Modal Operators Research Articles

Лексические средства выражения субъективной модальности в испанском языке

The study aims to determine peculiarities of using lexical means of expressing subjective modality in the novel “Niebla” by Miguel de Unamuno. The article is devoted to studying the functioning of lexical means of expressing subjective modality in the Spanish language. Various approaches to interpretation of the “modality” notion are considered. The use of the adverbs ‘tal vez’ and ‘acaso’ as modal operators for expressing subjective modality in the novel “Niebla” is analysed. Scientific novelty of the study lies in analysing different subjective-modal meanings conveyed by the adverbs ‘tal vez’ and ‘acaso’ using the novel “Niebla” as an example. As a result, it is proved that modal adverbs convey a whole range of speculative meanings in the Spanish language.

Quantified Temporal Alethic Boulesic Doxastic Logic

The paper develops a set of quantified temporal alethic boulesic doxastic systems. Every system in this set consists of five parts: a ‘quantified’ part, a temporal part, a modal (alethic) part, a boulesic part and a doxastic part. There are no systems in the literature that combine all of these branches of logic. Hence, all systems in this paper are new. Every system is defined both semantically and proof-theoretically. The semantic apparatus consists of a kind of T times W models, and the proof-theoretical apparatus of semantic tableaux. The ‘quantified part’ of the systems includes relational predicates and the identity symbol. The quantifiers are, in effect, a kind of possibilist quantifiers that vary over every object in the domain. The tableaux rules are classical. The alethic part contains two types of modal operators for absolute and historical necessity and possibility. According to ‘boulesic logic’ (the logic of the will), ‘willing’ (‘consenting’, ‘rejecting’, ‘indifference’ and ‘non-indifference’) is a kind of modal operator. Doxastic logic is the logic of beliefs; it treats ‘believing’ (and ‘conceiving’) as a kind of modal operator. I will explore some possible relationships between these different parts, and investigate some principles that include more than one type of logical expression. I will show that every tableau system in the paper is sound and complete with respect to its semantics. Finally, I consider an example of a valid argument and an example of an invalid sentence. I show how one can use semantic tableaux to establish validity and invalidity and read off countermodels. These examples illustrate the philosophical usefulness of the systems that are introduced in this paper.

Subject-oriented spatial logic

We present a modal logic for subject-oriented representation and reasoning about a two-dimensional space, which we call SOSL. The space is represented with the polar coordinate system where the subject occupies the central point and modal operators are interpreted by relations defined relatively to the position and orientation of the subject, namely ‘outwards’, ‘inwards’, ‘clockwise’, ‘counter-clockwise’, and the transitive closures of the first two. Such logic enables to express operators for the intrinsic relations: ‘in front’, ‘behind’, ‘to the left’, and ‘to the right’ of the subject, for the relative relations: ‘behind an object’, ‘between the subject and an object’, ‘to the left of an object’, and ‘to the right of an object’, and hybrid or distance operators. We prove that the satisfiability problem in SOSL is PSpace-complete, the same complexity holds over the classes of finite or infinite models, however, for models of fixed size the problem becomes NP-complete.

INQUISITIVE BISIMULATION

AbstractInquisitive modal logic, InqML, is a generalisation of standard Kripke-style modal logic. In its epistemic incarnation, it extends standard epistemic logic to capture not just the information that agents have, but also the questions that they are interested in. Technically, InqML fits within the family of logics based on team semantics. From a model-theoretic perspective, it takes us a step in the direction of monadic second-order logic, as inquisitive modal operators involve quantification over sets of worlds. We introduce and investigate the natural notion of bisimulation equivalence in the setting of InqML. We compare the expressiveness of InqML and first-order logic in the context of relational structures with two sorts, one for worlds and one for information states, and characterise inquisitive modal logic as the bisimulation invariant fragment of first-order logic over various natural classes of two-sorted structures.

Predicting the end: Epistemic change in Romance

This paper analyzes a verbal periphrasis in European Portuguese (EP) and Italian (I), formed by a verb meaning ‘to end’, a ‘by’-preposition and a verb in the infinitive form (V). We show that this periphrasis—for short, [end by V]—behaves differently from the aspectual terminative periphrasis formed with the verb end , by focusing on three semantic criteria. We argue that the meaning of [end by V] involves a specific type of epistemic modality related to expectation. We compare the periphrasis with the discourse particles afinal (EP) and alla fine (I), both etymologically related to a form meaning ‘end’, which can also express expectation-related modality and have been analyzed as epistemic modal operators sensitive to a temporal succession of epistemic states (Amaral and Del Prete 2016, 2017). To account for the semantic properties of [end by V], we propose a formal analysis of this periphrasis in which the ‘end’-verb, combined with the ‘by’-preposition, selects a property of eventualities P as argument and presupposes the existence of a sequence of events s →. The periphrasis makes a twofold semantic contribution: (i) it asserts that a P -eventuality occurs at the end of the sequence s →, and (ii) it implies that the occurrence of the P -eventuality at the end of s → is ranked below some alternative outcome on a likelihood/preference scale. We discuss the implications of our analysis for the study of epistemic modality. EARLY ACCESS

THE MODAL LOGICS OF KRIPKE–FEFERMAN TRUTH

AbstractWe 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 noncongruent modal logics whose internal logic is nonclassical with respect to this semantics.

The Metasemantics of Indefinite Extensibility

ABSTRACT Generality relativism is the view that any domain of quantification can always be expanded. The view promises to resolve a broad range of paradoxes, but, without an explanation of how domains expand, it sounds very mysterious. Proponents of linguistic versions of generality relativism try to demystify the view by likening domain expansions to semantic change. They think that domains expand when we re-interpret certain terms so that, upon re-interpretation, the quantifiers range over more things. This article makes trouble for linguistic approaches. According to the so-called charge of ineffability, generality relativism cannot be asserted in a coherent manner—given that asserting the view requires generalising over absolutely all domains of quantification. Generality relativists typically try to answer the charge with the help of modal operators; but linguistic approaches interpret these operators in semantically and syntactically idiosyncratic ways. I argue that, because of these commitments, linguistic approaches ultimately do not have a good response to the charge of ineffability.

Desire, belief, and semantic composition: variation in mood selection with desire predicates

Mood selection properties of desire verbs provide a rich source of evidence regarding the semantics of propositional attitudes. This paper approaches the topic by providing an analysis of crosslinguistic variation in the selection patterns of the desire verbs ‘want’ and ‘hope’, focusing on Spanish and French. There is no evidence that the meanings of ‘hope’ and ‘want’ differ between these languages, and yet in Spanish esperar ‘hope’ and querer ‘want’ both take subjunctive, while in French only vouloir ‘want’ selects subjunctive and esperer ‘hope’ strongly prefers the indicative. The inclination of ‘hope’ toward the indicative is manifest also in other Romance languages. Previous theories tie mood selection tightly to the verb’s modal backgrounds and do not anticipate such variation. We explain the consistency in mood selection with ‘want’ versus the variation with ‘hope’ in terms of two key ideas: (i) moods are modal operators that encode different degrees of modal necessity, and (ii) modal backgrounds can be manipulated by the grammar. In terms of (i), we argue, building on the comparison-based theory of mood, that the indicative is a strong necessity operator, while the subjunctive encodes a weaker necessity. Regarding (ii), we propose that two backgrounds may function as one under certain well-defined circumstances. Our proposal supports the decompositional approach to attitude verbs, where mood is responsible for the quantificational force traditionally attributed to the verb.

On a strong negation-based representation of modalities

In this study, the notion of a dual pair of modal operators is interpreted according to the algebraic criteria for necessity and possibility operators on De Morgan lattices presented by Cattaneo, Ciucci and Dubois, 2011. Here, a representation theorem is introduced which demonstrates that, in this algebraic model, a dual pair of modal operators can be represented by compositions of two strong negations, where one of them is stricter than the other. Then, the Pliant negation operator is utilized to derive dual modal operators. It is demonstrated that using the generator function of Dombi operators, the composition of two Pliant negations results in modal operators that have simple forms and easy-to-use characteristics. Next, we examine how the proposed modal operators are connected with the drastic necessity and possibility operators. Also, the necessary and sufficient condition for the distributivity of modal operators induced by compositions of strong negations over strict t-norms and strict t-conorms is presented. Lastly, a connection between the modal operators and hedges is highlighted.

The Philippine Online News Headlines on COVID-19 Pandemic: An Interpersonal Metafunction Analysis

This paper carries on a tentative interpersonal metafunctional analysis of the interpersonal metafunction online news headlines about COVID-19 pandemic. The analysis involved the differences in the mood type distribution, polarity, modality and other lexical characteristics. The findings reveal the dominance of the declarative mood texts tells the particular function of the online news headlines. It's informing the readers, rather than questioning or demanding them. Positive declarative clauses should convey information and convince readers with positive facts. A negative polarity does not imply a negative impression but confirms that a certain issue is inconsistent with what has become apparent. The use of “must” indicates that the speaker has a firm and strong conviction about a particular issue and that the readers are obliged to take the necessary steps in connection with what is believed. Word choices and the clause structure reflect the context of the texts and not the mood adjuncts or the modal verbal operators. Analysis of the functional structural mood text is very useful for understanding the interpersonal relationships between participants.

Fully divine and fully human: a bi-modal outline towards consistency

AbstractPhilosophers and theologians have long recognized apparent contradictions in the Christian doctrine of the incarnation wherein Jesus is both fully divine and fully human and thereby exemplifies all properties essential to being divine and all properties essential to being human, including, respectively, immutability and mutability. In this article, we outline a new account of such apparent contradictions, wherein each such pair of (modal) properties involves different modal operators.

Conjunctive and Disjunctive Limits: Abstract Logics and Modal Operators

Abstract Departing from basic concepts in abstract logics, this paper introduces two concepts: conjunctive and disjunctive limits. These notions are used to formalize levels of modal operators.

Belnap–Dunn Modal Logic with Value Operators

The language of Belnap–Dunn modal logic $${\mathscr {L}}_0$$ expands the language of Belnap–Dunn four-valued logic (having constant symbols for the values 0 and 1) with the modal operator $$\Box $$ . We introduce the polarity semantics for $${\mathscr {L}}_0$$ and its two expansions $${\mathscr {L}}_1$$ and $${\mathscr {L}}_2$$ with value operators. The local finitary consequence relation $$\models _4^k$$ in the language $${\mathscr {L}}_k$$ with respect to the class of all frames is axiomatized by a sequent system $$\mathsf {S}_k$$ where $$k=0, 1, 2$$ . We prove by using translations between sequents and formulas that these languages under the polarity semantics have the same expressive power on the level of frames with the language $${\mathscr {L}}_0$$ under the relational semantics for classical modal logic.

An epistemic approach to the formal specification of statistical machine learning

We propose an epistemic approach to formalizing statistical properties of machine learning. Specifically, we introduce a formal model for supervised learning based on a Kripke model where each possible world corresponds to a possible dataset and modal operators are interpreted as transformation and testing on datasets. Then, we formalize various notions of the classification performance, robustness, and fairness of statistical classifiers by using our extension of statistical epistemic logic. In this formalization, we show relationships among properties of classifiers, and relevance between classification performance and robustness. As far as we know, this is the first work that uses epistemic models and logical formulas to express statistical properties of machine learning, and would be a starting point to develop theories of formal specification of machine learning.

Measuring Inconsistency in Some Logics with Modal Operators

The first mention of the concept of an inconsistency measure for sets of formulas in first-order logic was given in 1978, but that paper presented only classifications for them. The first actual inconsistency measure with a numerical value was given in 2002 for sets of formulas in propositional logic. Since that time, researchers in logic and AI have developed a substantial theory of inconsistency measures. While this is an interesting topic from the point of view of logic, an important motivation for this work is also that some intelligent systems may encounter inconsistencies in their operation. This research deals primarily with propositional knowledge bases, that is, finite sets of propositional logic formulas. The goal of this paper is to extend the concept of inconsistency measure in a formal way to sets of formulas with the modal operators “necessarily” and “possibly” applied to propositional logic formulas. We use frames for the semantics, but in a way that is different from the way that frames are commonly used in modal logics, in order to facilitate measuring inconsistency. As a set of formulas may have different inconsistency measures for different frames, we define the concept of a standard frame that can be used for all finite sets of formulas in the language. We do this for two languages. The first language, AMPL, contains formulas where a prefix of operators is applied to a propositional logic formula. The second language, CMPL, adds connectives that can be applied to AMPL formulas in a limited way. We show how to extend propositional logic inconsistency measures to such sets of formulas. Finally, we define a new concept, weak inconsistency measure, and show how to compute it.

Contrast, Verum Focus and Anaphora: The Case ofet pourtant si/nonin French

This study examines the anaphoric status of the sequenceet pourtant si/nonin French. This sequence displays some properties not only of TP-Ellipsis but also of propositional anaphora. Consequently, the antecedent of this sequence can be recovered by means of either type of anaphoric process. I argue that the salient and relevant antecedent is constrained by the presence of a modalized environment. I claim that the discursive markerpourtantis assimilated to a modal operator (Jayez 1988, Martin 1987) expressingdiscourse contrastbetween two propositions anchored in two possible worlds that are not contradictory.Polarity Particles(POLPARTS) involved in this sequence are analyzed as emphasizing the truth of a proposition. As such, they are conveyingsemantic contrastbetween two polarities, that of a salient and accessible discourse antecedent and that of the missing part afteret pourtant si/non. This is how POLPARTS upgrade the Common Ground. I develop a focus-based account forVerum Focus, building on alternatives along the lines of Hardt & Romero (2004). I suggest that the scope of an epistemic operator (Romero & Han 2004) and the conditions of use are relevant in order to reconstruct the adequate antecedent, which is not possible in an analysis based solely on lexical insertion and upgrading the Question Under Discussion (qud) by conditions governing the felicitous use ofet pourtant si/non.

Measuring Inconsistency in Generalized Propositional Logic

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.

Boolean negation and non-conservativity I: Relevant modal logics

Abstract Many relevant logics can be conservatively extended by Boolean negation. Mares showed, however, that E is a notable exception. Mares’ proof is by and large a rather involved model-theoretic one. This paper presents a much easier proof-theoretic proof which not only covers E but also generalizes so as to also cover relevant logics with a primitive modal operator added. It is shown that from even very weak relevant logics augmented by a weak K-ish modal operator, and up to the strong relevant logic R with a S5 modal operator, all fail to be conservatively extended by Boolean negation. The proof, therefore, also covers Meyer and Mares’ proof that NR—R with a primitive S4-modality added—also fails to be conservatively extended by Boolean negation.

Kant, Linnaeus, and the economy of nature

Ecology arguably has roots in eighteenth-century natural histories, such as Linnaeus's economy of nature, which pressed a case for holistic and final-causal explanations of organisms in terms of what we'd now call their environment. After sketching Kant's arguments for the indispensability of final-causal explanation merely in the case of individual organisms, and considering the Linnaean alternative, this paper examines Kant's critical response to Linnaean ideas. I argue that Kant does not explicitly reject Linnaeus's holism. But he maintains that the indispensability of final-causal explanation depends on robust modal connections between types of organism and their functional parts; relationships in Linnaeus's economy of nature, by contrast, are relatively contingent. Kant's framework avoids strong metaphysical assumptions, is responsive to empirical evidence, and can be fruitfully compared with some contemporary approaches to biological organization.

A Hypersequent Solution to the Inferentialist Problem of Modality

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.