Showing papers on "Modal operator published in 2020"
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01 May 2020
TL;DR: A different view of modal discourse is proposed in which talk about what is “metaphysically necessary” does not aim to describe modal features of the world, but, rather, provides a particularly useful way of expressing constitutive semantic and conceptual rules in the object language.
Abstract: Modality presents notorious philosophical problems, including the epistemic problem of how we could come to know modal facts and metaphysical problems about how to place modal facts in the natural world. These problems arise from thinking of modal claims as attempts to describe modal features of this world that explain what makes them true. Here I propose a different view of modal discourse in which talk about what is “metaphysically necessary” does not aim to describe modal features of the world, but, rather, provides a particularly useful way of expressing constitutive semantic and conceptual rules in the object language. The result is a “modal normativist” view that enables us to avoid the epistemic problems of modality and mitigate the metaphysical worries, while also leaving open the possibility of a unified account of the function of modal language. Finally, I address a serious challenge: we have the norms we do in order to track the modal facts of the world, so that the order of explanation must go in the opposite direction. I close by showing how the normativist may answer that challenge.
40 citations
TL;DR: A family of intuitionistic non-normal modal logics, which can be seen as intuitionistic counterparts of classical ones, are defined and a semantic characterisation of their logics is defined in terms of neighbourhood models containing two distinct neighbourhood functions corresponding to the two modalities.
Abstract: We define a family of intuitionistic non-normal modal logics; they can be seen as intuitionistic counterparts of classical ones. We first consider monomodal logics, which contain only Necessity or Possibility. We then consider the more important case of bimodal logics, which contain both modal operators. In this case we define several interactions between Necessity and Possibility of increasing strength, although weaker than duality. We thereby obtain a lattice of 24 distinct bimodal logics. For all logics we provide both a Hilbert axiomatisation and a cut-free sequent calculus, on its basis we also prove their decidability. We then define a semantic characterisation of our logics in terms of neighbourhood models containing two distinct neighbourhood functions corresponding to the two modalities. Our semantic framework captures modularly not only our systems but also already known intuitionistic non-normal modal logics such as Constructive K (CK) and the propositional fragment of Wijesekera’s Constructive Concurrent Dynamic Logic.
13 citations
TL;DR: A new semantics that is based on the logic of here-and-there is proposed: it is added to the existing semantics of epistemic specifications, and which formalisms provide more intuitive results by pointing out some formal properties a semantics proposal should satisfy.
10 citations
10 Dec 2020
TL;DR: The results show that some of these problems posed by the English non-modal operators to the undergraduate level students of Hazara University, Mansehra, Pakistan were caused by the intervention of some of grammatical concepts like tense, aspect, back shifting and voice.
Abstract: This study focuses on the problems posed by the English non-modal operators to the undergraduate level students of Hazara University, Mansehra, Pakistan. The data was collected from hundred students selected through non-random and convenience sampling technique. A proficiency test was used as a tool for data collection. The test was focused on all the uses of non-modal operators. The results show that some of these problems were caused by the intervention of some of grammatical concepts like tense, aspect, back shifting and voice. While some grammatical operations like negation, interrogation and insertion/omission had no role and so were found comparatively easy. These operators when used after wh-word such as when, while, before and if posed difficulty for the subjects. Similarly, different forms such as nontensed form and uses such as dynamic and non-dynamic of non-modal operators were also problematic for the subjects. The highest frequency of error was found in the use of non-model operator for emphasis and surprise. However, the degree of difficulty posed by non-modal operators in idiomatic expressions was not significant.
9 citations
TL;DR: In this article, the authors propose a modal logic for reasoning with both graded preferences and propositions, in the style of van Benthem et al.'s classical modal logics for preferences.
8 citations
TL;DR: An enhanced method will exploit distribution specifications, an SOS-like framework defining the probabilistic behavior of processes, to decompose this particular form of lookahead, and it will show how the proposed decomposition method can be applied to derive congruence formats for the considered equivalences from their logical characterizations.
6 citations
TL;DR: This framework takes into account the four-valued nature of FDE by considering four partially defined modal operators corresponding to conditions for verifying and falsifying modal necessity and possibility operators.
Abstract: We develop a general theory of FDE-based modal logics. Our framework takes into account the four-valued nature of FDE by considering four partially defined modal operators corresponding to conditions for verifying and falsifying modal necessity and possibility operators. The theory comes with a uniform characterization for all obtained systems in terms of FDE-style formula-formula sequents. We also develop some correspondence theory and show how Hilbert-style axiom systems can be obtained in appropriate cases. Finally, we outline how different systems from the literature can be expressed in our framework.
5 citations
TL;DR: In this article, the authors show the role of beliefs, language and logic in the conditions of possibility of knowledge, based on a review of the relevance of the theory of possible worlds and the spaces of logical possibility in the construction and analysis of the scientific discourse.
Abstract: Knowledge construction has been a topic of interest to the philosophy of language, given the need to validate the aletic and apophantic form of statements that constitute such a discourse. For a long time, Classical logic has been in charge of validating the judgments of fact that constitute science; however, in the last few decades, logic modal studies have gained particular strength, with which it is intended to examine, among others, the epistemic statements and beliefs of cognizants from the use of modal operators, among them, those that set the boundaries of the theory of possible worlds. This article aims to show the role of beliefs, language and logic in the conditions of possibility of knowledge, based on a review of the relevance of the theory of possible worlds and the spaces of logical possibility in the Construction and analysis of the scientific discourse.
5 citations
TL;DR: It is shown how to extend propositional inconsistency measures to sets of formulas in any such generalized propositional logic, and it is shown that the addition of operators leads to a weak type of inconsistency.
Abstract: Consistency is one of the key concepts of logic; logicians have put a great deal of effort into proving the consistency of many logics. Understanding what causes inconsistency is also important; some logicians have developed paraconsistent logics that, unlike classical logics, allow some contradictions without making all formulas provable. Another direction of research studies inconsistency by measuring the amount of inconsistency of sets of formulas. While the initial attempt in 1978 was too ambitious in trying to do this for first-order logic, this research got a substantial boost when an inconsistency measure was proposed for propositional logic in 2002. Since then, researchers in logic and artificial intelligence (AI systems need the capability to deal with inconsistency) have made many interesting proposals and found related issues. Almost all of this work has been done for propositional logic. The purpose of this paper is to extend inconsistency measures to logics that also contain operators, such as modal operators. We use the terminology “generalized propositional logic” for such logics. We show how to extend propositional inconsistency measures to sets of formulas in any such generalized propositional logic. Examples are used to illustrate how various modal operators, including spatial and tense operators, fit into this framework. We also show that the addition of operators leads to a weak type of inconsistency. In all cases, the calculations for several inconsistency measures are given.
5 citations
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TL;DR: In this work, a fragment of trivalent Lukasiewicz logic is studied and the adequacy theorem with respect to the suitable algebraic structures is proven.
Abstract: MV-algebras are an algebraic semantics for Lukasiewicz logic and MV-algebras generated by a finite chain are Heyting algebras where the Godel implication can be written in terms of De Morgan and Moisil's modal operators. In our work, a fragment of trivalent Lukasiewicz logic is studied. The propositional and first-order logic is presented. The maximal consistent theories are studied as Monteiro's maximal deductive systems of the Lindenbaum-Tarski algebra, in both cases. Consequently, the adequacy theorem with respect to the suitable algebraic structures is proven.
5 citations
TL;DR: A novel version of possible worlds semantics featuring both classical and nonclassical worlds is introduced and the completeness of a family of noncongruent modal logics whose internal logic is non classical with respect to this semantics is established.
Abstract: We determine the modal logic of fixed-point models of truth and their axiomatizations by Solomon Feferman via Solovay-style completeness results. Given a fixed-point model $\mathcal{M}$, or an axiomatization $S$ thereof, we find a modal logic $M$ such that a modal sentence $\varphi$ is a theorem of $M$ if and only if the sentence $\varphi^*$ obtained by translating the modal operator with the truth predicate is true in $\mathcal{M}$ or a theorem of $S$ under all such translations. To this end, we introduce a novel version of possible worlds semantics featuring both classical and nonclassical worlds and establish the completeness of a family of non-congruent modal logics whose internal logic is subclassical with respect to this semantics.
TL;DR: In this article, the propositional quantifiers are added to a propositional modal language with two modal operators and the resulting language is interpreted over so-called products of Kripke frames whose accessibility relations are equivalence relations.
Abstract: Propositional quantifiers are added to a propositional modal language with two modal operators. The resulting language is interpreted over so-called products of Kripke frames whose accessibility relations are equivalence relations, letting propositional quantifiers range over the powerset of the set of worlds of the frame. It is first shown that full second-order logic can be recursively embedded in the resulting logic, which entails that the two logics are recursively isomorphic. The embedding is then extended to all sublogics containing the logic of so-called fusions of frames with equivalence relations. This generalizes a result due to Antonelli and Thomason, who construct such an embedding for the logic of such fusions.
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TL;DR: This paper introduces the notion of `commonly knowing whether', a non-standard version of standard common knowledge which is defined on the basis of `knowing whether', instead of standard ` knowing that', and gives five possible definitions.
Abstract: This paper introduces `commonly knowing whether', a non-standard version of classical common knowledge which is defined on the basis of `knowing whether', instead of classical `knowing that' After giving five possible definitions of this concept, we explore the logical relations among them both in the multi-agent case and in the single-agent case We focus on one definition and treat it as a modal operator It is found that the expressivity of this operator is incomparable with the classical common knowledge operator Moreover, some special properties of it over binary-tree models and KD45-models are investigated
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TL;DR: To better understand the rule of necessitation, six ways of defining deductive consequence in modal logic are summarized: omitted definition, classical definition, ternary definition, reduced definition, bounded definition, and deflationary definition are summarized.
Abstract: We discuss four common mistakes in the teaching and textbooks of modal logic. The first one is missing the axiom $\Diamond\varphi\leftrightarrow
eg\Box
eg\varphi$, when choosing $\Diamond$ as the primitive modal operator, misunderstanding that $\Box$ and $\Diamond$ are symmetric. The second one is forgetting to make the set of formulas for filtration closed under subformulas, when proving the finite model property through filtration, neglecting that $\Box\varphi$ and $\Diamond\varphi$ may be abbreviations of formulas. The third one is giving wrong definitions of canonical relations in minimal canonical models that are unmatched with the primitive modal operators. The final one is misunderstanding the rule of necessitation, without knowing its distinction from the rule of modus ponens. To better understand the rule of necessitation, we summarize six ways of defining deductive consequence in modal logic: omitted definition, classical definition, ternary definition, reduced definition, bounded definition, and deflationary definition, and show that the last three definitions are equivalent to each other.
TL;DR: All three modal negation logics characterized by a frame with a single state are formalized as extensions of $${{\textsf {N}}}_0$$ .
Abstract: The minimal weakening
$${{\textsf {N}}}_0$$
of Belnap-Dunn logic under the polarity semantics for negation as a modal operator is formulated as a sequent system which is characterized by the class of all birelational frames. Some extensions of
$${{\textsf {N}}}_0$$
with additional sequents as axioms are introduced. In particular, all three modal negation logics characterized by a frame with a single state are formalized as extensions of
$${{\textsf {N}}}_0$$
. These logics have the finite model property and they are decidable.
TL;DR: In this paper, the notion of quasi-modal operator in the variety of distributive nearlattices was introduced, which turns out to be a generalization of the necessity modal operator studied in [S. Celani and I. Calomino, 2019].
Abstract: We introduce the notion of quasi-modal operator in the variety of distributive nearlattices, which turns out to be a generalization of the necessity modal operator studied in [S. Celani and I. Calomino, Math. Slovaca 69 (2019), no. 1, 35–52]. We show that there is a one to one correspondence between a particular class of quasi-modal operators on a distributive nearlattice and the class of possibility modal operators on the distributive lattice of its finitely generated filters. Finally, we consider the concept of quasi-modal congruence, and we show that the lattice of quasi-modal congruences of a quasi-modal distributive nearlattice is isomorphic to the lattice of congruences of the lattice of finitely generated filters with a possibility modal operator.
TL;DR: In this paper, a hypersequent calculus for modal logics K, D, T, S4, B, and S5 has been proposed to underwrite an inferentialist theory of meaning.
Abstract: The standard inferentialist approaches to modal logic tend to suffer from not being able to uniquely characterize the modal operators, require that introduction and elimination rules be interdefined, or rely on the introduction of possible-world like indexes into the object language itself. In this paper I introduce a hypersequent calculus that is flexible enough to capture many of the standard modal logics and does not suffer from the above problems. It is therefore an ideal candidate to underwrite an inferentialist theory of meaning for modal operators. Here I treat specifically the modal logics K, D, T, S4, B, and S5. I show that the calculi are adequate for each set of models, and show that they meet a large set of criteria that are generally thought necessary for a calculus to underwrite a theory of meaning.
TL;DR: This work improves on a previous model by the first two named authors which not only had a flaw but was also considerably more complicated, and shows how to model the tick constant of CloTT using a semantic substitution.
Abstract: Clocked Type Theory (CloTT) is a type theory for guarded recursion useful for programming with coinductive types, allowing productivity to be encoded in types, and for reasoning about advanced programming language features using an abstract form of step-indexing. CloTT has previously been shown to enjoy a number of syntactic properties including strong normalisation, canonicity and decidability of the equational theory. In this paper we present a denotational semantics for CloTT useful, e.g., for studying future extensions of CloTT with constructions such as path types.
The main challenge for constructing this model is to model the notion of ticks on a clock used in CloTT for coinductive reasoning about coinductive types. We build on a category previously used to model guarded recursion with multiple clocks. In this category there is an object of clocks but no object of ticks, and so tick-assumptions in a context can not be modelled using standard tools. Instead we model ticks using dependent right adjoint functors, a generalisation of the category theoretic notion of adjunction to the setting of categories with families. Dependent right adjoints are known to model Fitch-style modal types, but in the case of CloTT, the modal operators constitute a family indexed internally in the type theory by clocks. We model this family using a dependent right adjoint on the slice category over the object of clocks. Finally we show how to model the tick constant of CloTT using a semantic substitution.
This work improves on a previous model by the first two named authors which not only had a flaw but was also considerably more complicated.
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TL;DR: In this article, the authors study propositional logical systems arising from the language of Johansson's minimal logic and obtained by weakening the requirements for the negation operator, and present their semantics as a variant of neighbourhood semantics.
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.
TL;DR: The canonical possible-world semantics for Justification Logic is extended to this fuzzy realm by considering fuzzy accessibility- and evaluation-functions evaluated over the minimum t-norm and strong completeness theorems for various fuzzy analogies of prominent extensions for basic Justification logic are established.
Abstract: Justification logics are special kinds of modal logics which provide a framework for reasoning about epistemic justifications. 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.
TL;DR: In this paper, the modal logic and modal algebra of the theory of modality in first-order languages are considered. But the modality theory is interpreted via a modal operator, and it depends on the model-theoretic language, their Kripke completeness, and expressibility of the modelality inside L.
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.
TL;DR: Experimental data relevant to understanding the modal free choice effect (Kamp, 1973) when there are more than two disjuncts under the relevant modal operator is presented.
Abstract: This paper presents experimental data relevant to understanding the modal free choice effect (Kamp, 1973) when there are more than two disjuncts under the relevant modal operator. The results sugge...
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TL;DR: It is found that the expressivity of this operator is incomparable with the classical common knowledge operator.
Abstract: This paper introduces `commonly knowing whether', a non-standard version of classical common knowledge which is defined on the basis of `knowing whether', instead of classical `knowing that'. After giving five possible definitions of this concept, we explore the logical relations among them both in the multi-agent case and in the single-agent case. We focus on one definition and treat it as a modal operator. It is found that the expressivity of this operator is incomparable with the classical common knowledge operator. Moreover, some special properties of it over binary-tree models and KD45-models are investigated.
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TL;DR: The syntax of Answer Set Programming is extended by using temporal operators from temporal modal logics, then infinite time can be considered and this extension constitutes a formalism called Temporal Equilibrium Logic, which is based on Equ equilibrium Logic (a logical characterisation of ASP).
Abstract: We present an overview on Temporal Logic Programming under the perspective of its application for Knowledge Representation and declarative problem solving. Such programs are the result of combining usual rules with temporal modal operators, as in Linear-time Temporal Logic (LTL). We focus on recent results of the non-monotonic formalism called Temporal Equilibrium Logic (TEL) that is defined for the full syntax of LTL, but performs a model selection criterion based on Equilibrium Logic, a well known logical characterization of Answer Set Programming (ASP). We obtain a proper extension of the stable models semantics for the general case of arbitrary temporal formulas. We recall the basic definitions for TEL and its monotonic basis, the temporal logic of Here-and-There (THT), and study the differences between infinite and finite traces. We also provide other useful results, such as the translation into other formalisms like Quantified Equilibrium Logic or Second-order LTL, and some techniques for computing temporal stable models based on automata. In a second part, we focus on practical aspects, defining a syntactic fragment called temporal logic programs closer to ASP, and explain how this has been exploited in the construction of the solver TELINGO.
01 Nov 2020
TL;DR: In this paper, a translation from the proposed sequential fuzzy description logic to a standard fuzzy description logics is defined, and a theorem for embedding the sequential description logic into the standard description logic is proved using this translation.
Abstract: Description logics are known to be a family of logic-based knowledge representation formalisms, and fuzzy description logics are expressive description logics for representing and handling fuzzy (vague or imprecise) knowledge bases. A sequential fuzzy description logic, which is introduced in this paper, is an extended fuzzy description logic where a sequence modal operator is introduced. In this paper, a translation from the proposed sequential fuzzy description logic to a standard fuzzy description logic is defined. Further, a theorem for embedding the sequential fuzzy description logic into the standard fuzzy description logic is proved using this translation. A theorem for relative decidability of the sequential fuzzy description logic with respect to the standard fuzzy description logic is established using the embedding theorem. The proposed logic and translation are intended for effective handling of fuzzy knowledge bases with sequential information (i.e., information expressed as sequences). Moreover, using the translation, existing methods and algorithms for the standard fuzzy description logic can be reused to effectively handle fuzzy knowledge bases with sequential information described by the sequential fuzzy description logic.
TL;DR: This work investigates the variant of epistemic logic S5 for reasoning about knowledge under hypotheses with a modal operator of necessity that can be parameterized with a hypothesis representing background assumptions and presents an axiomatization of the logic and its extension with the common knowledge operator and distributed knowledge operator.
Abstract: We investigate the variant of epistemic logic S5 for reasoning about knowledge under hypotheses. The logic is equipped with a modal operator of necessity that can be parameterized with a hypothesis representing background assumptions. The modal operator can be described as relative necessity and the resulting logic turns out to be a variant of Chellas’ Conditional Logic. We present an axiomatization of the logic and its extension with the common knowledge operator and distributed knowledge operator. We show that the logics are decidable, complete w.r.t. Kripke as well as topological structures. The topological completeness results are obtained by utilizing the Alexandroi¬€ connection between preorders and Alexandroi¬€ spaces.
Journal Article•
TL;DR: In this article, a denotational semantics for CloTT has been presented, e.g., for studying future extensions of CloTT with constructions such as path types, and the notion of ticks on a clock used in CloTT for coinductive reasoning about co-inductive types.
Abstract: Clocked Type Theory (CloTT) is a type theory for guarded recursion useful for programming with coinductive types, allowing productivity to be encoded in types, and for reasoning about advanced programming language features using an abstract form of step-indexing. CloTT has previously been shown to enjoy a number of syntactic properties including strong normalisation, canonicity and decidability of the equational theory. In this paper we present a denotational semantics for CloTT useful, e.g., for studying future extensions of CloTT with constructions such as path types.
The main challenge for constructing this model is to model the notion of ticks on a clock used in CloTT for coinductive reasoning about coinductive types. We build on a category previously used to model guarded recursion with multiple clocks. In this category there is an object of clocks but no object of ticks, and so tick-assumptions in a context can not be modelled using standard tools. Instead we model ticks using dependent right adjoint functors, a generalisation of the category theoretic notion of adjunction to the setting of categories with families. Dependent right adjoints are known to model Fitch-style modal types, but in the case of CloTT, the modal operators constitute a family indexed internally in the type theory by clocks. We model this family using a dependent right adjoint on the slice category over the object of clocks. Finally we show how to model the tick constant of CloTT using a semantic substitution.
This work improves on a previous model by the first two named authors which not only had a flaw but was also considerably more complicated.
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TL;DR: In this article, the authors study logics of bounded distributive residuated lattices with modal operators considering $Box$ and $Diamond$ in a noncommutative setting, and prove that any canonical logic is Kripke complete via discrete duality and canonical extensions.
Abstract: In this paper, we study logics of bounded distributive residuated lattices with modal operators considering $\Box$ and $\Diamond$ in a noncommutative setting. We introduce relational semantics for such substructural modal logics. We prove that any canonical logic is Kripke complete via discrete duality and canonical extensions. That is, we show that a modal extension of the distributive full Lambek calculus is the logic of its frames if its variety is closed under canonical extensions. After that, we establish a Priestley-style duality between residuated distributive modal algebras and topological Kripke structures based on Priestley spaces.
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TL;DR: In this article, the authors analyse logic of false belief in intuitionistic setting and discuss monotonicity of forcing, soundness, completeness and several other issues, and present also some simple systems in which confirmation of previously accepted formula is modelled.
Abstract: In this paper we analyse logic of false belief in intuitionistic setting. This logic, studied in its classical version by Steinsvold, Fan, Gilbert and Venturi, describes the following situation: a formula F is not satisfied in a given world, but we still believe in it (or we think that it should be accepted). Another interpretations are also possible: e.g. that we do not accept F but it is imposed on us by a kind of council or advisory board. From the mathematical point of view, the idea is expressed by an adequate form of modal operator W which is interpreted in relational frames with neighborhoods. We discuss monotonicity of forcing, soundness, completeness and several other issues. We present also some simple systems in which confirmation of previously accepted formula is modelled.
01 Aug 2020
TL;DR: A new direction of tow modifications of the two standard modal operators "necessity" and "possibility" is discussed and four new operators are introduced and studied.
Abstract: Ever since the first publication over intuitionistic fuzzinsess, the two standard modal operators "necessity" ( □ ) and "possibility" (◊) are introduced over intuitionistic fuzzy sets. In the years, these two operators are object of a lot of extensions and modifications. In the present paper, a new direction of tow modifications of the two standard modal operators is discussed. We introduce four new operators and study their properties.