Showing papers on "Modal operator published in 2014"

An Overview On Intuitionistic Fuzzy Sets

Abstract: We present a brief overview on Intuitionistic fuzzy sets which cuts across some definitions, operations, algebra, modal operators and normalization on Intuitionistic fuzzy set.

Fake Tense in Conditional Sentences: A Modal Approach

Abstract: Many languages allow for “fake” uses of their past tense marker: the marker: can occur in certain contexts without conveying temporal pastness. Instead it appears to bear a modal meaning. Iatridou (Linguist Inq 31(2):231–270, 2000) has dubbed this phenomenon Fake Tense. Fake Tense is particularly common to conditional constructions. This paper analyzes Fake Tense in English conditional sentences as a certain kind of ambiguity: the past tense morphology can mark the presence of a temporal operator, but it can also signal a specific modal operator. The ambiguity is proposed to be the result of recategorization: the Simple Past develops a second, modal meaning because of structural similarities between the temporal and the modal/epistemic domain. The proposal is spelled out in the generative semantics framework (Heim and Kratzer, Semantics in generative grammar, 1998), using the restrictor approach to conditionals (Kratzer, in: R. Bauerle et al. (eds.) Semantics from different points of view, 1979; in: A. von Stechow and D. Wunderlich (eds.) Semantics: an international handbook of contemporary research, 1991), and building on von Stechows et al.’s recent work on the English tense system (von Stechow, in: W. Klein (ed.) The expression of time in language, 2010; Romero and von Stechow, Tense: introduction, 2008).

Cross-linguistic variation in imperfectivity

Abstract: The paper examines variation in the interpretations of imperfectives in Slavic, Romance, and Je (Mẽbengokre). It develops a core modal analysis for an imperfective operator (IMPF) within situation semantics, coupled with language-specific constraints formally encoded in modal bases. Cross-linguistic contrasts in the interpretation of imperfectives are explained in terms of variation in modal bases for IMPF, lexicalization patterns, and its interactions with other operators. The proposal accounts for why Romance languages use imperfectives to make reference to past plans while most Slavic languages do not, as well as for narrative uses specific to Romance languages, and factual uses specific to some Slavic languages. The proposal also accounts for lexically specified aspectual operators in Mẽbengokre, as well as language-specific interaction between IMPF and other modal operators, as in the Bulgarian Renarrated Mood, and two different semantic instances of Slavic Involuntary States. Appealing to cross-linguistic evidence to argue for a view according to which IMPF makes significant semantic contributions in all occurrences, the paper shows how a modal analysis can account for well-known temporal properties of imperfectives. It also demonstrates that data from closely related as well as unrelated languages provide evidence for an invariant semantic core behind imperfectivity.

Reasoning about Knowledge and Strategies: Epistemic Strategy Logic

Abstract: In this paper we introduce Epistemic Strategy Logic (ESL), an extension of Strategy Logic with modal operators for individual knowledge. This enhanced framework allows us to represent explicitly and to reason about the knowledge agents have of their own and other agents' strategies. We provide a semantics to ESL in terms of epistemic concurrent game models, and consider the corresponding model checking problem. We show that the complexity of model checking ESL is not worse than (non-epistemic) Strategy Logic

Roles, Rigidity, and Quantification in Epistemic Logic

Abstract: Epistemic modal predicate logic raises conceptual problems not faced in the case of alethic modal predicate logic: Frege’s “Hesperus-Phosphorus” problem—how to make sense of ascribing to agents ignorance of necessarily true identity statements—and the related “Hintikka-Kripke” problem—how to set up a logical system combining epistemic and alethic modalities, as well as others problems, such as Quine’s “Double Vision” problem and problems of self-knowledge. In this paper, we lay out a philosophical approach to epistemic predicate logic, implemented formally in Melvin Fitting’s First-Order Intensional Logic, that we argue solves these and other conceptual problems. Topics covered include: Quine on the “collapse” of modal distinctions; the rigidity of names; belief reports and unarticulated constituents; epistemic roles; counterfactual attitudes; representational versus interpretational semantics; ignorance of co-reference versus ignorance of identity; two-dimensional epistemic models; quantification into epistemic contexts; and an approach to multi-agent epistemic logic based on centered worlds and hybrid logic.

The Expressive Power of Modal Dependence Logic

Abstract: We study the expressive power of various modal logics with team semantics. We show that exactly the properties of teams that are downward closed and closed under team k-bisimulation, for some finite k, are definable in modal logic extended with intuitionistic disjunction. Furthermore, we show that the expressive power of modal logic with intuitionistic disjunction and extended modal dependence logic coincide. Finally we establish that any translation from extended modal dependence logic into modal logic with intuitionistic disjunction increases the size of some formulas exponentially.

Modal Logics of Uncertainty with Two-Layer Syntax: A General Completeness Theorem

Abstract: Modal logics with two syntactical layers both governed by classical logic have been proposed as logics of uncertainty following Hamblin's seminal idea of reading the modal operator i¾?i¾? as 'probably i¾?', meaning that the probability of i¾? is bigger than a given threshold. An interesting departure from that classical paradigm has been introduced by Hajek with his fuzzy probability logic when, while still keeping classical logic as interpretation of the lower syntactical layer, he proposed to use Łukasiewicz logic in the upper one, so that the truth degree of i¾?i¾? could be directly identified with the probability of i¾?. Later, other authors have used the same formalism with different kinds of uncertainty measures and other pairs of logics, allowing for a treatment of uncertainty of vague events i.e. also changing the logic in the lower layer. The aim of this paper is to provide a general framework for two-layer modal logics that encompasses all the previously studied two-layer modal fuzzy logics, provides a general axiomatization and a semantics of measured Kripke frames, and prove a general completeness theorem.

Partiality and Adjointness in Modal Logic

Abstract: Following a proposal of Humberstone, this paper studies a semantics for modal logic based on partial “possibilities” rather than total “worlds.” There are a number of reasons, philosophical and mathematical, to find this alternative semantics attractive. Here we focus on the construction of possibility models with a finitary flavor. Our main completeness result shows that for a number of standard modal logics, we can build a canonical possibility model, wherein every logically consistent formula is satisfied, by simply taking each individual finite formula (modulo equivalence) to be a possibility, rather than each infinite maximally consistent set of formulas as in the usual canonical world models. Constructing these locally finite canonical models involves solving a problem in general modal logic of independent interest, related to the study of adjoint pairs of modal operators: for a given modal logic L, can we find for every formula φ a formula f(φ) such that for every formula ψ, φ → []ψ is provable in L if and only if f(φ) → ψ is provable in L? We answer this question for a number of standard modal logics, using model-theoretic arguments with world semantics. This second main result allows us to build for each logic a canonical possibility model out of the lattice of formulas related by provable implication in the logic.

Two-dimensional semantics and the nesting problem

Abstract: Graeme Forbes (2011) raises some problems for two-dimensional semantic theories. The problems concern nested environments: linguistic environments where sentences are nested under both modal and epistemic operators. Closely related problems involving nested environments have been raised by Scott Soames (2005) and Josh Dever (2007). Soames goes so far as to say that nested environments pose the �chief technical problem� for strong two-dimensionalism. We call the problem of handling nested environments within two-dimensional semantics �the nesting problem�. We show that the two-dimensional semantics for attitude ascriptions developed in Chalmers (2011a) has no trouble accommodating certain forms of the nesting problem that involve factive verbs such as �know� or �establish�. A certain form of the nesting problem involving apriority and necessity operators does raise an interesting puzzle, but we show how a generalized version of the nesting problem arises independently of two-dimensional semantics�it arises, in fact, for anyone who accepts the contingent a priori. We, then, provide a two-dimensional treatment of the apriority operator that fits the two-dimensional treatment of attitude verbs and apply it to the generalized nesting problem. We conclude that two-dimensionalism is not seriously threatened by cases involving the nesting of epistemic and modal operators.

The Undergeneration of Permutation Invariance as a Criterion for Logicality

Abstract: Permutation invariance is often presented as the correct criterion for logicality. The basic idea is that one can demarcate the realm of logic by isolating specific entities—logical notions or constants—and that permutation invariance would provide a philosophically motivated and technically sophisticated criterion for what counts as a logical notion. The thesis of permutation invariance as a criterion for logicality has received considerable attention in the literature in recent decades, and much of the debate is developed against the background of ideas put forth by Tarski in a 1966 lecture (Tarski 1966/1986). But as noted by Tarski himself in the lecture, the permutation invariance criterion yields a class of putative ‘logical constants’ that are essentially only sensitive to the number of elements in classes of individuals. Thus, to hold the permutation invariance thesis essentially amounts to limiting the scope of logic to quantificational phenomena, which is controversial at best and possibly simply wrong. In this paper, I argue that permutation invariance is a misguided approach to the nature of logic because it is not an adequate formal explanans for the informal notion of the generality of logic. In particular, I discuss some cases of undergeneration of the criterion, i.e. the fact that it excludes from the realm of logic operators that we have good reason to regard as logical, especially some modal operators.

Reasoning about uncertainty and explicit ignorance in generalized possibilistic logic

Abstract: Generalized possibilistic logic (GPL) is a logic for reasoning about the revealed beliefs of another agent. It is a two-tier propositional logic, in which propositional formulas are encapsulated by modal operators that are interpreted in terms of uncertainty measures from possibility theory. Models of a GPL theory represent weighted epistemic states and are encoded as possibility distributions. One of the main features of GPL is that it allows us to explicitly reason about the ignorance of another agent. In this paper, we study two types of approaches for reasoning about ignorance in GPL, based on the idea of minimal specificity and on the notion of guaranteed possibility, respectively. We show how these approaches naturally lead to different flavours of the language of GPL and a number of decision problems, whose complexity ranges from the first to the third level of the polynomial hierarchy.

Reasoning about Knowledge and Strategies: Epistemic Strategy Logic

Abstract: In this paper we introduce Epistemic Strategy Logic (ESL), an extension of Strategy Logic with modal operators for individual knowledge. This enhanced framework allows us to represent explicitly and to reason about the knowledge agents have of their own and other agents’ strategies. We provide a semantics to ESL in terms of epistemic concurrent game models, and consider the corresponding model checking problem. We show that the complexity of model checking ESL is not worse than (non-epistemic) Strategy Logic.

Information, Issues, and Attention

Abstract: A sentence is informative if there are possible worlds that are eliminated from the common ground by each of the proposed updates, and it is inquisitive if it proposes two or more alternative updates, requesting information from other participants in order to establish at least one of these updates. This chapter argues that this notion of meaning has an additional advantage. It discusses a recapitulation of inquisitive semantics, and presents the definition of the semantics. The chapter shows how attentive content can be captured in a natural extension of this framework. It examines pragmatic aspects of the interpretation of sentences that are not merely informative, but also inquisitive and/or attentive. The chapter describes the behaviour of might in certain embedded contexts, and argues that the semantic meaning of might sentences is strengthened in a particular way before being composed with the semantic meaning of the embedding operator. Keywords: attentive content; epistemic modal operator; informative content; inquisitive semantics

Dualities for modal N4-lattices

Abstract: We introduce a new Priestley-style topological duality for N4-lattices, which are the algebraic counterpart of paraconsistent Nelson logic. Our duality differs from the existing one, due to S. Odintsov, in that we only rely on Esakia duality for Heyting algebras and not on the duality for De Morgan algebras of Cornish and Fowler. A major advantage of our approach is that we obtain a simple description for our topological structures, which allows us to extend the duality to other algebraic structures such as N4-lattices with monotonic modal operators, and also to provide a neighborhood semantics for the non-normal modal logic corresponding to these algebras.

Belief as Willingness to Bet.

Abstract: We investigate modal logics of high probability having two unary modal operators: an operator $K$ expressing probabilistic certainty and an operator $B$ expressing probability exceeding a fixed rational threshold $c\geq\frac 12$. Identifying knowledge with the former and belief with the latter, we may think of $c$ as the agent's betting threshold, which leads to the motto "belief is willingness to bet." The logic $\mathsf{KB.5}$ for $c=\frac 12$ has an $\mathsf{S5}$ $K$ modality along with a sub-normal $B$ modality that extends the minimal modal logic $\mathsf{EMND45}$ by way of four schemes relating $K$ and $B$, one of which is a complex scheme arising out of a theorem due to Scott. Lenzen was the first to use Scott's theorem to show that a version of this logic is sound and complete for the probability interpretation. We reformulate Lenzen's results and present them here in a modern and accessible form. In addition, we introduce a new epistemic neighborhood semantics that will be more familiar to modern modal logicians. Using Scott's theorem, we provide the Lenzen-derivative properties that must be imposed on finite epistemic neighborhood models so as to guarantee the existence of a probability measure respecting the neighborhood function in the appropriate way for threshold $c=\frac 12$. This yields a link between probabilistic and modal neighborhood semantics that we hope will be of use in future work on modal logics of qualitative probability. We leave open the question of which properties must be imposed on finite epistemic neighborhood models so as to guarantee existence of an appropriate probability measure for thresholds $c eq\frac 12$.

Avicenna on Possibility and Necessity

Abstract: In this paper, I raise the following problem: How does Avicenna define modalities? What oppositional relations are there between modal propositions, whether quantified or not? After giving Avicenna's definitions of possibility, necessity and impossibility, I analyze the modal oppositions as they are stated by him. This leads to the following results: The relations between the singular modal propositions may be represented by means of a hexagon. Those between the quantified propositions may be represented by means of two hexagons that one could relate to each other.This is so because the exact negation of the bilateral possible, i.e. ‘necessary or impossible’ is given and applied to the quantified possible propositions.Avicenna distinguishes between the scopes of modality which can be either external (de dicto) or internal (de re). His formulations are external unlike al-Farab;’s ones. However his treatment of modal oppositions remains incomplete because not all the relations between the modal propositi...

Concurrent Dynamic Algebra

Abstract: We reconstruct Peleg's concurrent dynamic logic in the context of modal Kleene algebras. We explore the algebraic structure of its multirelational semantics and develop an abstract axiomatisation of concurrent dynamic algebras from that basis. In this axiomatisation, sequential composition is not associative. It interacts with concurrent composition through a weak distributivity law. The modal operators of concurrent dynamic algebra are obtained from abstract axioms for domain and antidomain operators; the Kleene star is modelled as a least fixpoint. Algebraic variants of Peleg's axioms are shown to be valid in these algebras and their soundness is proved relative to the multirelational model. Additional results include iteration principles for the Kleene star and a refutation of variants of Segerberg's axiom in the multirelational setting. The most important results have been verified formally with Isabelle/HOL.

Logics with Copy and Remove

Abstract: We propose a logic with the dynamic modal operators copy and remove. The copy operator replicates a given model, and the remove operator removes paths in a given model. We show that the product update by an action model with Boolean pre-conditions in dynamic epistemic logic decomposes in copy and remove operations. We also show that copy and remove operators of path of length 1 can be expressed by action models. We investigate the complexity of the satisfiability problem of syntactic fragments of the logic with copy and remove operations.

Knowability as potential knowledge

Abstract: The thesis that every truth is knowable is usually glossed by decomposing knowability into possibility and knowledge. Under elementary assumptions about possibility and knowledge, considered as modal operators, the thesis collapses the distinction between truth and knowledge (as shown by the so-called Fitch-argument). We show that there is a more plausible interpretation of knowability—one that does not decompose the notion in the usual way—to which the Fitch-argument does not apply. We call this the potential knowledge-interpretation of knowability. We compare our interpretation with the rephrasal of knowability proposed by Edgington and Rabinowicz and Segerberg, inserting an actuality-operator. This proposal shares some key features with ours but suffers from requiring specific transworld-knowledge. We observe that potential knowledge involves no transworld-knowledge. We describe the logic of potential knowledge by providing models for interpreting the new operator. Finally we show that the knowability thesis can be added to elementary conditions on potential knowledge without collapsing modal distinctions.

Modes of adjointness

Abstract: The fact that many modal operators are part of an adjunction is probably folklore since the discovery of adjunctions. On the other hand, the natural idea of a minimal propositional calculus extended with a pair of adjoint operators seems to have been formulated only very recently. This recent research, mainly motivated by applications in computer science, concentrates on technical issues related to the calculi and not on the significance of adjunctions in modal logic. It then seems a worthy enterprise (both for these contemporary topical pursuits and also for historical interest) to trace the concept of adjunction back to the origins of the algebraic semantics of modal logic and to make explicit its ubiquity in this branch of mathematics.

Constructive empiricism, partial structures and the modal interpretation of quantum mechanics

Abstract: Van Fraassen's modal interpretation of non-relativistic quantum mechanics is articulated to support an anti-realist account of quantum theory. However, given the particular form of van Fraassen's anti-realism (constructive empiricism), two problems arise when we try to make it compatible with the modal interpretation: one difficulty concerns the tension between the need for modal operators in the modal interpretation and van Fraassen's skepticism regarding real modality in nature; another addresses the need for the truth predicate in the modal interpretation and van Fraassen's rejection of truth as the aim of science. After examining these two problems, I suggest a formal framework in which they can be accommodated – using da Costa and French's partial structures approach – and indicate a variant of van Fraassen's modal interpretation that does not face these difficulties. Quanta 2014; 3: 1–15.

The future in Greek and Italian: metaphysical and epistemic dimensions

Abstract: While the question of whether future morphemes in languages denote temporal or modal operators has been central in formal semantics, most analyses agree that such morphemes convey modality, and do not simply make reference to future times. The modality is often assumed to be purely metaphysical (e.g. Thomason 1984, Kaufmann, 2005). In this paper, we present novel data from Greek and Italian showing a systematic availability of purely epistemic readings with the future morphemes (FUT) alongside the predictive readings. We propose a fully Kratzerian account (following closely Portner 2009), and argue for a common semantic core. FUT is nonveridical in both cases: the modal space is partitioned intop and:p worlds, and FUT universally quantifies over the Bestp worlds established by the ordering sources, which are reasonability and knowledge relevant to the sentence (called the future criterion). With universal quantification over Best worlds an underlying bias is revealed towards those worlds; therefore in our analysis the future is both weak (nonveridical metaphysical and epistemic space) and strong, because of the bias. Our analysis enriches the metaphysical modality of the future with epistemic components, captures the common core of the predictive and epistemic FUT, and provides simple tools for dealing both with the novel facts of Greek and Italian, as well as apparent Moore paradoxical effects observed with future expressions and MUST.

On the relative succinctness of modal logics with union, intersection and quantification

Abstract: In the study of knowledge representation formalisms, there is a current interest in the question of how different formal languages compare in their ability to compactly express semantic properties. Recently, French et al. have shown that modal logics with a modality for public announcement, for everybody knows, and for somebody knows are all exponentially more succinct than basic modal logic. In this paper we compare the above mentioned logics not with basic modal logic but with each other and also with modal logics that have a modality for distributed knowledge. Interestingly, modal logic with such a modality is more expressive than the other modal logics mentioned, but still we can show that some of those weaker logics are exponentially more succinct than the former. Additionally, we prove that the opposite is also possible: indeed, we show that modal logic with a modality for distributed knowledge is more succinct than modal logic with a modality for everybody knows.

Modality and Axiomatic Theories of Truth I: Friedman-Sheard

Abstract: In this investigation we explore a general strategy for constructing modal theories where the modal notion is conceived as a predicate. The idea of this strategy is to develop modal theories over axiomatic theories of truth. In this first paper of our two part investigation we develop the general strategy and then apply it to the axiomatic theory of truth Friedman-Sheard. We thereby obtain the theory Modal Friedman-Sheard. The theory Modal Friedman-Sheard is then discussed from three different perspectives. First, we show that Modal Friedman-Sheard preserves theoremhood modulo translation with respect to modal operator logic. Second, we turn to semantic aspects and develop a modal semantics for the newly developed theory. Third, we investigate whether the modal predicate of Modal Friedman-Sheard can be understood along the lines of a proposal of Kripke, namely as a truth predicate modified by a modal operator.

The Adequacy of Genuine Modal Realism

Abstract: What are the requirements on an adequate genuine modal realist analysis of modal discourse? One is material adequacy: the modal realist must provide for each candidate analysandum an analysans in the language of counterpart theory which by his lights has the same truth value as the candidate analysandum. Must the material biconditional joining these be necessarily true? This is the requirement of strict adequacy. It is not satisfied if Lewis�s 1968 scheme provides the analysis. John Divers puts forward a modification, which identifies cases of �advanced modalizing� in which the modal operator is semantically redundant. Even with this modification modal realist analyses of statements of modal discourse will be strictly inadequate. Strict adequacy can be achieved by extending the redundancy interpretation to all de dicto modal statements. The price is the denial of de dicto contingency. But perhaps material adequacy is enough. If the modal realist has a systematic means of replacing every sentence of quantified modal logic which he considers true by a sentence of counterpart theory that he considers true, perhaps he need do no more. Still, traditionally philosophical analysis aims at strict adequacy so it is as well to know that this is a test the modal realist analysis fails unless he abandons de dicto contingency.

Logical Operators for Ontological Modeling

Abstract: We show that logic has more to offer to ontologists than standard first order and modal operators. We first describe some operators of linear logic which we believe are particularly suitable for ontological modeling, and suggest how to interpret them within an ontological framework. After showing how they can coexist with those of classical logic, we analyze three notions of artifact from the literature to conclude that these linear operators allow for reducing the ontological commitment needed for their formalization, and even simplify their logical formulation.

Modality and Axiomatic Theories of Truth II: Kripke-Feferman

Abstract: In this second and last paper of the two part investigation on "Modality and Axiomatic Theories of Truth" we apply a general strategy for constructing modal theories over axiomatic theories of truth to the theory Kripke-Feferman. This general strategy was developed in the first part of our investigation. Applying the strategy to Kripke-Feferman leads to the theory Modal Kripke-Feferman which we discuss from the three perspectives that we had already considered in the first paper, where we discussed the theory Modal Friedman-Sheard. That is, we first show that Modal Kripke-Feferman preserves theoremhood modulo translation with respect to modal operator logic. Second, we develop a modal semantics fitting the newly developed theory. Third, we investigate whether the modal predicate of Modal Kripke-Feferman can be understood along the lines of a proposal of Kripke, namely as a truth predicate modified by a modal operator.

Three-Valued Modal Logic for Qualitative Comparative Policy Analysis with Crisp-Set QCA

Abstract: Contradictory and missing outcomes are problems common to many qualitative comparative studies, based on the methodology of crisp-set QCA. They also occur in public policy analyses, e.g. if important background variables are omitted or outcomes of new policies are technically censored. As a new solution to these problems, this article proposes the use of three-valued modal logic, originally introduced by the Polish philosopher Jan Lukasiewicz (1970). In addition to true and false, indeterminate is the third truth-value in this alternative approach, which serves to code missing or contradictory data. Moreover, modal operators allow a differentiation between strict and possible triggers and inhibitors of policy outcomes. The advantages of three-valued modal logic in crisp-set QCA are illustrated by an empirical example from comparative welfare policy analysis. Its conclusions allow comparisons with the corresponding results from a conventional crisp-set QCA of the same data-set.

Verification of non-uniform and unbounded artifact-centric systems: decidability through abstraction

Abstract: The formal verification of Artifact-centric (AC) systems is a subject of growing interest in the Service Oriented Computing (SOC) community, which can benefit from techniques developed for Multi-agent systems and knowledge reasoning and representation. In the present contribution we consider the verification of AC systems that do not necessarily satisfy boundedness and uniformity, the typical assumptions used to prove decidability of the model checking problem in this setting. We provide a partial model checking procedure for agent-based AC systems against a first-order temporal logic that includes modal operators for agent knowledge. Interestingly, we obtain this result by introducing a counterpart semantics for first-order modal logic, and by defining notions of simulation and abstraction for this setting. This allows us to generate finite abstractions of infinite-state AC systems, even when these are not bounded nor uniform, thus enabling us to perform verification also in cases not covered by the current state-of-the-art.