We consider some modal languages with a modal operator D whose semantics is based on the relation of inequality. Basic logical properties such as definability, expressive power and completeness are studied. Also, some connections with a number of other recent proposals to extend the standard modal language are pointed at.

This much-needed book provides a thorough account of temporal logic, one of the most important areas of logic in computer science today. The book begins with a solid introduction to semantical and axiomatic approaches to temporal logic. It goes on to cover predicate temporal logic, meta-languages, general theories of axiomatization, many dimensional systems, propositional quantifiers, expressive power, Henkin dimension, temporalization of other logics, and decidability results. With its inclusion of cutting-edge results and unifying methodologies, this book is an indispensable reference (...) for both the pure logician and the theoretical computer scientist. (shrink)

T × W logic is a combination of tense and modal logic for worlds or histories with the same time order. It is the basis for logics of causation, agency and conditionals, and therefore an important tool for philosophical logic. Semantically it has been defined, among others, by R. H. Thomason. Using an operator expressing truth in all worlds, first discussed by C. M. Di Maio and A. Zanardo, an axiomatization is given and its completeness proved via D. Gabbay’s irreflexivity (...) lemma. Given this lemma the proof is more or less straight forward. At the end an alternative axiomatization is sketched in which Di Maio’s and Zanardo’s operator is replaced by a version of “actually”. (shrink)

The semantical structures called T x W frames were introduced in (Thomason, 1984) for the Ockhamist temporal-modal language, $[Unrepresented Character]_{o}$ , which consists of the usual propositional language augmented with the Priorean operators P and F and with a possibility operator ◇. However, these structures are also suitable for interpreting an extended language, $[Unrepresented Character]_{so}$ , containing a further possibility operator $\lozenge^{s}$ which expresses synchronism among possibly incompatible histories and which can thus be thought of as a cross-history 'simultaneity' operator. (...) In the present paper we provide an infinite set of axioms in $[Unrepresented Character]_{so}$ , which is shown to be strongly complete for T x W-validity. Von Kutschera (1997) contains a finite axiomatization of T x W-validity which however makes use of the Gabbay Irreflexivity Rule (Gabbay, 1981). In order to avoid using this rule, the proof presented here develops a new technique to deal with reflexive maximal consistent sets in Henkin-style constructions. (shrink)

T × W logic is a combination of tense and modal logic for worlds or histories with the same time order. It is the basis for logics of causation, agency and conditionals, and therefore an important tool for philosophical logic. Semantically it has been defined, among others, by R. H. Thomason. Using an operator expressing truth in all worlds, first discussed by C. M. Di Maio and A. Zanardo, an axiomatization is given and its completeness proved via D. Gabbay’s irreflexivity (...) lemma. Given this lemma the proof is more or less straight forward. At the end an alternative axiomatization is sketched in which Di Maio’s and Zanardo’s operator is replaced by a version of actually. (shrink)

John Horty effectively develops deontic logic (the logic of ethical concepts like obligation and permission) against the background of a formal theory of agency. He incorporates certain elements of decision theory to set out a new deontic account of what agents ought to do under various conditions over extended periods of time. Offering a conceptual rather than technical emphasis, Horty's framework allows a number of recent issues from moral theory to be set out clearly and discussed from a uniform point (...) of view. (shrink)

A structure is described that can serve as a foundation for a semantics for a modal agentive construction such as sees to it that Q ([ stit: Q]). The primitives are Tree,,Instant, Agent, choice. Eleven simple postulates governing this structure are set forth and motivated. Tree and encode a picture of branching time consisting of moments gathered into maximal chains called histories. Instant imposes a time-like ordering. Agent consists of agents, and choice assigns to each agent and each moment in (...) Tree a set of possible choices, where each possible choice is a set of histories. All of these ingredients are referred to in the semantics suggested for [ stit: Q]. The most complex part of the discussion is the motivation for the definition of what it means for a typically non-terminating chain of moments jointly to witness the truth of [ stit: Q] at a moment.The paper begins with an informal account of the Refref conjecture, which says that the only way to refrain from refraining from seeing to something is to see to it. The paper ends with a consideration of an argument of Prior's that in a certain sense contemplation and action are inconsistent. (shrink)

The main purpose of this paper is to prove that in every stit semantic structure that contains a busy choice sequence, neither does doing imply refraining from refraining from doing, nor does refraining from refraining from doing imply doing.

Based on a notion of "companions to stit formulas" applied in other papers dealing with astit logics, we introduce "choice formulas" and "nested choice formulas" to prove the completeness theorems for dstit logics in a language with the dstit operator as the only non-truth-functional operator. The main logic discussed in this paper is the basic logic of dstit with multiple agents, other logics discussed include the basic logic of dstit with a single agent and some logics of dstit with multiple (...) agents each of which corresponds to a semantic condition concerning the number of possible choices for agents. (shrink)

The purpose of this paper is to present some results instit theory, a theory of agency proposed by N. Belnap and M. Perloff. We will establish a correspondence between the numbers ofstit modalities and the complexity degrees ofbusy choice sequences in semantic structures, and consequently, a correspondence between the number of modes of actions/inactions instit theory and the complexity degrees ofbusy choice sequences in semantic structures.

The purpose of this paper is to prove the decidability ofstit theory (a logic of seeing to it that) with a single agent andRefref Equivalence. This result is obtained through an axiomatization of the theory and a proof that it has thefinite model property. A notion ofcompanions to stit formulas is introduced and extensively used in the proof.

We present in this paper an axiomatization of Belnap and Perloff's stit theory (a logic of "seeing to it that") with a single agent. The idea of the proof is to apply the notion of companion sets--the same notion as used in another paper by the author that showed the decidability of stit theory with a single agent and Refref equivalence.

In recent years combinations of tense and modality have moved intothe focus of logical research. From a philosophical point of view, logical systems combining tense and modality are of interest because these logics have a wide field of application in original philosophical issues, for example in the theory of causation, of action, etc. But until now only methods yielding completeness results for propositional languages have been developed. In view of philosophical applications, analogous results with respect to languages of predicate logic (...) are desirable, and in this paper I present two such results. The main developments in this area can be split into two directions, differing in the question whether the ordering of time is world-independent or not. Semantically, this difference appears in the discussion whether T x W-frames or Kamp-frames (resp. Ockham-frames) provide a suitable semantics for combinations of tense and modality. Here, two calculi are presented, the first adequate with respect to Kamp-semantics, the second to T x Wsemantics. (Both calculi contain an appropriate version of Gabbay's irreflexivity rule.) Furthermore, the proposed constructions of canonical frames simplify some of those which have hitherto been discussed. (shrink)

We give a sound and complete axiomatization for the full computation tree logic, CTL*, of R-generable models. This solves a long standing open problem in branching time temporal logic.

We investigate an enrichment of the propositional modal language L with a "universal" modality ■ having semantics x ⊧ ■φ iff ∀y(y ⊧ φ), and a countable set of "names" - a special kind of propositional variables ranging over singleton sets of worlds. The obtained language ℒ $_{c}$ proves to have a great expressive power. It is equivalent with respect to modal definability to another enrichment ℒ(⍯) of ℒ, where ⍯ is an additional modality with the semantics x ⊧ ⍯φ (...) iff Vy(y ≠ x → y ⊧ φ). Model-theoretic characterizations of modal definability in these languages are obtained. Further we consider deductive systems in ℒ $_{c}$ . Strong completeness of the normal ℒ $_{c}$ logics is proved with respect to models in which all worlds are named. Every ℒ $_{c}$ -logic axiomatized by formulae containing only names (but not propositional variables) is proved to be strongly frame-complete. Problems concerning transfer of properties ([in]completeness, filtration, finite model property etc.) from ℒ to ℒ $_{c}$ are discussed. Finally, further perspectives for names in multimodal environment are briefly sketched. (shrink)

We consider some modal languages with a modal operator $D$ whose semantics is based on the relation of inequality. Basic logical properties such as definability, expressive power and completeness are studied. Also, some connections with a number of other recent proposals to extend the standard modal language are pointed at.

Recent theories of agency (sees to it that) of Nuel Belnap and Michael Perloff are examined, particularly in the context of an early proposal of the author.

The conditional,if an agent did something, then the agent could have done otherwise, is analyzed usingstit theory, which is a logic of seeing to it that based on agents making choices in the context of branching time. The truth of the conditional is found to be a subtle matter that depends on how it is interpreted (e.g., on what otherwise refers to, and on the difference between could and might) and also on whether or not there are busy choosers that (...) can make infinitely many choices in a finite span of time. (shrink)

The semantical structures called T×W frames were introduced in (Thomason, 1984) for the Ockhamist temporal-modal language, ℒO, which consists of the usual propositional language augmented with the Priorean operators P and F and with a possibility operator ⋄. However, these structures are also suitable for interpreting an extended language, ℒSO, containing a further possibility operator ⋄s which expresses synchronism among possibly incompatible histories and which can thus be thought of as a cross-history ‘simultaneity’ operator. In the present paper we provide (...) an infinite set of axioms in ℒSO, which is shown to be strongly complete forT ×W-validity. Von Kutschera (1997) contains a finite axiomatization of T×W-validity which however makes use of the Gabbay Irreflexivity Rule (Gabbay, 1981). In order to avoid using this rule, the proof presented here develops a new technique to deal with reflexive maximal consistent sets in Henkin-style constructions. (shrink)

Here is an important new theory of human action, a theory that assumes actions are founded on choices made by agents who face an open future.