Logic of nondeterministic information

In the paper we define a class of languages for representation o knowledge in those application areas when a complete information about a domain is not available. In the languages we introduce modal operators determined by accessibility relations depending on parameters.     

Expressive completeness in modal language

Conclusion The logics of the modal operators and of the quantifiers show striking analogies. The analogies are so extensive that, when a special class of entities (possible worlds) is postulated, natural and non-arbitrary translation procedures can be defined from the language with the modal operators into a purely quantificational one, under which the necessity and possibility operators translate into universal and existential quantifiers. In view of this I would be willing to classify the modal operators as disguised quantifiers, and I think that wholehearted acceptance of modal language should be considered to carry ontological commitment to something like possible worldsConsidered as two languages for describing the same subject matter, modal and purely quantificational languages show interesting differences. The operator variables of the purely quantificational languages give them more power than the modal languages, but at least some of the functions performed by the apparatus of operator variables are also performed, in a more primitive and less versatile way, by actuality operators in modal languages.A final note. Quine has written much on the inter-relations of quantifiers, identity, and the concept of existence. These, he holds, form a tightly knit conceptual system which has been evolved to a high point of perfection, but which might conceivably change yet further.²⁹ He has also dropped hints about the possibility of a simpler, primitive or defective version of the system, in which the quantifiers are not backed up in their accustomed way by the concept of identity. He has dubbed the resulting concept a pre-individuative concept of existence, or a concept of entity without identity. What would a pre-individuative concept of existence be like? Quine has sometimes suggested that one might be embodied in the use of mass nouns, but the identity concept is used in connection with stuff as well as with things: is that the same coffee that was in the cup last night? I would submit that modality provides a better case. In view of the comparative weakness of modal languages, compared to the explicitly quantificational ones Quine takes as canonical, there is surely a sense in which the concept of existence embodied in that disguised existential quantifier, the possibility operator, is a defective one. And as we have seen, one of the differences between modal operators and explicit quantifiers is that modal operators cannot be joined with the identity predicate in the way quantifiers with operator variables can. Surely, then, there is a sense in which ordinary speech, as opposed to the metaphysical theorizing of a Leibniz or a David Lewis, conceives of possible worlds as entities without identity.As should become obvious on reading it, this paper is inspired by the work of David Lewis, particularly his classic Counterpart Theory and Quantified Modal Logic, Journal of Philosophy 7, March 1968. I would like to thank Lewis and the referee for this journal for encouragement and advice. An earlier version of the paper was formulated in terms of Lewis's counterpart theory rather than in terms of individuals being in more than one world, but, since I considered only the case in which the counterpart relation was an equivalence, I Felt that the added complexity was not justified. Doing things in a counterpart-theoretic framework does produce two new classes of sentences of the non-modal language which lack translations in the modal: speaking of the properties individuals have simpliciter rather than of those they have at a world allows us to discuss relations obtaining between individuals in different worlds (e.g. the longer than relation obtaining between actual yachts and their counterparts), and the assumption that no individual is in more than one world allows a tricky way of asserting that there are at most n worlds without using the identity predicate between terms for worlds. Otherwise, given the assumption that the counterpart relation was an equivalence with at most one member of each equivalence class being in each world, the transition from the counterpart theoretic framework to the current one was perfectly straightforward.     

Coinductive models and normal forms for modal logics (or how we learned to stop worrying and love coinduction)

We present a coinductive definition of models for modal logics and show that it provides a homogeneous framework in which it is possible to include different modal languages ranging from classical modalities to operators from hybrid and memory logics. Moreover, results that had to be proved separately for each different language—but whose proofs were known to be mere routine—now can be proved in a general way. We show, for example, that we can have a unique definition of bisimulation for all these languages, and prove a single invariance-under-bisimulation theorem.We then use the new framework to investigate normal forms for modal logics. The normal form we introduce may have a smaller modal depth than the original formula, and it is inspired by global modalities like the universal modality and the satisfiability operator from hybrid logics. These modalities can be extracted from under the scope of other operators. We provide a general definition of extractable modalities and show how to compute extracted normal forms. As it is the case with other classical normal forms—e.g., the conjunctive normal form of propositional logic—the extracted normal form of a formula can be exponentially bigger than the original formula, if we require the two formulas to be equivalent. If we only require equi-satisfiability, then every modal formula has an extracted normal form which is only polynomially bigger than the original formula, and it can be computed in polynomial time.     

Completeness and Decidability Results for Some Propositional Modal Logics Containing “Actually” Operators

The addition of actually operators to modal languages allows us to capture important inferential behaviours which cannot be adequately captured in logics formulated in simpler languages. Previous work on modal logics containing actually operators has concentrated entirely upon extensions of KT5 and has employed a particular model-theoretic treatment of them. This paper proves completeness and decidability results for a range of normal and nonnormal but quasi-normal propositional modal logics containing actually operators, the weakest of which are conservative extensions of K, using a novel generalisation of the standard semantics.     

Term-Modal Logics

Many powerful logics exist today for reasoning about multi-agent systems, but in most of these it is hard to reason about an infinite or indeterminate number of agents. Also the naming schemes used in the logics often lack expressiveness to name agents in an intuitive way.To obtain a more expressive language for multi-agent reasoning and a better naming scheme for agents, we introduce a family of logics called term-modal logics. A main feature of our logics is the use of modal operators indexed by the terms of the logics. Thus, one can quantify over variables occurring in modal operators. In term-modal logics agents can be represented by terms, and knowledge of agents is expressed with formulas within the scope of modal operators.This gives us a flexible and uniform language for reasoning about the agents themselves and their knowledge. This article gives examples of the expressiveness of the languages and provides sequent-style and tableau-based proof systems for the logics. Furthermore we give proofs of soundness and completeness with respect to the possible world semantics.     

On Modal Logics Characterized by Models with Relative Accessibility Relations: Part II

This work is divided in two papers (Part I and Part II). In Part I, we introduced the class of Rare-logics for which the set of terms indexing the modal operators are hierarchized in two levels: the set of Boolean terms and the set of terms built upon the set of Boolean terms. By investigating different algebraic properties satisfied by the models of the Rare-logics, reductions for decidability were established by faithfully translating the Rare-logics into more standard modal logics (some of them contain the universal modal operator).In Part II, we push forward the results from Part I. For Rare-logics with nominals (present at the level of formulae and at the level of modal expressions), we show that the constructions from Part I can be extended although it is technically more involved. We also characterize a class of standard modal logics for which the universal modal operator can be eliminated as far as satifiability is concerned. Although the previous results have a semantic flavour, we are also able to define proof systems for Rare-logics from existing proof systems for the corresponding standard modal logics. Last, but not least, decidability results for Rare-logics are established uniformly, in particular for information logics derived from rough set theory.Since this paper is the continuation of Part I, we do not recall here the definitions of Part I although we refer to them.     

A General Framework for $$ {FDE}$$ FDE -Based Modal Logics

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.

     

Comparing Fixed-Point and Revision Theories of Truth

In response to the liar’s paradox, Kripke developed the fixed-point semantics for languages expressing their own truth concepts. (Martin and Woodruff independently developed this semantics, but not to the same extent as Kripke.) Kripke’s work suggests a number of related fixed-point theories of truth for such languages. Gupta and Belnap develop their revision theory of truth in contrast to the fixed-point theories. The current paper considers three natural ways to compare the various resulting theories of truth, and establishes the resulting relationships among these theories. The point is to get a sense of the lay of the land amid a variety of options. Our results will also provide technical fodder for the methodological remarks of the companion paper to this one.     

Propositional Epistemic Logics with Quantification Over Agents of Knowledge

The paper presents a family of propositional epistemic logics such that languages of these logics are extended by quantification over modal (epistemic) operators or over agents of knowledge and extended by predicate symbols that take modal (epistemic) operators (or agents) as arguments. Denote this family by \({\mathcal {P}\mathcal {E}\mathcal {L}}_{({ QK})}\). There exist epistemic logics whose languages have the above mentioned properties (see, for example Corsi and Orlandelli in Stud Log 101:1159–1183, 2013; Fitting et al. in Stud Log 69:133–169, 2001; Grove in Artif Intell 74(2):311–350, 1995; Lomuscio and Colombetti in Proceedings of ATAL 1996. Lecture Notes in Computer Science (LNCS), vol 1193, pp 71–85, 1996). But these logics are obtained from first-order modal logics, while a logic of \({\mathcal {P}\mathcal {E}\mathcal {L}}_{({ QK})}\) can be regarded as a propositional multi-modal logic whose language includes quantifiers over modal (epistemic) operators and predicate symbols that take modal (epistemic) operators as arguments. Among the logics of \({\mathcal {P}\mathcal {E}\mathcal {L}}_{({ QK})}\) there are logics with a syntactical distinction between two readings of epistemic sentences: de dicto and de re (between ‘knowing that’ and ‘knowing of’). We show the decidability of logics of \({\mathcal {P}\mathcal {E}\mathcal {L}}_{({ QK})}\) with the help of the loosely guarded fragment (LGF) of first-order logic. Namely, we generalize LGF to a higher-order decidable loosely guarded fragment. The latter fragment allows us to construct various decidable propositional epistemic logics with quantification over modal (epistemic) operators. The family of this logics coincides with \({\mathcal {P}\mathcal {E}\mathcal {L}}_{({ QK})}\). There are decidable propositional logics such that these logics implicitly contain quantification over agents of knowledge, but languages of these logics are usual propositional epistemic languages without quantifiers and predicate symbols (see Grove and Halpern in J Log Comput 3(4):345–378, 1993). Some logics of \({\mathcal {P}\mathcal {E}\mathcal {L}}_{({ QK})}\) can be regarded as counterparts of logics defined in Grove and Halpern (J Log Comput 3(4):345–378, 1993). We prove that the satisfiability problem for these logics of \({\mathcal {P}\mathcal {E}\mathcal {L}}_{({ QK})}\) is Pspace-complete using their counterparts in Grove and Halpern (J Log Comput 3(4):345–378, 1993).     

Ontology-free modal semantics

The problem with model-theoretic modal semantics is that it provides only the formal beginnings of an account of the semantics of modal languages. In the case of non-modal language, we bridge the gap between semantics and mere model theory, by claiming that a sentence is true just in case it is true in an intended model. Truth in a model is given by the model theory, and an intended model is a model which has as domain the actual objects of discourse, and which relates these objects in an appropriate manner. However, the same strategy applied to the modal case seems to require an intended modal model whose domain includes mere possibilia.Building on recent work by Christopher Menzel (Nous 1990), I give an account of model-theoretic semantics for modal languages which does not require mere possibilia or intensional entities of any kind. Menzel has offered a representational account of model-theoretic modal semantics that accords with actualist scruples, since it does not require possibilia. However, Menzel's view is in the company of other actualists who seek to eliminate possible worlds, but whose accounts tolerate other sorts of abstract, intensional entities, such as possible states of affairs. Menzel's account crucially depends on the existence of properties and relations in intension.I offer a purely extensional, representational account and prove that it does all the work that Menzel's account does. The result of this endeavor is an account of model-theoretic semantics for modal languages requiring nothing but pure sets and the actual objects of discourse. Since ontologically beyond what is prima facie presupposed by the model theory itself. Thus, the result is truly an ontology-free model-theoretic semantics for modal languages. That is to say, getting genuine modal semantics out of the model theory is ontologically cost-free. Since my extensional account is demonstrably no less adeguate, and yet is at the same time more ontologically frugal, it is certainly to be preferred.Special thanks to Brian Chellas, Charles Chihara, Harry Deutsch, Bernard Linsky, Kirk Ludwig, Christopher Menzel and Gila Sher for helpful discussion. My thanks also to an anonymous referee for this Journal for kind words and attention to detail. Portions of this paper were presented at the 1993 meeting of the Society for Exact Philosophy in Toronto, and at the 1994 conference of the Association for Symbolic Logic in Gainesville, Florida. Thanks to all who attended those sessions.     

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.

     

Subjunctivity and cross-world predication

The main goal of this paper is to present and compare two approaches to formalizing cross-world comparisons like ??John might have been taller than he is?? in quantified modal logics. One is the standard method employing degrees and graded positives, according to which the example just given is to be paraphrased as something like ??The height that John has is such that he might have had a height greater than it,?? which is amenable to familiar formalization strategies with respect to quantified modal logic. The other approach, based on subjunctive modal logic, mimics the mixed indicative-subjunctive patterns typical of cross-world comparisons in many natural languages by means of explicit mood markers. This latter approach is new and should, for various reasons, appeal to linguists and philosophers. Along the way, I argue that attempts to capture cross-world comparison by means of sentential operators are either inadequate or subject to substantive logical and philosophical objections.     

SUPERVALUATION FIXED-POINT LOGICS OF TRUTH

Michael Kremer defines fixed-point logics of truth based on Saul Kripke’s fixed point semantics for languages expressing their own truth concepts. Kremer axiomatizes the strong Kleene fixed-point logic of truth and the weak Kleene fixed-point logic of truth, but leaves the axiomatizability question open for the supervaluation fixed-point logic of truth and its variants. We show that the principal supervaluation fixed point logic of truth, when thought of as consequence relation, is highly complex: it is not even analytic. We also consider variants, engendered by a stronger notion of ‘fixed point’, and by variant supervaluation schemes. A ‘logic’ is often thought of, not as a consequence relation, but as a set of sentences – the sentences true on each interpretation. We axiomatize the supervaluation fixed-point logics so conceived.     

Quantifiers as modal operators

Montague, Prior, von Wright and others drew attention to resemblances between modal operators and quantifiers. In this paper we show that classical quantifiers can, in fact, be regarded as S5-like operators in a purely propositional modal logic. This logic is axiomatized and some interesting fragments of it are investigated.This paper is based in part on material in Chapter IV of [9]. I have benefited greatly from comments of Anthony M. Ungar.     

A modal type theory for formalizing trusted communications

This paper introduces a multi-modal polymorphic type theory to model epistemic processes characterized by trust, defined as a second-order relation affecting the communication process between sources and a receiver. In this language, a set of senders is expressed by a modal prioritized context, whereas the receiver is formulated in terms of a contextually derived modal judgement. Introduction and elimination rules for modalities are based on the polymorphism of terms in the language. This leads to a multi-modal non-homogeneous version of a type theory, in which we show the embedding of the modal operators into standard group knowledge operators.     

The Logic of “Most” and “Mostly”

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.     

‘Now’ and ‘Then’ in Tense Logic

According to Hans Kamp and Frank Vlach, the two-dimensional tense operators “now” and “then” are ineliminable in quantified tense logic. This is often adduced as an argument against tense logic, and in favor of an extensional account that makes use of explicit quantification over times. The aim of this paper is to defend tense logic against this attack. It shows that “now” and “then” are eliminable in quantified tense logic, provided we endow it with enough quantificational structure. The operators might not be redundant in some other systems of tense logic, but this merely indicates a lack of quantificational resources and does not show any deep-seated inability of tense logic to express claims about time. The paper closes with a brief discussion of the modal analogue of this issue, which concerns the role of the actuality operator in quantified modal logic.     

First-Order Classical Modal Logic

The paper focuses on extending to the first order case the semantical program for modalities first introduced by Dana Scott and Richard Montague. We focus on the study of neighborhood frames with constant domains and we offer in the first part of the paper a series of new completeness results for salient classical systems of first order modal logic. Among other results we show that it is possible to prove strong completeness results for normal systems without the Barcan Formula (like FOL + K)in terms of neighborhood frames with constant domains. The first order models we present permit the study of many epistemic modalities recently proposed in computer science as well as the development of adequate models for monadic operators of high probability. Models of this type are either difficult of impossible to build in terms of relational Kripkean semantics [40].We conclude by introducing general first order neighborhood frames with constant domains and we offer a general completeness result for the entire family of classical first order modal systems in terms of them, circumventing some well-known problems of propositional and first order neighborhood semantics (mainly the fact that many classical modal logics are incomplete with respect to an unmodified version of either neighborhood or relational frames). We argue that the semantical program that thus arises offers the first complete semantic unification of the family of classical first order modal logics.     

Modality,Quantification, and Many Vlach-Operators

Consider two standard quantified modal languages \(\textbf{\textsf{A}}\) and \(\textbf{\textsf{P}}\) whose vocabularies comprise the identity predicate and the existence predicate, each endowed with a standard S5 Kripke semantics where the models have a distinguished actual world, which differ only in that the quantifiers of \(\textbf{\textsf{A}}\) are actualist while those of \(\textbf{\textsf{P}}\) are possibilist. Is it possible to enrich these languages in the same manner, in a non-trivial way, so that the two resulting languages are equally expressive—i.e., so that for each sentence of one language there is a sentence of the other language such that given any model, the former sentence is true at the actual world of the model iff the latter is? Forbes (1989) shows that this can be done by adding to both languages a pair of sentential operators called Vlach-operators, and imposing a syntactic restriction on their occurrences in formulas. As Forbes himself recognizes, this restriction is somewhat artificial. The first result I establish in this paper is that one gets sameness of expressivity by introducing infinitely many distinct pairs of indexed Vlach-operators. I then study the effect of adding to our enriched modal languages a rigid actuality operator. Finally, I discuss another means of enriching both languages which makes them expressively equivalent, one that exploits devices introduced in Peacocke (1978). Forbes himself mentions that option but does not prove that the resulting languages are equally expressive. I do, and I also compare the Peacockian and the Vlachian methods. In due course, I introduce an alternative notion of expressivity and I compare the Peacockian and the Vlachian languages in terms of that other notion.