Modal Language Research Articles

A Simple Modal Logic for Reasoning in Multigranulation Rough Set Model

The notions of strong/weak approximations have been studied extensively in recent years. These approximations are based on a structure of the form ( W,{R i } i∈ N ), called the multiple-source approximation system, where R i is an equivalence relation on W , and N is an initial segment of the set N of natural numbers. We propose and explore a simple modal language and semantics that can be used to reason about the strong/weak approximations of concepts. Moreover, our study is not confined to collections of equivalence relations only, but other types of relations are also considered. This study is important, keeping in view the notions of generalized approximation spaces with relations other than equivalence.

Propositional Quantification in Bimodal $$\mathbf {S5}$$S5

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.

Characterizing right inverses for spatial constraint systems with applications to modal logic

Spatial constraint systems are algebraic structures from concurrent constraint programming to specify spatial and epistemic behavior in multi-agent systems. In this paper spatial constraint systems are used to give an abstract characterization of the notion of normality in modal logic and to derive right inverse/reverse operators for modal languages. In particular, a necessary and sufficient condition for the existence of right inverses is identified and the abstract notion of normality is shown to correspond to the preservation of finite suprema. Furthermore, a taxonomy of normal right inverses is provided, identifying the greatest normal right inverse as well as the complete family of minimal right inverses. These results are applied to existing modal languages such as the weakest normal modal logic, Hennessy–Milner logic, and linear-time temporal logic. Some implications of these results are also discussed in the context of modal concepts such as bisimilarity and inconsistency invariance.

A quantitative approach to reasoning about incomplete knowledge

In this paper, we aim to present a quantitative approach to reasoning about incomplete information. The study is conducted in MEL, a minimal epistemic logic relating modal languages to uncertainty theories. The proposed approach leads to two types of epistemic truth degrees of a proposition. Some related properties are derived. By means of a more general probability distribution on the set of epistemic states, two randomized versions of epistemic truth degrees are obtained. The connection between the notion of local probabilistic epistemic truth degree and belief function is also established. Based upon the fundamental notion of the global epistemic truth degree, the notion of epistemic similarity degree is also proposed and a kind of pseudo-metric used for approximate reasoning in MEL is thus derived. The obtained results provide a useful supplement to the existing study in the sense that it offers a quantitative approach instead of the qualitative manner in the literature.

Neighborhood semantics for modal many-valued logics

The majority of works on modal many-valued logics consider Kripke-style possible worlds frames as the principal semantics despite their well-known axiomatizability issues when considering non-Boolean accessibility relations. The present work explores a more general semantical picture, namely a many-valued version of the classical neighborhood semantics. We present it in two levels of generality. First, we work with modal languages containing only the two usual unary modalities, define neighborhood frames over algebras of the logic FLew with operators, and show their relation with the usual Kripke semantics (this is actually the highest level of generality where one can give a straightforward definition of the Kripke-style semantics). Second, we define generalized neighborhood frames for arbitrary modal languages over a given class of algebras for an arbitrary protoalgebraic logic and, assuming certain additional conditions, axiomatize the logic of all such frames (which generalizes the completeness theorem of the classical modal logic E with respect to classical neighborhood frames).

KD is nullary

In the ordinary modal language, KD is the modal logic determined by the class of all serial frames. In this paper, we demonstrate that KD is nullary.

Higher-Order Contingentism, Part 3: Expressive Limitations

Two expressive limitations of an infinitary higher-order modal language interpreted on models for higher-order contingentism – the thesis that it is contingent what propositions, properties and relations there are – are established: First, the inexpressibility of certain relations, which leads to the fact that certain model-theoretic existence conditions for relations cannot equivalently be reformulated in terms of being expressible in such a language. Second, the inexpressibility of certain modalized cardinality claims, which shows that in such a language, higher-order contingentists cannot express what is communicated using various instances of talk of ‘possible things’, such as ‘there are uncountably many possible stars’.

Definability in the class of all -frames – computability and complexity

In the basic modal language and in the basic modal language with the added universal modality , first-order definability of all formulas over the class of all frames is shown. Also, it is shown that the problems of modal definability of first-order sentences over the class of all frames in the languages and are both PSPACE-complete.

LOGICS FOR PROPOSITIONAL CONTINGENTISM

AbstractRobert Stalnaker has recently advocated propositional contingentism, the claim that it is contingent what propositions there are. He has proposed a philosophical theory of contingency in what propositions there are and sketched a possible worlds model theory for it. In this paper, such models are used to interpret two propositional modal languages: one containing an existential propositional quantifier, and one containing an existential propositional operator. It is shown that the resulting logic containing an existential quantifier is not recursively axiomatizable, as it is recursively isomorphic to second-order logic, and a natural candidate axiomatization for the resulting logic containing an existential operator is shown to be incomplete.

A logic of goal-directed knowing how

In this paper, we propose a decidable single-agent modal logic for reasoning about goal-directed “knowing how”, based on ideas from linguistics, philosophy, modal logic, and automated planning in AI. We first define a modal language to express “I know how to guarantee \(\varphi \) given \(\psi \)” with a semantics based not on standard epistemic models but on labeled transition systems that represent the agent’s knowledge of his own abilities. The semantics is inspired by conformant planning in AI. A sound and complete proof system is given to capture valid reasoning patterns, which highlights the compositional nature of “knowing how”. The logical language is further extended to handle knowing how to achieve a goal while maintaining other conditions.

Still in the Mood: The Versatility of Subjunctive Markers in Modal Logic

We investigate and compare two major approaches to enhancing the expressive capacities of modal languages, namely the addition of subjunctive markers on the one hand, and the addition of scope-bearing actuality operators, on the other. It turns out that the subjunctive marker approach is not only every bit as versatile as the actuality operator approach, but that it in fact outperforms its rival in the context of cross-world predication.

On the Modal Logic of Subset and Superset: Tense Logic over Medvedev Frames

Viewing the language of modal as a language for describing directed graphs, a natural type of directed graph to study modally is one where the nodes are sets and the edge relation is the subset or superset relation. A well-known example from the literature on intuitionistic is the class of Medvedev frames$${\langle W, R\rangle}$$?W,R? where W is the set of nonempty subsets of some nonempty finite set S, and xRy iff $${x\supseteq y}$$x?y, or more liberally, where $${\langle W, R\rangle}$$?W,R? is isomorphic as a directed graph to $${\langle \wp(S)\setminus\{\emptyset\},\supseteq\rangle}$$??(S)\{?},??. Prucnal (Stud Logica 38(3):247---262, 1979) proved that the modal of Medvedev frames is not finitely axiomatizable. Here we continue the study of Medvedev frames with extended modal languages. Our results concern definability. We show that the class of Medvedev frames is definable by a formula in the language of tense logic, i.e., with a converse modality for quantifying over supersets in Medvedev frames, extended with any one of the following standard devices: nominals (for naming nodes), a difference modality (for quantifying over those y such that $${x\not= y}$$x?y), or a complement modality (for quantifying over those y such that $${x\not\supseteq y}$$x?y). It follows that either the of Medvedev frames in one of these tense languages is finitely axiomatizable--which would answer the open question of whether Medvedev's (Sov Math Dokl 7:857---860, 1966) logic of finite problems is decidable--or else the minimal logics in these languages extended with our defining formulas are the beginnings of infinite sequences of frame-incomplete logics.

One'sModus Ponens: Modality, Coherence and Logic

Recently, there has been a shift away from traditional truth‐conditional accounts of meaning towards non‐truth‐conditional ones, e.g., expressivism, relativism and certain forms of dynamic semantics. Fueling this trend is some puzzling behavior of modal discourse. One particularly surprising manifestation of such behavior is the alleged failure of some of the most entrenched classical rules of inference; viz.,modus ponensandmodus tollens. These revisionary, non‐truth‐conditional accounts tout these failures, and the alleged tension between the behavior of modal vocabulary and classical logic, as data in support of their departure from tradition, since the revisionary semantics invalidate some of these patterns. I, instead, offer a semantics for modality with the resources to accommodate the puzzling data while preserving classical logic, thus affirming the tradition that modals express ordinary truth‐conditional content. My account shows that the real lesson of the apparent counterexamples is not the one the critics draw, but rather one they missed: namely, that there are linguistic mechanisms, reflected in the logical form, that affect the interpretation of modal language in a context in a systematic and precise way, which have to be captured by any adequate semantic account of the interaction between discourse context and modal vocabulary. The semantic theory I develop specifies these mechanisms and captures precisely how they affect the interpretation of modals in a context, and do so in a way that both explains the appearance of the putative counterexamples and preserves classical logic.

Certification of Prefixed Tableau Proofs for Modal Logic

Different theorem provers tend to produce proof objects in different formats and this is especially the case for modal logics, where several deductive formalisms (and provers based on them) have been presented. This work falls within the general project of establishing a common specification language in order to certify proofs given in a wide range of deductive formalisms. In particular, by using a translation from the modal language into a first-order polarized language and a checker whose small kernel is based on a classical focused sequent calculus, we are able to certify modal proofs given in labeled sequent calculi, prefixed tableaux and free-variable prefixed tableaux. We describe the general method for the logic K, present its implementation in a prolog-like language, provide some examples and discuss how to extend the approach to other normal modal logics

Models and reality*

AbstractKripke models, interpreted realistically, have difficulty making sense of the thesis that there might have existed things that do not in fact exist, since a Kripke model in which this thesis is true requires a model structure in which there are possible worlds with domains that contain things that do not exist. This paper argues that we can use Kripke models as representational devices that allow us to give a realistic interpretation of a modal language. The method of doing this is sketched, with the help of an analogy with a Galilean relativist theory of spatial properties and relations.

Williamsonian modal epistemology, possibility-based

AbstractWilliamsonian modal epistemology (WME) is characterized by two commitments: realism about modality, and anti-exceptionalism about our modal knowledge. Williamson's own counterfactual-based modal epistemology is the best known implementation of WME, but not the only option that is available. I sketch and defend an alternative implementation which takes our knowledge of metaphysical modality to arise, not from knowledge of counterfactuals, but from our knowledge of ordinary possibility statements of the form ‘x can F’. I defend this view against a criticism indicated in Williamson's own work, and argue that it is better connected to the semantics of modal language.

Summary

What is the relation between modal logic and modal metaphysics? As a technical discipline, modal logic is in effect a branch of applied mathematics that studies extensions of standard formal languages by additional sentence operators, independently of any particular interpretation of those operators. Modal metaphysics studies the nature and extent of possibility and necessity, where those terms are understood in a strong non-epistemic ‘metaphysical’ sense: something is necessary just in case it would have obtained no matter what, and possible just in case it does not necessarily not obtain. Informally, metaphysicians can read the operators □ and ◊ of modal logic as expressing such necessity and possibility, although that reading plays no official role in the technical development. When the modal language also contains quantifiers, metaphysicians will naturally read them as unrestricted. That metaphysical reading of the modal operators and quantifiers will be assumed in what follows. Metaphysicians want to know which formulas of the modal language are metaphysically universal , in the sense of being true on every interpretation of the non-logical constants and free variables in them. The metaphysically universal formulas, whichever they are, constitute the core metaphysical theory of the most general structure of modal reality. We can use modal logic as a technical aid in studying that theory.

Text Mining to Uncover the Intensity of SEC Comment Letters and Its Association with the Probability of 10-K Restatement

ABSTRACTThe SEC comment letter is the correspondence between SEC staff and SEC filers about the filers' public information disclosures. The intensity of comment letters in terms of the use of strong/weak modal language can reflect perceived deficiencies in the reviewed filings. This paper uses text mining to examine the intensity of SEC comment letters. A measure of intensity based on the modality of comment letters is developed. Empirical analysis is conducted on a sample of initial comment letters related to 10-K filings. Results show that intensity is positively associated with the probability of a restatement of the reviewed 10-K filings and that this association is robust using both the original Loughran and McDonald (2011) word lists and the modified word lists.

A Semantical Analysis of Second-Order Propositional Modal Logic

This paper is aimed as a contribution to the use of formal modal languages in Artificial Intelligence. We introduce a multi-modal version of Second-order Propositional Modal Logic (SOPML), an extension of modal logic with propositional quantification, and illustrate its usefulness as a specification language for knowledge representation as well as temporal and spatial reasoning. Then, we define novel notions of (bi)simulation and prove that these preserve the interpretation of SOPML formulas. Finally, we apply these results to assess the expressive power of SOPML.

Higher-Order Contingentism, Part 1: Closure and Generation

This paper is a study of higher-order contingentism – the view, roughly, that it is contingent what properties and propositions there are. We explore the motivations for this view and various ways in which it might be developed, synthesizing and expanding on work by Kit Fine, Robert Stalnaker, and Timothy Williamson. Special attention is paid to the question of whether the view makes sense by its own lights, or whether articulating the view requires drawing distinctions among possibilities that, according to the view itself, do not exist to be drawn. The paper begins with a non-technical exposition of the main ideas and technical results, which can be read on its own. This exposition is followed by a formal investigation of higher-order contingentism, in which the tools of variable-domain intensional model theory are used to articulate various versions of the view, understood as theories formulated in a higher-order modal language. Our overall assessment is mixed: higher-order contingentism can be fleshed out into an elegant systematic theory, but perhaps only at the cost of abandoning some of its original motivations.