Showing papers on "Modal operator published in 2018"

A unified analysis of the future as epistemic modality the view from Greek and Italian

Abstract: We offer an analysis of the Greek and Italian future morphemes as epistemic modal operators. The main empirical motivation comes from the fact that future morphemes have systematic purely epistemic readings-not only in Greek and Italian, but also in Dutch, German, and En-glish will. The existence of epistemic readings suggests that the future expressions quantify over epistemic, not metaphysical alternatives. We provide a unified analysis for epistemic and predic-tive readings as epistemic necessity, and the shift between the two is determined compositionally by the lower tense. Our account thus acknowledges a systematic interaction between modality and tense-but the future itself is a pure modal, not a mixed temporal/modal operator. We show that the modal base of the future is nonveridical, i.e. it includes p and ¬p worlds, parallel to epistemic modals such as must, and present arguments that future morphemes have much in common with epistemic modals and predicates of personal taste. We identify, finally, a subclass of epistemic futures which are ratificational, and argue that will is a member of this class.

Internal Universes in Models of Homotopy Type Theory

Abstract: We show that universes of fibrations in various models of homotopy type theory have an essentially global character: they cannot be described in the internal language of the presheaf topos from which the model is constructed. We get around this problem by extending the internal language with a modal operator for expressing properties of global elements. In this setting we show how to construct a universe that classifies the Cohen-Coquand-Huber-M\"ortberg (CCHM) notion of fibration from their cubical sets model, starting from the assumption that the interval is tiny - a property that the interval in cubical sets does indeed have. This leads to a completely internal development of models of homotopy type theory within what we call crisp type theory.

The semantic roots of positive polarity: epistemic modal verbs and adverbs in English, Greek and Italian

Abstract: Epistemic modal verbs and adverbs of necessity are claimed to be positive polarity items. We study their behavior by examining modal spread, a phenomenon that appears redundant or even anomalous, since it involves two apparent modal operators being interpreted as a single modality. We propose an analysis in which the modal adverb is an argument of the MUST modal, providing a meta-evaluation $$\mathcal {O}$$ which ranks the Ideal, stereotypical worlds in the modal base as better possibilities than the Non-Ideal worlds in it. MUST and possibility modals differ in that the latter have an empty $$\mathcal {O}$$ , a default that can be negotiated. Languages vary in the malleability of this parameter. Positive polarity is derived as a conflict between the ranking imposed by $$\mathcal {O}$$ —which requires that the Ideal worlds be better possibilities than Non-Ideal worlds—and the effect of higher negation which renders the Ideal set non-homogenous. Applying the ordering over such a non-homogeneous set would express preference towards both p and $$\lnot p $$ worlds thus rendering the sentence uninformative. Negative polarity MUST and possibility modals, on the other hand, contain an empty $$\mathcal {O}$$ , application of higher negation therefore poses no problem. This account is the first to connect modal spread to positive polarity of necessity modals, and captures the properties of both in a unified analysis.

Epistemic Logic Programs with World View Constraints

Abstract: An epistemic logic program is a set of rules written in the language of Epistemic Specifications, an extension of the language of answer set programming that provides for more powerful introspective reasoning through the use of modal operators K and M. We propose adding a new construct to Epistemic Specifications called a world view constraint that provides a universal device for expressing global constraints in the various versions of the language. We further propose the use of subjective literals (literals preceded by K or M) in rule heads as syntactic sugar for world view constraints. Additionally, we provide an algorithm for finding the world views of such programs.

Gamma graph calculi for modal logics

Abstract: We describe Peirce’s 1903 system of modal gamma graphs, its transformation rules of inference, and the interpretation of the broken-cut modal operator. We show that Peirce proposed the normality rule in his gamma system. We then show how various normal modal logics arise from Peirce’s assumptions concerning the broken-cut notation. By developing an algebraic semantics we establish the completeness of fifteen modal logics of gamma graphs. We show that, besides logical necessity and possibility, Peirce proposed an epistemic interpretation of the broken-cut modality, and that he was led to analyze constructions of knowledge in the style of epistemic logic.

There is more to negation than modality

Abstract: There is a relatively recent trend in treating negation as a modal operator. One such reason is that doing so provides a uniform semantics for the negations of a wide variety of logics and arguably speaks to a longstanding challenge of Quine put to non-classical logics. One might be tempted to draw the conclusion that negation is a modal operator, a claim Francesco Berto (Mind, 124(495), 761–793, 2015) defends at length in a recent paper. According to one such modal account, the negation of a sentence is true at a world x just in case all the worlds at which the sentence is true are incompatible with x. Incompatibility is taken to be the key notion in the account, and what minimal properties a negation has comes down to which minimal conditions incompatibility satisfies. Our aims in this paper are twofold. First, we wish to point out problems for the modal account that make us question its tenability on a fundamental level. Second, in its place we propose an alternative, non-modal, account of negation as a contradictory-forming operator that we argue is superior to, and more natural than, the modal account.

Internal Universes in Models of Homotopy Type Theory

Abstract: We begin by recalling the essentially global character of universes in various models of homotopy type theory, which prevents a straightforward axiomatization of their properties using the internal language of the presheaf toposes from which these model are constructed. We get around this problem by extending the internal language with a modal operator for expressing properties of global elements. In this setting we show how to construct a universe that classifies the Cohen-Coquand-Huber-Mortberg (CCHM) notion of fibration from their cubical sets model, starting from the assumption that the interval is tiny - a property that the interval in cubical sets does indeed have. This leads to an elementary axiomatization of that and related models of homotopy type theory within what we call crisp type theory.

A Survey of Advances in Epistemic Logic Program Solvers

Abstract: Recent research in extensions of Answer Set Programming has included a renewed interest in the language of Epistemic Specifications, which adds modal operators K ("known") and M ("may be true") to provide for more powerful introspective reasoning and enhanced capability, particularly when reasoning with incomplete information. An epistemic logic program is a set of rules in this language. Infused with the research has been the desire for an efficient solver to enable the practical use of such programs for problem solving. In this paper, we report on the current state of development of epistemic logic program solvers.

A temporal modal defeasible logic for formalizing social commitments in dialogue and argumentation models

Abstract: In this paper, we extend a temporal defeasible logic with a modal operator Committed to formalize commitments that agents undertake as a consequence of communicative actions (speech acts) during dialogues. We represent commitments as modal sentences. The defeasible dual of the modal operator Committed is a modal operator called Exempted. The logical setting makes the social-commitment based semantics of speech acts verifiable and practical; it is possible to detect if, and when, a commitment is violated and/or complied with. One of the main advantages of the proposed system is that it allows for capturing the nonmonotonic behavior of the commitments induced by the relevant speech acts.

Do modals take propositions or sets of propositions? Evidence from Japanese darou

Abstract: The Japanese modal particle darou can take either a declarative or an interrogative prejacent (Hara 2006; Hara & Davis 2013). We point out, however, that its interrogative-embedding use cannot be reduced to its declarative-embedding use. This is problematic under the standard assumption that modal operators always apply to propositions, but not under more recent proposals which take modal op- erators to apply to sets of propositions. We develop a detailed account of darou, capturing its non-reductive nature as well as its puzzling interaction with intonation (Hara 2015).

On the Complexity of Modal Separation Logics

Abstract: We introduce a modal separation logic MSL whose models are memory states from separation logic and the logical connectives include modal operators as well as separating conjunction and implication from separation logic. With such a combination of operators, some fragments of MSL can be seen as genuine modal logics whereas some others capture standard separation logics, leading to an original language to speak about memory states. We analyse the decidability status and the computational complexity of several fragments of MSL, leading to surprising results, obtained by designing proof methods that take into account the modal and separation features of MSL. For example, the satisfiability problem for the fragment of MSL with 3, the inequality modality = and separating conjunction * is shown Tower-complete whereas the restriction either to 3 and * or to = and * is only NP-complete.

On the Most Extended Modal Operator of First Type over Interval-Valued Intuitionistic Fuzzy Sets

Abstract: The definition of the most extended modal operator of first type over interval-valued intuitionistic fuzzy sets is given, and some of its basic properties are studied.

A note on strong axiomatization of G\"odel Justification Logic

Abstract: Justification Logics are special kinds of modal logics which provide a framework for reasoning about epistemic justification. For this, they extend classical boolean propositional logic by a family of necessity-style modal operators "$t:$", indexed over $t$ by a corresponding set of justification terms, which thus explicitly encode the justification for the necessity assertion in the syntax. With these operators, one can therefore not only reason about modal effects on propositions but also about dynamics inside the justifications themselves. We replace this classical boolean base with Godel logic, one of the three most prominent fuzzy logics, i.e. special instances of many-valued logics, taking values in the unit interval $[0,1]$, which are intended to model inference under vagueness. We extend the canonical possible-world semantics for Justification Logic to this fuzzy realm by considering fuzzy accessibility- and evaluation-functions evaluated over the minimum t-norm and establish strong completeness theorems for various fuzzy analogies of prominent extensions for basic Justification Logic.

On modal logics of model-theoretic relations.

Abstract: Given a class $\mathcal C$ of models, a binary relation ${\mathcal R}$ between models, and a model-theoretic language $L$, we consider the modal logic and the modal algebra of the theory of $\mathcal C$ in $L$ where the modal operator is interpreted via $\mathcal R$. We discuss how modal theories of $\mathcal C$ and ${\mathcal R}$ depend on the model-theoretic language, their Kripke completeness, and expressibility of the modality inside $L$. We calculate such theories for the submodel and the quotient relations. We prove a downward Lowenheim--Skolem theorem for first-order language expanded with the modal operator for the extension relation between models.

On interval-valued intuitionistic fuzzy modal operators

Abstract: An survey of the existing interval-valued intuitionistic fuzzy modal operators is given. Eight new operators are introduced that extend the older ones. Some of their basic properties are discussed. Open problems are formulated.

Putting the Agents back in the Domain: A Two-Sorted Term-Modal Logic.

Abstract: In the present paper syntax and semantics will be presented for an expansion of ordinary n-agent QML with constant domain, non-rigid constants, rigid variables and including both functions, relations, and equality. Further, the number of agents will be specified axiomatically thus ensuring maximal flexibility wrt. the cardinality of the set of agents. Domain, variables, and constants will be partitioned in an agent-part and an object-part and the syntax will be expanded to include strings in which indexes of modal operators are quantified over as wff's of the language. This will enhance expressiveness regarding the epistemic status of agents. Such a term-modal version of the logic K is shown to be sound and complete wrt. the class of (appropriate) frames, and a term-version of S4 is shown to be sound and complete wrt. the class of (appropriate) frames in which the relations are transitive. It should be noted that completeness is shown via the framework of canonical models and thus allows for non-complicated generalizations to other logics than the term-versions of K and S4.

On modal operators over the generalized intuitionistic fuzzy set

Abstract: In recent years, different operators (modal, topological, level, negation and other types) have been defined over intuitionistic fuzzy sets. In this note, modal type of operators over an extended generalized intuitionistic fuzzy set GIFS B are proposed. Some of the basic properties of the new operators are derived.

Generics and Weak Necessity

Abstract: A prevailing thought is that generics have a covert modal operator at logical form. I claim that if this is right, the covert generic modality is a weak necessity modal. In this paper, I provide ev...

Dealing with conceptual indeterminacy: A framework based on supervaluation semantics

Abstract: It is widely accepted that most natural language terms do not have precise universally agreed definitions that fix their meanings. Instead, humans use terms in a variety of ways that adapt to different contexts and points of view. In this paper we present a framework based on Supervaluation Semantics for interpreting languages in the presence of semantic variability. This work builds on supervaluationist accounts, which explain linguistic indeterminacy in terms of a collection of possible precise interpretations of the terms of the language. We extend the basic supervaluation semantics by adding the notion of standpoint. A multi-modal logical language for describing standpoints is presented. The language includes a modal operator ⬜s for each standpoint s, such that ⬜s φ means that proposition φ is unequivocally true according to standpoint s — i.e. φ is true at all precisifications compatible with s. We show how it can be used to represent logical properties and connections between alternative ways of describing a domain and different accounts of the semantics of terms.

A Study of Subminimal Logics of Negation and Their Modal Companions

Abstract: We study propositional logical systems arising from the language of Johansson’s minimal logic and obtained by weakening the requirements for the negation operator. We present their semantics as a variant of neighbourhood semantics. We use duality and completeness results to show that there are uncountably many subminimal logics. We also give model-theoretic and algebraic definitions of filtration for minimal logic and show that they are dual to each other. These constructions ensure that the propositional minimal logic has the finite model property. Finally, we define and investigate bi-modal companions with non-normal modal operators for some relevant subminimal systems, and give infinite axiomatizations for these bi-modal companions.

The Logic of “Most” and “Mostly”

Abstract: The paper suggests a modal predicate logic that deals with classical quantification and modalities as well as intermediate operators, like “most” and “mostly”. Following up the theory of generalized quantifiers, we will understand them as two-placed operators and call them determiners. Quantifiers as well as modal operators will be constructed from them. Besides the classical deduction, we discuss a weaker probabilistic inference “therefore, probably” defined by symmetrical probability measures in Carnap’s style. The given probabilistic inference relates intermediate quantification to singular statements: “Most S are P” does not logically entail that a particular individual S is also P, but it follows that this is probably the case, where the probability is not ascribed to the propositions but to the inference. We show how this system deals with single case expectations while predictions of statistical statements remain generally problematic.

Simple example of weak modal logic based on intuitionistic core.

Abstract: In this paper we present simple example of propositional logic which has one modal operator and is based on intuitionistic core. This system is very weak in modal sense - e.g. rules of regularity or monotonicity do not hold. It has complete semantics composed of possible worlds equipped with neighborhoods and pre-order relation. We discuss certain restrictions imposed on those structures. Also, we present characterization of axiom 4 known from logic S4.

The First-Person Singular Imperative In Chukchi

Abstract: In this paper, I consider the semantics and distribution of the first-person singular from in the Chukchi language. The main aim of the present study is to describe different contexts in which this form can be used and provide a formal analysis to its syntactic and semantic properties. I show that the distribution of this form is non-trivial and challenging for the current theories of the Imperative. In addition to standard uses in root (non-embedded) contexts, this form can appear dependent clauses with a desiderative predicate and in rationale clauses. Although the availability of embedded of Imperative forms have been reported for a number of languages (for an overview see [Kaufmann 2014]), I am not aware of any studies that would state the possibility of using such forms in rationale clauses. Taking as a starting point the theory developed in [Stegovec 2018], I propose and examine a hypothesis that the productivity of the form in root contexts is due to the fact the imperative modal operator can have event antecedents

Possibilistic Reasoning About Actions in Agent Systems

Abstract: In reasoning about games, we can understand players' behaviors according to their belief, action, and preference. While modal logic can be easily used to represent and reason about agents' beliefs and knowledge if we adopt an epistemic reading of modal operators, reasoning about action requires the extension of modalities. Dynamic logic is one of the earliest attempt along this direction. The original motivation of dynamic logic is to reason about program. However, it can be applied to any structural set of actions. In this paper, we propose a graded propositional dynamic logic (gPDL) for possibilistic reasoning about regular program.

A Modal and Relevance Logic for Qualitative Spatial Reasoning

Abstract: Boolean contact algebras constitute a suitable algebraic theory for qualitative spatial reasoning. They are Boolean algebras with an additional contact relation grasping the topological aspect of spatial entities. In this paper we present a logic that combines propositional logic, relevance logic, and modal logic to reason about Boolean contact algebras. This is done in two steps. First, we use the relevance logic operators to obtain a logic suitable for Boolean algebras. Then we add modal operators that are based on the contact relation. In both cases we present axioms that are equivalent to requirement that every frame for the logic is indeed a Boolean algebra resp. Boolean contact algebra. We also provide a natural deduction system for this logic by defining introduction and eliminations rules for each logical operator. The system is shown to be sound. Furthermore, we sketch an implementation of the natural deduction system in the functional programming language and interactive theorem prover Coq.

Actualism and modal semantics

Abstract: According to actualism, modal reality is constructed out of valuations (combinations of truth values for all propositions). According to possibilism, modal reality consists in a set of possible worlds, conceived as independent objects that assign truth values to propositions. According to possibilism, accounts of modal reality can intelligibly disagree with each other even if they agree on which valuations are contained in modal reality. According to actualism, these disagreements (possibilist disagreements) are completely unintelligible. An essentially actualist semantics for modal propositional logic specifies which sets of valuations are compatible with the meanings of the truth-functional connectives and modal operators without drawing on formal resources that would enable us to represent possibilist disagreements. The paper discusses the availability of an essentially actualist semantics for modal propositional logic. I argue that the standard Kripkean semantics is not essentially actualist and that other extant approaches also fail to provide a satisfactory essentially actualist semantics. I end by describing an essentialist actualist semantics for modal propositional logic.