system for the logic is presented and corresponding soundness and completeness theorems are proved. The logic is also shown to be decidable.