This paper deals with structures ${\langle{\bf T}, I\rangle}$ in which T is a tree and I is a function assigning each moment a partition of the set of histories passing through it. The function I is called indistinguishability and generalizes the notion of undividedness. Belnap’s choices are particular indistinguishability functions. Structures ${\langle{\bf T}, I\rangle}$ provide a semantics for a language ${\mathcal{L}}$ with tense and modal operators. The first part of the paper investigates the set-theoretical properties of the set of indistinguishability (...) classes, which has a tree structure. The significant relations between this tree and T are established within a general theory of trees. The aim of second part is testing the expressive power of the language ${\mathcal{L}}$ . The natural environment for this kind of investigations is Belnap’s seeing to it that (stit). It will be proved that the hybrid extension of ${\mathcal{L}}$ (with a simultaneity operator) is suitable for expressing stit concepts in a purely temporal language. (shrink)

We propose two alternatives to Xu’s axiomatization of Chellas’s STIT. The first one simplifies its presentation, and also provides an alternative axiomatization of the deliberative STIT. The second one starts from the idea that the historic necessity operator can be defined as an abbreviation of operators of agency, and can thus be eliminated from the logic of Chellas’s STIT. The second axiomatization also allows us to establish that the problem of deciding the satisfiability of a STIT formula without temporal operators (...) is NP-complete in the single-agent case, and is NEXPTIME-complete in the multiagent case, both for the deliberative and Chellas’s STIT. (shrink)

In this paper I propose and motivate a logic of the interdefined concepts of making true and control, understood as intensional propositional operators to be indexed to an agent. While bearing a resemblance to earlier logics in the tradition, the motivations, semantics, and object language theory differ on crucial points. Applying this logic to widespread formal theories of agency, I use it as a framework to argue against the ubiquitous assumption that the strongest actions or options available to a given (...) agent must always be pairwise incompatible. The conclusion is that this assumption conflicts with failures of higher order control of agents over their degree or precision of control, failures exhibited by such imperfect agents as ourselves. I discuss models in this setting for understanding such imperfectly self-controlling agents. In an appendix, I prove several relevant results about the logic described, including soundness and completeness both for it and for certain natural extensions. (shrink)

We introduce a multimodal framework of deontic action logic which encodes the interaction between two fundamental procedures in normative reasoning: conceptual classification and deontic classification. The expressive power of the framework is noteworthy, since it combines insights from agency logic and dynamic logic, allowing for a representation of many kinds of normative conflicts. We provide a semantic characterization for three axiomatic systems of increasing strength, showing how our approach can be modularly extended in order to get different levels of analysis (...) of normative reasoning. Finally, we discuss ways in which the framework can be used to capture other formalisms proposed in the literature, as well as to model searching problems in Artificial Intelligence. (shrink)

The so-called theory of karma is one of the distinguishing aspects of Hinduism and other non-Hindu south-Asian traditions. At the same time that the theory can be seen as closely connected with the freedom of will and action that we humans supposedly have, it has many times been said to be determinist and fatalist. The purpose of this paper is to analyze in some deepness the relations that are between the theory of karma on one side and determinism, fatalism and (...) free-will on the other side. In order to do that, I shall use what has been described as the best formal approach we have to indeterminism: branching time theory. More specifically, I shall introduce a branching time semantic framework in which, among other things, statements such as “state of affairs e is a karmic effect of agent a”, “a wills it to be the case that e” and “e is inevitable” could be properly represented. (shrink)

Philosophical accounts of collective intentionality typically rely on members to form a personal intention of sorts, viewed as a mental state. This tendency is opposed by recent economic literature on team-directed reasoning, which focuses on the reasoning process leading up to the formation of the members’ intentions. Our formal analysis bridges these paradigms and criticizes the team- directed reasoning account on two counts: first, team-directed reasoning is supposed to transcend traditional game and decision theory by adopting a certain collectivistic reasoning (...) method. However, we show that team-directed reasoning yields the same action recommendations as a certain I-mode we-intention type. Accordingly, an important part of we-mode reasoning can be reduced to I-mode reasoning with certain preferences. Second, contrary to the claims of team-directed-reasoning theorists, we refute that team-directed reasoning surpasses pro-group intentions in selecting cooperatively rational solutions. That is, in some scenarios team-directed reasoning fails to guarantee successful cooperation whereas pro-group intentions succeed in doing so. We therefore propose to revise team-directed reasoning and introduce a third we-intention type, called participatory intentions. We prove that participatory intentions guarantee that a best group action is performed whenever either team-directed reasoning or pro-group intentions do. (shrink)

In 1926, Ernst Mally, an Austrian logician, has introduced a system of deontic logic in which he has proposed three fundamental distinctions which proved to be important in the context of the further development of the logic of norms. It is argued that in his philosophical considerations Mally has introduced a number of important distinctions concerning the very concept of norm, but by getting them confused in introducing the subsequent formalisms he failed to formally preserve them. In some of his (...) philosophically made distinctions Mally apparently foresaw contemporary trends in logic of norms. To some extent this particular feature of Mally’s system open wide opportunities to reconstruct –– with the corresponding renovations — his illformed Deontik into many nowadays known systems of logic of norms and thus provides a fertile ground for this kind of research. (shrink)

This article presents modal versions of resource-conscious logics. We concentrate on extensions of variants of linear logic with one minimal non-normal modality. In earlier work, where we investigated agency in multi-agent systems, we have shown that the results scale up to logics with multiple non-minimal modalities. Here, we start with the language of propositional intuitionistic linear logic without the additive disjunction, to which we add a modality. We provide an interpretation of this language on a class of Kripke resource models (...) extended with a neighbourhood function: modal Kripke resource models. We propose a Hilbert-style axiomatisation and a Gentzen-style sequent calculus. We show that the proof theories are sound and complete with respect to the class of modal Kripke resource models. We show that the sequent calculus admits cut elimination and that proof-search is in PSPACE. We then show how to extend the results when non-commutative connectives are added to the language. Finally, we put the l.. (shrink)

In this paper we propose a method for modeling social influence within the STIT approach to action. Our proposal consists in extending the STIT language with special operators that allow us to represent the consequences of an agent’s choices over the rational choices of another agent.

We provide a Kripke semantics for a STIT logic with the "next" operator. As the atemporal group STIT is undecidable and unaxiomatizable, we are interested in strict fragments of atemporal group STIT. First we prove that the satisfiability problem of a formula of the fragment made up of individual coalitions plus the grand coalition is also NEXPTIME-complete. We then generalize this result to a fragment where coalitions are in a given lattice. We also prove that if we restrict the language (...) to nested coalitions the satisfiability problem is NP-complete if the number of agents is fixed and PSPACE-complete if the number of agents is variable. Finally we embed individual STIT with the "next" operator into a fragment of atemporal group STIT. (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)

In this paper we prove the uncompactness of every stit logic that contains a generalized refref conditional and is a sublogic of the stit logic with refref equivalence, a syntactical condition of uncompactness that covers infinitely many stit logics. This result is established through the uncompactness of every stit logic whose semantic structures contain no chain of busy choice sequences with cardinality , where is any natural number . The basic idea in the proof is to apply the notion of (...) companions to stit sentences in finding busy choice sequences in structures, and to make use of a relation between chains of busy choice sequences and generalized refref conditionals in connecting the two conditions of uncompactness mentioned above. (shrink)

“Philosophy of action” is a recognized specialty in contemporary philosophy, and the literature on action is fairly extensive: see, for instance, (Care & Landesman 1968; Goldman 1970; Hornsby 1980). The relation of actions to their effects is formulated most clearly in the more specialized literature on the logic of action; see (Belnap & Perloff 1988; Chellas 1992; Czelakowski 1996; Segerberg 1982).

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.

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.

I explore some issues in the logics and dialectics of practical modalities connected with the Consequence Argument (CA) considered as the best argument for the incompatibility of free will and determinism. According to Lewis (1981) in one of the possible senses of (in)ability, the argument is not valid; however, understood in the other of its possible senses, the argument is not sound. This verdict is based on the assessment of the modal version of the argument, where the crucial notion is (...) power necessity (“no choice” operator), while Lewis analyses the version where the central notion is the locution “cannot render false.”Lewis accepts closure of the relevant (in)ability operator under entailment but not closure under implication. His strategy has a seemingly strange corollary: a free predetermined agent is able (in a strong, causal sense) to falsity the conjunction of history and law. I compare a Moorean position with respect to radical skepticism and knowledge closure with ability closure and propose to explain Lewis’s strategy in the framework of his Moorean stance. (shrink)

We propose a new formalization of legal competences, and in particular for the Hohfeldian categories of power and immunity, through a deontic reinterpretation of dynamic epistemic logic. We argue that this logic explicitly captures the norm-changing character of legal competences while providing a sophisticated reduction of the latter to static normative positions. The logic is completely axiomatizable, and we apply it to a concrete case in German contract law to illustrate that it can capture the distinction between legal ability and (...) legal permissibility. (shrink)

We review the stit semantic framework and the main stit operators that have been proposed by Belnap, Perloff and Horty , a theory that has recently attracted the attention of the multiagent community. We discuss the problem of how to model the notion of collective agency, both in the sense of a joint action of a group of agents and as the agency of collective entities, like organisations. We show how we can define the direct and immediate effects of agents’ (...) actions, both in the achievement sense and in the deliberative sense, without any assumptions on the nature of time and considering the particular case of a discrete time. Finally, we also show how we can define a kind of dynamic logic operator that allows us to express what would obtain, if a direct “sees to it” action is performed, seeing it as a particular case of formulae of the form [a: φ]Ψ, that have the following informal meaning: “if agent a selects an action that leads to the truth of φ, after such action is performed, Ψ is the case”. (shrink)

I present a variant of with time, called, interpreted in standard Kripke semantics. On the syntactic level, is nothing but the extension of atemporal individual by: the future tense and past tense operators, and the operator of group agency for the grand coalition. A sound and complete axiomatisation for is given. Moreover, it is shown that supports reasoning about interesting normative concepts such as the concepts of achievement obligation and commitment.

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)

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)

Topic of the paper is Q-logic - a logic of agency in its temporal and modal context. Q-logic may be considered as a basal logic of agency since the most important stitoperators discussed in the literature can be defined or axiomatized easily within its semantical and syntactical framework. Its basic agent dependent operator, the Q-operator (also known as Δ- or cstit-operator), which has been discussed independently by E v. Kutschera and B. E Chellas, is investigated here in respect of its (...) relation to other temporal and modal operators. The main result of the paper, then, is a completeness result for a calculus of Q-logic with respect to a semantics defined on the tree-approach to agency as introduced and developed by, among others, E v. Kutschera and N. D. Belnap. (shrink)

This volume seeks to further the use of formal methods in clarifying one of the central problems of philosophy: that of our free human agency and its place in our indeterministic world. It celebrates the important contributions made in this area by Nuel Belnap, American logician and philosopher. Philosophically, indeterminism and free action can seem far apart, but in Belnap’s work, they are intimately linked. This book explores their philosophical interconnectedness through a selection of original research papers that build forth (...) on Belnap’s logical and philosophical work. Some contributions take the form of critical discussions of Belnap's published work, some develop points made in his publications in new directions, and others provide additional insights on the topics of indeterminism and free action. In Nuel Belnap’s work on indeterminism and free action, three formal frameworks figure prominently: the simple branching histories framework known as "branching time;" its relativistic spatio-temporal extension, branching space-times; and the “seeing to it that” logic of agency. As those frameworks provide the formal background for the contributed papers, the volume introduction gives an overview of the current state of their development. It also introduces case-intensional first order logic, a general intensional logic offering resources for a first-order extension of the mentioned frameworks and a recent research focus of Belnap’s. The volume also contains an extended biographical interview with Nuel Belnap. (shrink)

This paper has a twofold aim. The first is to offer a precise definition of soft fact. Without such definition it is impossible to assess the Ockhamist solution to the problem of divine foreknowledge and human freedom in an accurate way. The second purpose is to identify the costs of such a solution, distinguishing them from some of the other costs usually ascribed to Ockhamism, which Ockhamism does not actually need to pay. In particular, it is argued that Ockhamism is (...) committed to the view that a true future exists and to a form of backward causation. (shrink)

This chapter presents a new semantics for inductive empirical knowledge. The epistemic agent is represented concretely as a learner who processes new inputs through time and who forms new beliefs from those inputs by means of a concrete, computable learning program. The agent’s belief state is represented hyper-intensionally as a set of time-indexed sentences. Knowledge is interpreted as avoidance of error in the limit and as having converged to true belief from the present time onward. Familiar topics are re-examined within (...) the semantics, such as inductive skepticism, the logic of discovery, Duhem’s problem, the articulation of theories by auxiliary hypotheses, the role of serendipity in scientific knowledge, Fitch’s paradox, deductive closure of knowability, whether one can know inductively that one knows inductively, whether one can know inductively that one does not know inductively, and whether expert instruction can spread common inductive knowledge—as opposed to mere, true belief—through a community of gullible pupils. (shrink)

This paper has a twofold aim. The first is to offer a precise definition of soft fact. Without such definition it is impossible to assess the Ockhamist solution to the problem of divine foreknowledge and human freedom in an accurate way. The second purpose is to identify the costs of such a solution, distinguishing them from some of the other costs usually ascribed to Ockhamism, which Ockhamism does not actually need to pay. In particular, it is argued that Ockhamism is (...) committed to the view that a true future exists and to a form of backward causation. (shrink)

Stit theory (a logic of seeing-to-it-that) is applied to cases involving many agents. First treated are complex nestings of stits involving distinct agents. The discussion is driven by the logical impossibility of "a sees to it that b sees to it that Q" in the technical sense, even though that seems to make sense in everyday language, Of special utility are the concepts of "forced choice", of the creation of deontic states, and of probabilities, Second, joint agency, both plain and (...) strict (every participant is essential) is given a rigorous treatment. A central theorem is that strict joint agency is itself agentive. (shrink)

In Ockhamist branching-time logic [Prior 67], formulas are meant to be evaluated on a specified branch, or history, passing through the moment at hand. The linguistic counterpart of the manifoldness of future is a possibility operator which is read as `at some branch, or history (passing through the moment at hand)'. Both the bundled-trees semantics [Burgess 79] and the $\langle moment, history\rangle$ semantics [Thomason 84] for the possibility operator involve a quantification over sets of moments. The Ockhamist frames are (3-modal) (...) Kripke structures in which this second-order quantification is represented by a first-order quantification. The aim of the present paper is to investigate the notions of modal definability, validity, and axiomatizability concerning 3-modal frames which can be viewed as generalizations of Ockhamist frames. (shrink)

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.

O propósito desse artigo é fornecer um tratamento formal para um tipo de ação até o momento negligenciada nas lógicas modais filosóficas da ação: ação em plano. Ao fazer isso nós seguimos a abordagem padrão nas lógicas modais da ação exemplificados pelos trabalhos de Belnap, Chellas and Pörn. Como nós acreditamos que existe uma relação forte entre plano, tempo e indeterminismo, nós usamos a teoria do tempo ramificado para investigar as características básicas da ação em plano. Além de introduzir uma (...) semântica do tempo ramificado, nós também apresentamos um cálculo correto e completo com relação a essa semântica. DOI:10.5007/1808-1711.2010v14n3p421. (shrink)

Agency can be construed as both the manner in which autonomous individuals embark on particular courses of action (or inaction), and the relationship between such agents and the outcomes of the courses of action on which they embark. A promising strategy for understanding both senses of agency consists in the combination of a modal logic of agency and branching time semantics. Such is the strategy behind stit theory, the theory of agentive action developed by Nuel Belnap and others. However, stit (...) theoretic evaluations of the agentive relationship between agents and outcomes that are uncertain—due to either the presence of indeterminism, or the possible intervention of other agents—yield counterintuitive results. This paper develops a pair of alternative operators (the “act” operators) for modeling agency with respect to uncertain outcomes. Unlike the stit-theoretic model, the act-theoretic model of agency with respect to a particular state of affairs does not require that the state of affairs be realized in every possible history. If the state of affairs in fact obtains in the actual history, and its obtaining was dependent on the agent’s pursuing a particular course of action, then the agent is deemed agentive under act theory. (shrink)

This paper presents a short survey of recent developments in stit theories, with an emphasis on combinations of stit and deontic logic, and those of stit and epistemic logic.

This paper explores the analysis of ability, where ability is to be understood in the epistemic sense—in contrast to what might be called a causal sense. There are plenty of cases where an agent is able to perform an action that guarantees a given result even though she does not know which of her actions guarantees that result. Such an agent possesses the causal ability but lacks the epistemic ability. The standard analysis of such epistemic abilities relies on the notion (...) of action types—as opposed to action tokens—and then posits that an agent has the epistemic ability to do something if and only if there is an action type available to her that she knows guarantees it. We show that these action types are not needed: we present a formalism without action types that can simulate analyzes of epistemic ability that rely on action types. Our formalism is a standard epistemic extension of the theory of “seeing to it that”, which arose from a modal tradition in the logic of action. (shrink)

Hamblin’s Action-State Semantics provides a sound philosophical foundation for understanding the character of the imperative. Taking this as our inspiration, in this paper we present a logic of action, which we call ST, that captures the clear ontological distinction between being responsible for the achievement of a state of affairs and being responsible for the performance of an action. We argue that a relativised modal logic of type RT founded upon a ternary relation over possible worlds integrated with a basic (...) tense logic captures intuitions of the Hamblinian model of imperatives. The logic implements a direct mapping of each of Hamblin’s key concepts: strategies, partial strategies and wholehearted satisfaction. (shrink)

We describe a new way in which theories about the deontic status of actions can be represented in terms of the standard two-sorted first-order extensional predicate calculus. Some of the resulting formal theories are easy to implement in Prolog; one prototype implementation—R. M. Lee's deontic expert shell DX—is briefly described.