We present tableau systems and sequent calculi for the intuitionistic analoguesIK, ID, IT, IKB, IKDB, IB, IK4, IKD4, IS4, IKB4, IK5, IKD5, IK45, IKD45 andIS5 of the normal classical modal logics. We provide soundness and completeness theorems with respect to the models of intuitionistic logic enriched by a modal accessibility relation, as proposed by G. Fischer Servi. We then show the disjunction property forIK, ID, IT, IKB, IKDB, IB, IK4, IKD4, IS4, IKB4, IK5, IK45 andIS5. We also investigate the relationship (...) of these logics with some other intuitionistic modal logics proposed in the literature. (shrink)

We give sound and complete tableau and sequent calculi for the prepositional normal modal logics S4.04, K4B and G 0(these logics are the smallest normal modal logics containing K and the schemata A A, A A and A ( A); A A and AA; A A and ((A A) A) A resp.) with the following properties: the calculi for S4.04 and G 0are cut-free and have the interpolation property, the calculus for K4B contains a restricted version of the cut-rule, the (...) so-called analytical cut-rule.In addition we show that G 0is not compact (and therefore not canonical), and we proof with the tableau-method that G 0is characterised by the class of all finite, (transitive) trees of degenerate or simple clusters of worlds; therefore G 0is decidable and also characterised by the class of all frames for G 0. (shrink)

This volume is a collection of papers presented at the colloquium, and it testifies to the growing importance of game theory as a tool that can capture concepts ...

Free-variable semantic tableaux are a well-established technique for first-order theorem proving where free variables act as a meta-linguistic device for tracking the eigenvariables used during proof search. We present the theoretical foundations to extend this technique to propositional modal logics, including non-trivial rigorous proofs of soundness and completeness, and also present various techniques that improve the efficiency of the basic naive method for such tableaux.

The language of standard propositional modal logic has one operator (? or ?), that can be thought of as being determined by the quantifiers ? or ?, respectively: for example, a formula of the form ?F is true at a point s just in case all the immediate successors of s verify F.This paper uses a propositional modal language with one operator determined by a generalized quantifier to discuss a simple connection between standard invariance conditions on modal formulas and generalized (...) quantifiers: the combined generalized quantifier conditions of conservativity and extension correspond to the modal condition of invariance under generated submodels, and the modal condition of invariance under bisimulations corresponds to the generalized quantifier being a Boolean combination of ? and ? (shrink)

A system of modal logic with the operator is proposed, and proved complete. In contrast with a previous one by Stalnaker and Thomason, this system does not require two categories of singular terms.

I present here a modal extension of T called KTLM which is, by several measures, the simplest modal extension of T yet presented. Its axiom uses only one sentence letter and has a modal depth of 2. Furthermore, KTLM can be realized as the logical union of two logics KM and KTL which each have the finite model property (f.m.p.), and so themselves are complete. Each of these two component logics has independent interest as well.

In this article we deal with Glivenko type theorems for intuitionistic modal logics over Prior's MIPC. We examine the problems which appear in proving Glivenko type theorems when passing from the intuitionistic propositional logic Intto MIPC. As a result we obtain two different versions of Glivenko's theorem for logics over MIPC. Since MIPCcan be thought of as a one-variable fragment of the intuitionistic predicate logic Q-Int, one of the versions of Glivenko's theorem for logics over MIPCis closely related to that (...) for intermediate predicate logics obtained by Umezawa [27] and Gabbay [15]. Another one is rather surprising. (shrink)

In this paper we consider an intuitionistic variant of the modal logic S4 (which we call IS4). The novelty of this paper is that we place particular importance on the natural deduction formulation of IS4— our formulation has several important metatheoretic properties. In addition, we study models of IS4— not in the framework of Kirpke semantics, but in the more general framework of category theory. This allows not only a more abstract definition of a whole class of models but also (...) a means of modelling proofs as well as provability. (shrink)

We investigate uniform interpolants in propositional modal logics from the proof-theoretical point of view. Our approach is adopted from Pitts’ proof of uniform interpolationin intuitionistic propositional logic [15]. The method is based on a simulation of certain quantifiers ranging over propositional variables and uses a terminating sequent calculus for which structural rules are admissible.

This modern, advanced textbook reviews modal logic, a field which caught the attention of computer scientists in the late 1970's.

In this note we show that the classical modal technology of Sahlqvist formulas gives quick proofs of the completeness theorems in [8] (D. Gregory, Completeness and decidability results for some propositional modal logics containing actually operators, Journal of Philosophical Logic 30(1): 57–78, 2001) and vastly generalizes them. Moreover, as a corollary, interpolation theorems for the logics considered in [8] are obtained. We then compare Gregory's modal language enriched with an actually operator with the work of Arthur Prior now known under (...) the name of hybrid logic. This analysis relates the actually axioms to standard hybrid axioms, yields the decidability results in [8], and provides a number of complexity results. Finally, we use a bisimulation argument to show that the hybrid language is strictly more expressive than Gregory's language. (shrink)

ABSTRACT: The modal systems of the Stoic logician Chrysippus and the two Hellenistic logicians Philo and Diodorus Cronus have survived in a fragmentary state in several sources. From these it is clear that Chrysippus was acquainted with Philo’s and Diodorus’ modal notions, and also that he developed his own in contrast of Diodorus’ and in some way incorporated Philo’s. The goal of this paper is to reconstruct the three modal systems, including their modal definitions and modal theorems, and to make (...) clear the exact relations between them; moreover, to elucidate the philosophical reasons that may have led Chrysippus to modify his predessors’ modal concept in the way he did. It becomes apparent that Chrysippus skillfully combined Philo’s and Diodorus’ modal notions, with making only a minimal change to Diodorus’ concept of possibility; and that he thus obtained a modal system of modalities (logical and physical) which fit perfectly fit into Stoic philosophy. (shrink)

ABSTRACT: Part 1 discusses the Stoic notion of propositions (assertibles, axiomata): their definition; their truth-criteria; the relation between sentence and proposition; propositions that perish; propositions that change their truth-value; the temporal dependency of propositions; the temporal dependency of the Stoic notion of truth; pseudo-dates in propositions. Part 2 discusses Stoic modal logic: the Stoic definitions of their modal notions (possibility, impossibility, necessity, non-necessity); the logical relations between the modalities; modalities as properties of propositions; contingent propositions; the relation between the Stoic (...) modal notions and those of Diodorus Cronus and Philo of Megara; the role of ‘external hindrances’ for the modalities; the temporal dependency of the modalities; propositions that change their modalities; the principle that something possible can follow from something impossible; the interpretations of the Stoic modal system by B. Mates, M. Kneale, M. Frede, J. Vuillemin and M. Mignucci are evaluated. -/- For a much shorter English version of Part 1 of the book see my ‘Stoic Logic’, in K. Algra et al. (eds), The Cambridge History of Hellenistic Philosophy, Cambridge 1999, 92-157. For a shorter, updated, English version of Part 2 of the book see my 'Chrysippus' Modal Logic and its Relation to Philo and Diodorus', in K. Doering / Th. Ebert (eds) Dialektiker und Stoiker (Stuttgart 1993) 63-84. (shrink)

The theory of belief revision deals with (rational) changes in beliefs in response to new information. In the literature a distinction has been drawn between belief revision and belief update (see [6]). The former deals with situations where the objective facts describing the world do not change (so that only the beliefs of the agent change over time), while the letter allows for situations where both the facts and the doxastic state of the agent change over time. We focus on (...) belief revision and propose a temporal framework that allows for iterated revision. We model the notion of “minimal” or “conservative” belief revision by considering logics of increasing strength. We move from one logic to the next by adding one or more axioms and show that the corresponding logic captures more stringent notions of minimal belief revision. The strongest logic that we propose provides a full axiomatization of the well-known AGM theory of belief revision. (shrink)

Two views of game theory are discussed: (1) game theory as a description of the behavior of rational individuals who recognize each other’s rationality and reasoning abilities, and (2) game theory as an internally consistent recommendation to individuals on how to act in interactive situations. It is shown that the same mathematical tool, namely modal logic, can be used to explicitly model both views.

The logic of common belief does not always re‡ect that of individual beliefs. In particular, even when the individual belief operators satisfy the KD45 logic, the common belief operator may fail to satisfy axiom 5. That is, it can happen that neither is A commonly believed nor is it common belief that A is not commonly believed. We identify the intersubjective restrictions on individual beliefs that are incorporated in axiom 5 for common belief.

This book, written by one of the most distinguished of contemporary philosophers of mathematics, is a fully rewritten and updated successor to the author's earlier The Unprovability of Consistency (CUP, 1979). Modal logic is concerned with the notions of necessity and possibility. What George Boolos does is to show how the concepts, techniques and methods of modal logic shed brilliant light on the most important logical discovery of the twentieth century: the incompleteness theorems of Kurt Godel and the 'self referential' (...) sentences constructed in their proof. The book explores the effects of reinterpreting the notions of necessity and possibility to near probability and consistency. It contains the first application of quantified modal logic to formal probability, and shows the results of applying modal logic to formal provability. (shrink)

The Unprovability of Consistency is concerned with connections between two branches of logic: proof theory and modal logic. Modal logic is the study of the principles that govern the concepts of necessity and possibility; proof theory is, in part, the study of those that govern provability and consistency. In this book, George Boolos looks at the principles of provability from the standpoint of modal logic. In doing so, he provides two perspectives on a debate in modal logic that has persisted (...) for at least thirty years between the followers of C. I. Lewis and W. V. O. Quine. The author employs semantic methods developed by Saul Kripke in his analysis of modal logical systems. The book will be of interest to advanced undergraduate and graduate students in logic, mathematics and philosophy, as well as to specialists in those fields. (shrink)

This paper deals with the system of modal logicGL, in particular with a formulation of it in terms of sequents. We prove some proof theoretical properties ofGL that allow to get the cut-elimination theorem according to Gentzen's procedure, that is, by double induction on grade and rank.

Kripke-style models with two accessibility relations, one intuitionistic and the other modal, are given for analogues of the modal systemK based on Heyting's prepositional logic. It is shown that these two relations can combine with each other in various ways. Soundness and completeness are proved for systems with only the necessity operator, or only the possibility operator, or both. Embeddings in modal systems with several modal operators, based on classical propositional logic, are also considered. This paper lays the ground for (...) an investigation of intuitionistic analogues of systems stronger thanK. A brief survey is given of the existing literature on intuitionistic modal logic. (shrink)

In the paper (Braüner, 2001) we gave a minimal condition for the existence of a homophonic theory of truth for a modal or tense logic. In the present paper we generalise this result to arbitrary modal logics and we also show that a modal logic permits the existence of a homophonic theory of truth if and only if it permits the definition of a so-called master modality. Moreover, we explore a connection between the master modality and hybrid logic: We show (...) that if attention is restricted to bidirectional frames, then the expressive power of the master modality is exactly what is needed to translate the bounded fragment of first-order logic into hybrid logic in a truth preserving way. We believe that this throws new light on Arthur Prior's fourth grade tense logic. (shrink)

The aim of the work is to provide a language to reason about Closed Interactions, i.e. all those situations in which the outcomes of an interaction can be determined by the agents themselves and in which the environment cannot interfere with they are able to determine. We will see that two different interpretations can be given of this restriction, both stemming from Pauly Representation Theorem. We will identify such restrictions and axiomatize their logic. We will apply the formal tools to (...) reason about games and their regulation. (shrink)

This is part two of a complete exposition of Logic, in which there is a radically new synthesis of Aristotelian-Scholastic Logic with modern Logic. Part II is the presentation of the theory of propositions. Simple, composite, atomic, compound, modal, and tensed propositions are all examined. Valid consequences and propositional logical identities are rigorously proven. Modal logic is rigorously defined and proven. This is the first work of Logic known to unite Aristotelian logic and modern logic using scholastic logic as the (...) instrument. (shrink)

We present an extension of the mosaic method aimed at capturing many-dimensional modal logics. As a proof-of-concept, we define the method for logics arising from the combination of linear tense operators with an “orthogonal” S5-like modality. We show that the existence of a model for a given set of formulas is equivalent to the existence of a suitable set of partial models, called mosaics, and apply the technique not only in obtaining a proof of decidability and a proof of completeness (...) for the corresponding Hilbert-style axiomatization, but also in the development of a mosaic-based tableau system. We further consider extensions for dealing with the case when interactions between the two dimensions exist, thus covering a wide class of bundled Ockhamist branching-time logics, and present for them some partial results, such as a non-analytic version of the tableau system. (shrink)

Specifies an axiomatization of the system S13 of modal logic. Referenced in Cocchiarella & Freund "Modal Logic: an Introduction to its Syntax and Semantics", Oxford University Press, 2008.

Discusses alternative interpretations of the modal operators, for the modal propositional logic S5.

For a novice this book is a mathematically-oriented introduction to modal logic, the discipline within mathematical logic studying mathematical models of reasoning which involve various kinds of modal operators. It starts with very fundamental concepts and gradually proceeds to the front line of current research, introducing in full details the modern semantic and algebraic apparatus and covering practically all classical results in the field. It contains both numerous exercises and open problems, and presupposes only minimal knowledge in mathematics. A specialist (...) can use the book as a source of references. Results and methods of many directions in propositional modal logic, from completeness and duality to algorithmic problems, are collected and systematically presented in one volume. (shrink)

A textbook on modal logic, intended for readers already acquainted with the elements of formal logic, containing nearly 500 exercises. Brian F. Chellas provides a systematic introduction to the principal ideas and results in contemporary treatments of modality, including theorems on completeness and decidability. Illustrative chapters focus on deontic logic and conditionality. Modality is a rapidly expanding branch of logic, and familiarity with the subject is now regarded as a necessary part of every philosopher's technical equipment. Chellas here offers an (...) up-to-date and reliable guide essential for the student. (shrink)

A powerful challenge to some highly influential theories, this book offers a thorough critical exposition of modal realism, the philosophical doctrine that many possible worlds exist of which our own universe is just one. Chihara challenges this claim and offers a new argument for modality without worlds.

This introduction to the basic forms of deductive inference as evaluated by methods of modern symbolic logic is de­signed for sophomore-junior-level stu­dents ready to specialize in the study of deductive logic. It can be used also for an introductory logic course. The inde­pendence of many sections allows the instructor utmost flexibility. The text consists of eight chapters, the first six of which are designed to intro­duce the student to basic topics of sen­tence and predicate logic. The last two chapters extend (...) the procedures of the first six to alethic modal logic, the logic of imperatives, and deontic logic. Throughout the text there is an attempt to relate symbolic techniques to issues in the philosophy of logic. (shrink)

Supervaluationism is a well known theory of vagueness. Subvaluationism is a less well known theory of vagueness. But these theories cannot be taken apart, for they are in a relation of duality that can be made precise. This paper provides an introduction to the subvaluationist theory of vagueness in connection to its dual, supervaluationism. A survey on the supervaluationist theory can be found in the Compass paper of Keefe (2008); our presentation of the theory in this paper will be short (...) to get rapidly into the logical issues. This paper is relatively self-contained. A modest background on propositional modal logic is, though not strictly necessary, advisable. The reader might find useful the Compass papers Kracht (2011) and Negri (2011) (though these papers cover issues of more complexity than what is demanded to follow this paper). (shrink)

In this text, a variety of modal logics at the sentential, first-order, and second-order levels are developed with clarity, precision and philosophical insight.

We introduce a system PSI for a strict implication operator called Priorean strict implication. The semantics for PSI is based on partial Kripke models without accessibility relations. PSI is proved sound and complete with respect to that semantics, and Prior's system Q and related systems are shown to be fragments of PSI or of a mild extension of it.

Standard possible world semantics for propositional modal languages ignore truth-value gaps. However, simple considerations suggest that it should not be so. In Section 1, I identify what I take to be a correct truth-clause for necessity under the assumption that some possible worlds are incomplete (i.e., "at" which some propositions lack a truth-value). In Section 2, I build a world semantics, the semantics of TV-models, for standard modal propositional languages, which agrees with the truth-clause for necessity previously identified. Sections 3–5 (...) are devoted to systematic concerns. In particular, in Section 4, Prior’s system Q (propositional version) is given a TV-models semantics and proved adequate (i.e., sound and complete) with respect to it. (shrink)

A (normal) system of prepositional modal logic is said to be complete iff it is characterized by a class of (Kripke) frames. When we move to modal predicate logic the question of completeness can again be raised. It is not hard to prove that if a predicate modal logic is complete then it is characterized by the class of all frames for the propositional logic on which it is based. Nor is it hard to prove that if a propositional modal (...) logic is incomplete then so is the predicate logic based on it. But the interesting question is whether a complete propositional modal logic can have an incomplete extension. In 1967 Kripke announced the incompleteness of a predicate extension of S4. The purpose of the present article is to present several such systems. In the first group it is the systemswith the Barcan Formula which are incomplete, while those without are complete. In the second group it is thosewithout the Barcan formula which are incomplete, while those with the Barcan Formula are complete. But all these are based on propositional systems which are characterized by frames satisfying in each case a single first-order sentence. (shrink)

Discrepancies between an agents goals and beliefs play an important, if implicit, role in determining what a rational agent is motivated to do. This is most obvious in cases where an agent achieves a complex goal incrementally and must deliberate anew as each milestone is reached. In such cases the concept of goal/belief discrepancy defines an appropriate space to which a degree-of-achievement yardstick can be applied. This paper presents soundness and completeness results concerning a logic for reasoning about goal/belief discrepancy, (...) and it is suggested that a certain species of goal/belief discrepancy captures the concept of desire. (shrink)

A selection function based semantics is offered for the 'can' of ability based on the idea that 'John can run a four minute mile' is true iff John would do so under the right conditions, meaning that he would do so under at least one appropriately chosen test condition. Completeness is proved for an axiom system and semantics based on this idea, and the logic turns out to be interestingly different from any standard system of modal logic.

Can we find propositions that cannot rationally be denied in any possible world without assuming the existence of that same proposition, and so involving ourselves in a contradiction? In other words, can we find transworld propositions needing no further foundation or justification? Basically, three differing positions can be imagined: firstly, a relativist position, according to which ultimately founded propositions are impossible; secondly, a meta-relativist position, according to which ultimately founded propositions are possible but unnecessary; and thirdly, an absolute position, according (...) to which such propositions are necessary. In this short essay I show that under the premise of modal logic S5 with constant domain there are ultimately founded propositions and that their existence is even necessary, and I will give some reasons for the superiority of S5 over other logics. (shrink)

In this work we propose an encoding of Reiter’s Situation Calculus solution to the frame problem into the framework of a simple multimodal logic of actions. In particular we present the modal counterpart of the regression technique. This gives us a theorem proving method for a relevant fragment of our modal logic.

We show the completeness of a Hilbert-style system LK defined by M. Valiev involving the knowledge operator K dedicated to the reasoning with incomplete information. The completeness proof uses a variant of Makinson's canonical model construction. Furthermore we prove that the theoremhood problem for LK is co-NP-complete, using techniques similar to those used to prove that the satisfiability problem for propositional S5 is NP-complete.

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. (shrink)

This work is divided in two papers (Part I and Part II). In Part I, we study a class of polymodal logics (herein called 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 are established by faithfully translating the (...) Rare-logics into more standard modal logics. The main idea of the translation consists in eliminating the Boolean terms by taking advantage of the components construction and in using various properties of the classes of semilattices involved in the semantics. The novelty of our approach allows us to prove new decidability results (presented in Part II), in particular for information logics derived from rough set theory and we open new perspectives to define proof systems for such logics (presented also in Part II). (shrink)

We give a set of postulates for the minimal normal modal logicK + without negation or any kind of implication. The connectives are simply , , , . The postulates (and theorems) are all deducibility statements . The only postulates that might not be obvious are.

<span class='Hi'></span> We present a class of normal modal calculi PFD,<span class='Hi'></span> whose syntax is endowed with operators M r <span class='Hi'></span>(and their dual ones,<span class='Hi'></span> L r)<span class='Hi'></span>, one for each r <span class='Hi'></span>[0,1]<span class='Hi'></span>: if a is sentence,<span class='Hi'></span> M r is to he read the probability that a is true is strictly greater than r and to he evaluated as true or false in every world of a F-restricted probabilistic kripkean model.<span class='Hi'></span> Every such a model is (...) a kripkean model,<span class='Hi'></span> enriched by a family of regular <span class='Hi'></span>(see below)<span class='Hi'></span> probability evaluations with range in a fixed finite subset F of <span class='Hi'></span>[0,1]<span class='Hi'></span>: there is one such a function for every world w,<span class='Hi'></span> P F(w,<span class='Hi'></span>-)<span class='Hi'></span>, and this allows to evaluate M ra as true in the world w iff p F(w,<span class='Hi'></span> )<span class='Hi'></span> r.For every fixed F as before,<span class='Hi'></span> suitable axioms and rules are displayed,<span class='Hi'></span> so that the resulting system P FD is complete and compact with respect to the class of all the F-restricted probabilistic kripkean models. (shrink)

We give a sound and complete propositional S5 tableau system of a particularly simple sort, having an easy completeness proof. It sheds light on why the satisfiability problem for S5 is less complex than that for most other propositional modal logics. We believe the system remains complete when quantifier rules are added. If so, it would allow us to get partway to an interpolation theorem for first-order S5, a theorem that is known to fail in general.

One can add the machinery of relation symbols and terms to a propositional modal logic without adding quantifiers. Ordinarily this is no extension beyond the propositional. But if terms are allowed to be non-rigid, a scoping mechanism (usually written using lambda abstraction) must also be introduced to avoid ambiguity. Since quantifiers are not present, this is not really a first-order logic, but it is not exactly propositional either. For propositional logics such as K, T and D, adding such machinery produces (...) a decidable logic, but adding it to S5 produces an undecidable one. Further, if an equality symbol is in the language, and interpreted by the equality relation, logics from K4 to S5 yield undecidable versions. (Thus transitivity is the villain here.) The proof of undecidability consists in showing that classical first-order logic can be embedded. (shrink)

Justification logics are logics of knowledge in which explicit reasons are formally represented. Standard logics of knowledge have justification logic analogs. Connecting justification logics and logics of knowledge are Realization Theorems. In this paper we give a new, constructive proof of the Realization Theorem connecting S5 and its justification analog, JS5. This proof is, I believe, the simplest in the literature.

An interpolation theorem holds for many standard modal logics, but first order S5 is a prominent example of a logic for which it fails. In this paper it is shown that a first order S5 interpolation theorem can be proved provided the logic is extended to contain propositional quantifiers. A proper statement of the result involves some subtleties, but this is the essence of it.

In this paper we will provide a modal-to-modal translational embedding of E into K, simplifying a similar result which is obtainable using a novel translation due to S.K. Thomason.

The purpose of the present note is to advertise an interesting conjecture concerning a well-known translation in modal logic, by confirming a (highly restricted) special case of the conjecture.

Two-dimensional semantics is a theory in the philosophy of language that provides an account of meaning which is sensitive to the distinction between necessity and apriority. Usually, this theory is presented in an informal manner. In this thesis, I take first steps in formalizing it, and use the formalization to present some considerations in favor of two-dimensional semantics. To do so, I define a semantics for a propositional modal logic with operators for the modalities of necessity, actuality, and apriority that (...) captures the relevant ideas of two-dimensional semantics. I use this to show that some criticisms of two-dimensional semantics that claim that the theory is incoherent are not justified. I also axiomatize the logic, and compare it to the most important proposals in the literature that define similar logics. To indicate that two-dimensional semantics is a plausible semantic theory, I give an argument that shows that all theorems of the logic can be philosophically justified independently of two-dimensional semantics. (shrink)

This paper is concerned with a propositional modal logic with operators for necessity, actuality and apriority. The logic is characterized by a class of relational structures defined according to ideas of epistemic two-dimensional semantics, and can therefore be seen as formalizing the relations between necessity, actuality and apriority according to epistemic two-dimensional semantics. We can ask whether this logic is correct, in the sense that its theorems are all and only the informally valid formulas. This paper gives outlines of two (...) arguments that jointly show that this is the case. The first is intended to show that the logic is informally sound, in the sense that all of its theorems are informally valid. The second is intended to show that it is informally complete, in the sense that all informal validities are among its theorems. In order to give these arguments, a number of independently interesting results concerning the logic are proven. In particular, the soundness and completeness of two different proof systems with respect to the semantics is proven (Theorems 2.11. and 2.15.), as well as a normal form theorem (Theorem 3.23.), an elimination theorem for the actuality operator (Corollary 3.27.), and the decidability of the logic (Corollary 3.28.). (shrink)

Epistemic two-dimensional semantics is a theory in the philosophy of language that provides an account of meaning which is sensitive to the distinction between necessity and apriority. While this theory is usually presented in an informal manner, I take some steps in formalizing it in this paper. To do so, I define a semantics for a propositional modal logic with operators for the modalities of necessity, actuality, and apriority that captures the relevant ideas of epistemic two-dimensional semantics. I also describe (...) some properties of the logic that are interesting from a philosophical perspective, and apply it to the so-called nesting problem. (shrink)

In this work we develop goal-directed deduction methods for the implicational fragment of several modal logics. We give sound and complete procedures for strict implication of K, T, K4, S4, K5, K45, KB, KTB, S5, G and for some intuitionistic variants. In order to achieve a uniform and concise presentation, we first develop our methods in the framework of Labelled Deductive Systems [Gabbay 96]. The proof systems we present are strongly analytical and satisfy a basic property of cut admissibility. We (...) then show that for most of the systems under consideration the labelling mechanism can be avoided by choosing an appropriate way of structuring theories. One peculiar feature of our proof systems is the use of restart rules which allow to re-ask the original goal of a deduction. In case of K, K4, S4 and G, we can eliminate such a rule, without loosing completeness. In all the other cases, by dropping such a rule, we get an intuitionistic variant of each system. The present results are part of a larger project of a goal directed proof theory for non-classical logics; the purpose of this project is to show that most implicational logics stem from slight variations of a unique deduction method, and from different ways of structuring theories. Moreover, the proof systems we present follow the logic programming style of deduction and seem promising for proof search [Gabbay and Reyle 84, Miller et al. 91]. (shrink)

We define an interpretation of modal languages with polyadic operators in modal languages that use monadic operators (diamonds) only. We also define a simulation operator which associates a logic $\Lambda^{sim}$ in the diamond language with each logic Λ in the language with polyadic modal connectives. We prove that this simulation operator transfers several useful properties of modal logics, such as finite/recursive axiomatizability, frame completeness and the finite model property, canonicity and first-order definability.

The category-theoretic nature of general frames for modal logic is explored. A new notion of "modal map" between frames is defined, generalizing the usual notion of bounded morphism/p-morphism. The category Fm of all frames and modal maps has reflective subcategories CHFm of compact Hausdorff frames, DFm of descriptive frames, and UEFm of ultrafilter enlargements of frames. All three subcategories are equivalent, and are dual to the category of modal algebras and their homomorphisms. An important example of a modal map that (...) is typically not a bounded morphism is the natural insertion of a frame A into its ultrafilter enlargement EA. This map is used to show that EA is the free compact Hausdorff frame generated by A relative to Fm. The monad E of the resulting adjunction is examined and its Eilenberg-Moore category is shown to be isomorphic to CHFm. A categorical equivalence between the Kleisli category of E and UEFm is defined from a construction that assigns to each frame A a frame A* that is "image-closed" in the sense that every point-image {b : aRb} in A is topologically closed. A* is the unique image-closed frame having the same ultrafilter enlargement as A. (shrink)

Qualitative Reasoning (QR) is an area of research within Artificial Intelligence that automates reasoning and problem solving about the physical world. QR research aims to deal with representation and reasoning about continuous aspects of entities without the kind of precise quantitative information needed by conventional numerical analysis techniques. Order-of-magnitude Reasoning (OMR) is an approach in QR concerned with the analysis of physical systems in terms of relative magnitudes. In this paper we consider the logic OMR_N for order-of-magnitude reasoning with the (...) bidirectional negligibility relation. It is a multi-modal logic given by a Hilbert-style axiomatization that reflects properties and interactions of two basic accessibility relations (strict linear order and bidirectional negligibility). Although the logic was studied in many papers, nothing was known about its decidability. In the paper we prove decidability of OMR N by showing that the logic has the strong finite model property. (shrink)

We introduce an Automatic Theorem Prover (ATP) of a dual tableau system for a relational logic for order of magnitude qualitative reasoning, which allows us to deal with relations such as negligibility, non-closeness and distance. Dual tableau systems are validity checkers that can serve as a tool for verification of a variety of tasks in order of magnitude reasoning, such as the use of qualitative sum of some classes of numbers. In the design of our ATP, we have introduced some (...) heuristics, such as the so called phantom variables, which improve the efficiency of the selection of variables used un the proof. (shrink)

Qualitative description of the movement of objects can be very important when there are large quantity of data or incomplete information, such as in positioning technologies and movement of robots. We present a first step in the combination of fuzzy qualitative reasoning and quantitative data obtained by human interaction and external devices as GPS, in order to update and correct the qualitative information. We consider a Propositional Dynamic Logic which deals with qualitative velocity and enables us to represent some reasoning (...) tasks about qualitative properties. The use of logic provides a general framework which improves the capacity of reasoning. In this way, we can infer additional information by using axioms and the logic apparatus. In this paper we present sound and complete relational dual tableau that can be used for verification of validity of formulas of the logic in question. (shrink)