Set Of Possible Worlds Research Articles

The Dynamification of Modal Dependence Logic

We examine the transitions between sets of possible worlds described by the compositional semantics of Modal Dependence Logic, and we use them as the basis for a dynamic version of this logic. We give a game theoretic semantics, a (compositional) transition semantics and a power game semantics for this new variant of modal Dependence Logic, and we prove their equivalence; and furthermore, we examine a few of the properties of this formalism and show that Modal Dependence Logic can be recovered from it by reasoning in terms of reachability. Then we show how we can generalize this approach to a very general formalism for reasoning about transformations between pointed Kripke models.

Algebraic foundations for the semantic treatment of inquisitive content

In classical logic, the proposition expressed by a sentence is construed as a set of possible worlds, capturing the informative content of the sentence. However, sentences in natural language are not only used to provide information, but also to request information. Thus, natural language semantics requires a logical framework whose notion of meaning does not only embody informative content, but also inquisitive content. This paper develops the algebraic foundations for such a framework. We argue that propositions, in order to embody both informative and inquisitive content in a satisfactory way, should be defined as non-empty, downward closed sets of possibilities, where each possibility in turn is a set of possible worlds. We define a natural entailment order over such propositions, capturing when one proposition is at least as informative and inquisitive as another, and we show that this entailment order gives rise to a complete Heyting algebra, with meet, join, and relative pseudo-complement operators. Just as in classical logic, these semantic operators are then associated with the logical constants in a first-order language. We explore the logical properties of the resulting system and discuss its significance for natural language semantics. We show that the system essentially coincides with the simplest and most well-understood existing implementation of inquisitive semantics, and that its treatment of disjunction and existentials also concurs with recent work in alternative semantics. Thus, our algebraic considerations do not lead to a wholly new treatment of the logical constants, but rather provide more solid foundations for some of the existing proposals.

Haecceitism, Chance, and Counterfactuals

Antihaecceitists believe that all facts about specific individuals—such as the fact that Fred exists, or that Katie is tall—globally supervene on purely qualitative facts. Haecceitists deny that. The issue is not only of interest in itself, but receives additional importance from its intimate connection to the question of whether all fundamental facts are qualitative or whether they include facts about which specific individuals there are and how qualitative properties and relations are distributed over them. Those who think that all fundamental facts are qualitative are arguably committed to antihaecceitism. The goal of this paper is to point out some problems for antihaecceitism (and therefore for the thesis that all fundamental facts are qualitative). The article focuses on two common assumptions about possible worlds: (i) Sets of possible worlds are the bearers of objective physical chance. (ii) Counterfactual conditionals can be defined by appeal to a relation of closeness between possible worlds. The essay tries to show that absurd consequences ensue if either of these assumptions is combined with antihaecceitism. Then it considers a natural response by the antihaecceitist, which is to deny that worlds play the role described in (i) and (ii). Instead, the reply continues, we can introduce a new set of entities that are defined in terms of worlds and that behave the way worlds do on the haecceitist position. That allows the antihaecceitist to formulate antihaecceitist friendly versions of (i) and (ii) by replacing the appeal to possible worlds with reference to the newly introduced entities. This maneuver invites an obvious reply, however. If the new entities are the things that play the role we typically associate with worlds, as partially described by (i) and (ii), then it is natural to conclude that they really are the entities we talk about when we speak of worlds, so that haecceitism is true after all.

Are propositions sets of possible worlds?

Are propositions sets of possible worlds?

Updating as Communication*

On many traditional theories of belief, your belief state is represented by an assignment of credences to propositions, or sets of possible worlds. If you are rational, your credence distribution will be a probability measure. Traditional theories of belief fit with a standard Bayesian theory of rational belief change: on learning a proposition, you must update your belief state by conditionalizing your credence distribution on the proposition you learn. That is, you must update by assigning 0 credence to those worlds incompatible with what you learn, and re-normalizing your credence distribution over the remaining worlds. Following Quine 1969, Lewis 1979 argues that we should instead represent your belief state by an assignment of credences to sets of centered worlds: world-timeindividual triples. For instance, if you have .5 credence that it is 3:00pm, your belief state should be represented by a measure that assigns .5 to the set of centered worlds with that time coordinate. Unlike traditional theories of belief, Lewis’s theory does not fit with a standard Bayesian theory of rational belief change. For instance, Bayesian conditionalization preserves certainties. If you update by conditionalizing on the set of centered worlds you learn, it follows that if you are ever certain that it is 3:00pm, you must always remain certain that it is 3:00pm. But clearly this is not what rationality requires. If we agree with Lewis about how to represent belief states, we must develop another set of principles governing rational belief change.

Structures and circumstances: two ways to fine-grain propositions

This paper discusses two distinct strategies that have been adopted to provide fine-grained propositions; that is, propositions individuated more finely than sets of possible worlds. One strategy takes propositions to have internal structure, while the other looks beyond possible worlds, and takes propositions to be sets of circumstances, where possible worlds do not exhaust the circumstances. The usual arguments for these positions turn on fineness-of-grain issues: just how finely should propositions be individuated? Here, I compare the two strategies with an eye to the fineness-of-grain question, arguing that when a wide enough range of data is considered, we can see that a circumstance-based approach, properly spelled out, outperforms a structure-based approach in answering the question. (Part of this argument involves spelling out what I take to be a reasonable circumstance-based approach.) An argument to the contrary, due to Soames, is also considered.

On fineness of grain

A central job for propositions is to be the objects of the attitudes. Propositions are the things we doubt, believe and suppose. Some philosophers have thought that propositions are sets of possible worlds. But many have become convinced that such an account individuates propositions too coarsely. This raises the question of how finely propositions should be individuated. An account of how finely propositions should be individuated on which they are individuated very finely is sketched. Objections to the effect that the account individuates propositions too finely are raised and responses to the objections are provided. It is also shown that theories that try to individuate propositions less finely have serious problems.

On Speaker Identifiability

In this paper, I investigate the notion of speaker identifiability, a term that is strongly associated with the pragmatic approach to specificity. Following Haspelmath 1997, I provide evidence from Russian for the linguistic relevance of speaker identifiability. In particular, I discuss two series of existential indefinites, koe - items and - to items, which are inherently specified as identifiable or not identifiable to the speaker. This specification is shown to be independent of such phenomena as the free-choice effect or narrow scope relative to another operator in the logical form of the sentence. I propose a formal analysis of speaker identifiability formulated within the framework of possible-world semantics. According to this account, an NP is speaker-identifiable if and only if it picks up the same individual in every possible world that is compatible with the speaker's worldview. Speaker identifiability is analyzed as a condition on the relative scope of an existential operator that ranges over individuals and a universal quantifier which quantifies over a set of possible worlds introduced by the context. I also argue that the speaker (non-)identifiability meaning component contributed by the investigated items constitutes a conventional implicature.

Existential rigidity and many modalities in order-sorted logic

Order-sorted logic is a useful tool for knowledge representation and reasoning because it enables representation of sorted terms and formulas along with partially ordered sorts (called sort-hierarchy). However, this logic cannot represent more complex sorted expressions when they are true in any possible world (as rigid) or some possible worlds (as modality) such as time, space, belief, or situation. In this study, we extend order-sorted logic by introducing existential rigidity and many modalities. In the extended logic, sorted modal formulas are interpreted over the Cartesian product of sets of possible worlds. We present a new labeled tableau calculus to check the (un)satisfiability and validity of sorted modal formulas.

Epistemic privacy

We present a novel definition of privacy in the framework of offline (retroactive) database query auditing. Given information about the database, a description of sensitive data, and assumptions about users' prior knowledge, our goal is to determine if answering a past user's query could have led to a privacy breach. According to our definition, an audited property A is private, given the disclosure of property B , if no user can gain confidence in A by learning B , subject to prior knowledge constraints. Privacy is not violated if the disclosure of B causes a loss of confidence in A . The new notion of privacy is formalized using the well-known semantics for reasoning about knowledge, where logical properties correspond to sets of possible worlds (databases) that satisfy these properties. Database users are modeled as either possibilistic agents whose knowledge is a set of possible worlds, or as probabilistic agents whose knowledge is a probability distribution on possible worlds. We analyze the new privacy notion, show its relationship with the conventional approach, and derive criteria that allow the auditor to test privacy efficiently in some important cases. In particular, we prove characterization theorems for the possibilistic case, and study in depth the probabilistic case under the assumption that all database records are considered a-priori independent by the user, as well as under more relaxed (or absent) prior-knowledge assumptions. In the probabilistic case we show that for certain families of distributions there is no efficient algorithm to test whether an audited property A is private given the disclosure of a property B , assuming P ≠ NP . Nevertheless, for many interesting families, such as the family of product distributions, we obtain algorithms that are efficient both in theory and in practice.

Double preference relations for generalised belief change

Many belief change formalisms employ plausibility orderings over the set of possible worlds to determine how the beliefs of an agent ought to be modified after the receipt of a new epistemic input. While most such possible world semantics rely on a single ordering, we investigate the use of an additional preference ordering—representing, for instance, the epistemic context the agent finds itself in—to guide the process of belief change. We show that the resultant formalism provides a unifying semantics for a wide variety of belief change operators. By varying the conditions placed on the second ordering, different families of known belief change operators can be captured, including AGM belief contraction and revision, Rott and Pagnucco's severe withdrawal, the systematic withdrawal of Meyer et al., as well as the linear liberation and σ-liberation operators of Booth et al. Our approach also identifies novel classes of belief change operators worthy of further investigation.

“Assertion” and intentionality

Robert Stalnaker argues that his causal-pragmatic account of the problem of intentionality commits him to a coarse-grained conception of the contents of mental states, where propositions are represented as sets of possible worlds. Stalnaker also accepts the “direct reference” theory of names, according to which co-referring names have the same content. Stalnaker’s view of content is thus threatened by Frege’s Puzzle. Stalnaker’s classic paper “Assertion” is intended to provide a response to this threat. In this paper, I evaluate Stalnaker’s claim that the causal-pragmatic account of intentionality commits one to a coarse-grained conception of the contents of mental states, and argue that the apparatus laid out in “Assertion” is not sufficiently comprehensive to account for all versions of Frege’s Puzzle.

Proposition-valued random variables as information

The notion of a proposition as a set of possible worlds or states occupies central stage in probability theory, semantics and epistemology, where it serves as the fundamental unit both of information and meaning. But this fact should not blind us to the existence of prospects with a different structure. In the paper I examine the use of random variables—in particular, proposition-valued random variables—in these fields and argue that we need a general account of rational attitude formation with respect to them.

Two Types of Belief Report

Ascriptions of belief and other doxastic propositional attitudes are commonly interpreted as quantifying over a set of possible worlds constituting doxastic alternatives for the belief experiencer. Katz (2000, 2003, 2008) has argued that belief predicates and other stative attitude predicates, along with stative predicates generally, lack a Davidsonian event argument and therefore do not report on any eventuality (event or state). Hacquard (2010), in contrast, assumes that all attitude ascriptions describe an event corresponding to the mental state of the attitude experiencer. The present investigation suggests that the strengths of doxastic predicates can be modeled by generalized quantifiers over the doxastic alternative set, permitting us to formulate and test predictions based on standard interactions of these quantifiers with negation when these ascriptions are negated. This provides a middle ground between Katz and Hacquard, whereby some belief ascriptions are interpreted as nothing more than a quantified condition over a doxastic alternative set, while others attribute a Davidsonian belief state to the experiencer. In the latter case, the condition involving quantification over doxastic alternatives is an essential content condition which serves to individuate the eventuality described by the belief report, and to identify it across possible worlds.

A Remark on Superintuitionistic Predicate Logics of Kripke Frames with Constant and with Nested Domains

The superintuitionistic predicate logics (without or with equality) of all predicate Kripke frames with nested domains over a fixed poset W (a set of possible worlds) are embeddable in the logic (without equality) of all Kripke frames with constant domains over W. Therefore, Takano's result [13] on finite axiomatizability of the logic of Kripke frames with constant domains over the set of real numbers implies the recursive axiomatizability of the corresponding logics with nested domains. Other consequences are mentioned as well.

Teaching and Learning Guide for: Recent Work on Propositions

Teaching and Learning Guide for: Recent Work on Propositions

Logic of change, change of logic

Understanding human behaviour, and indeed human beings more generally, requires an understanding of their attitudes: their beliefs, desires and preferences of course, but also a plethora of other attitudes or attitudinal factors, such as their presuppositions, their assumptions, their intentions, their attention, their awareness and their emotional states. But people’s attitudes change over time: to take the least controversial example, their beliefs change on learning new facts. It is not sufficient to understand the role human attitudes play at any particular moment; an understanding of how different attitudes change is also required. Formal models—be they from logic or mathematics—have historically played an important role in the modelling and study of some of these attitudes. Notable examples include the Bayesian model of (partial) beliefs as probability functions (de Finetti 1937; Ramsey 1931), the modal logician’s representation of belief and knowledge in terms of accessibility relations or partitions on sets of possible worlds (Hintikka 1962; Aumann 1976), typical models of preferences as order relations over sets of

Interpretation of De Finetti coherence criterion in Łukasiewicz Logic

De Finetti gave a natural definition of “coherent probability assessment” β : E → [ 0 , 1 ] of a set E = { X 1 , … , X m } of “events” occurring in an arbitrary set W ⊆ [ 0 , 1 ] E of “possible worlds”. In the particular case of yes–no events, (where W ⊆ { 0 , 1 } E ), Kolmogorov axioms can be derived from his criterion. While De Finetti’s approach to probability was logic-free, we construct a theory Θ in infinite-valued Łukasiewicz propositional logic, and show: (i) a possible world of W is a valuation satisfying Θ , (ii) β is coherent iff it is a convex combination of valuations satisfying Θ , (iii) iff β agrees on E with a state of the Lindenbaum MV-algebra of Θ , (iv) iff β ( X i ) = ∫ W x i d μ , i = 1 , … , m for some Borel probability measure μ on W . Thus Łukasiewicz semantics, MV-algebraic (finitely additive) states, and (countably additive) Borel probability measures provide a universal representation of coherent assessments of events occurring in any conceivable set of possible worlds.

IMPOSSIBLE WORLDS AND PROPOSITIONS: AGAINST THE PARITY THESIS

Accounts of propositions as sets of possible worlds have been criticized for conflating distinct impossible propositions. In response to this problem, some have proposed to introduce impossible worlds to represent distinct impossibilities, endorsing the thesis that impossible worlds must be of the same kind; this has been called the parity thesis. I show that this thesis faces problems, and propose a hybrid account which rejects it: possible worlds are taken as concrete Lewisian worlds, and impossibilities are represented as set-theoretic constructions out of them. This hybrid account (1) distinguishes many intuitively distinct impossible propositions; (2) identifies impossible propositions with extensional constructions; (3) avoids resorting to primitive modality, at least so far as Lewisian modal realism does.

An overview of the asymmetric bipolar representation of positive and negative information in possibility theory

The paper emphasizes the interest of distinguishing between negative and positive information. While pieces of knowledge rule out what is known to be impossible, pieces of evidence, or observations, point out situations that are actually possible, i.e., do exist. But what is supposedly not impossible may never be observed. This applies as well to the modelling of the preferences of an agent when some potential choices are rejected since unfeasible or not tolerable, while other options are really satisfactory if they are available, leaving room for alternatives to which the agent is indifferent. Accumulating negative information removes possible worlds and is basically conjunctive (as done in classical logic), while positive information augments the set of possible worlds and is disjunctive or even cumulative by nature. This second type of information has been largely neglected by the logical tradition. Both types of information may be pervaded with graded uncertainty when modelling knowledge, or may be a matter of degree when handling preferences. The two types of information can be accommodated in the framework of possibility theory. The bipolar nature of information can shed new light on the revision of knowledge or preference. It is also highly relevant when making inference, or for improving the expressivity of flexible queries.