We investigate extensions of dependence logic with generalized quantifiers. We also introduce and investigate the notion of a generalized atom. We define a system of semantics that can accommodate variants of dependence logic, possibly extended with generalized quantifiers and generalized atoms, under the same umbrella framework. The semantics is based on pairs of teams, or double teams. We also devise a game-theoretic semantics equivalent to the double team semantics. We make use of the double team semantics by defining a logic (...) $$\hbox {DC}^2$$ DC 2 which canonically fuses together two-variable dependence logic $$\hbox {D}^2$$ D 2 and two-variable logic with counting quantifiers $$\hbox {FOC}^2$$ FOC 2 . We establish that the satisfiability and finite satisfiability problems of $$\hbox {DC}^2$$ DC 2 are complete for $$\hbox {NEXPTIME}$$ NEXPTIME. (shrink)

A proposition is noncontingent, if it is necessarily true or it is necessarily false. In an epistemic context, ‘a proposition is noncontingent’ means that you know whether the proposition is true. In this paper, we study contingency logic with the noncontingency operator? but without the necessity operator 2. This logic is not a normal modal logic, because?→ is not valid. Contingency logic cannot define many usual frame properties, and its expressive power is weaker than that of basic modal logic over (...) classes of models without reflexivity. These features make axiomatizing contingency logics nontrivial, especially for the axiomatization over symmetric frames. In this paper, we axiomatize contingency logics over various frame classes using a novel method other than the methods provided in the literature, based on the ‘almost-definability’ schema AD proposed in our previous work. We also present extensions of contingency logic with dynamic operators. Finally, we compare our work to the related work in the fields of contingency logic and ignorance logic, where the two research communities have similar results but are apparently unaware of each other’s work. One goal of our paper is to bridge this gap. (shrink)

We describe a logic which is the same as first-order logic except that it allows control over the information that passes down from formulas to subformulas. For example the logic is adequate to express branching quantifiers. We describe a compositional semantics for this logic; in particular this gives a compositional meaning to formulas of the 'information-friendly' language of Hintikka and Sandu. For first-order formulas the semantics reduces to Tarski's semantics for first-order logic. We prove that two formulas have the same (...) interpretation in all structures if and only if replacing an occurrence of one by an occurrence of the other in a sentence never alters the truth-value of the sentence in any structure. (shrink)

The properties of the ${\forall^{1}}$ quantifier defined by Kontinen and Väänänen in [13] are studied, and its definition is generalized to that of a family of quantifiers ${\forall^{n}}$ . Furthermore, some epistemic operators δ n for Dependence Logic are also introduced, and the relationship between these ${\forall^{n}}$ quantifiers and the δ n operators are investigated.The Game Theoretic Semantics for Dependence Logic and the corresponding Ehrenfeucht- Fraissé game are then adapted to these new connectives.Finally, it is proved that the ${\forall^{1}}$ quantifier (...) is not uniformly definable in Dependence Logic, thus answering a question posed by Kontinen and Väänänen in the above mentioned paper. (shrink)

Modal dependence logic was introduced recently by Väänänen. It enhances the basic modal language by an operator = (). For propositional variables p 1, . . . , p n , = (p 1, . . . , p n-1, p n ) intuitively states that the value of p n is determined by those of p 1, . . . , p n-1. Sevenster (J. Logic and Computation, 2009) showed that satisfiability for modal dependence logic is complete for nondeterministic (...) exponential time.In this paper we consider fragments of modal dependence logic obtained by restricting the set of allowed propositional connectives. We show that satisfiability for poor man’s dependence logic, the language consisting of formulas built from literals and dependence atoms using ${\wedge, \square, \lozenge}$ (i. e., disallowing disjunction), remains NEXPTIME-complete. If we only allow monotone formulas (without negation, but with disjunction), the complexity drops to PSPACE-completeness.We also extend Väänänen’s language by allowing classical disjunction besides dependence disjunction and show that the satisfiability problem remains NEXPTIME-complete. If we then disallow both negation and dependence disjunction, satisfiability is complete for the second level of the polynomial hierarchy. Additionally we consider the restriction of modal dependence logic where the length of each single dependence atom is bounded by a number that is fixed for the whole logic. We show that the satisfiability problem for this bounded arity dependence logic is PSPACE-complete and that the complexity drops to the third level of the polynomial hierarchy if we then disallow disjunction.In this way we completely classify the computational complexity of the satisfiability problem for all restrictions of propositional and dependence operators considered by Väänänen and Sevenster. (shrink)

In 1890 Christian von Ehrenfels published his classic paper "Über 'Gestaltqualitäten'", the first systematic investigation of the philosophy and psychology of Gestalt. Ehrenfels thereby issued an important challenge to the psychological atomism that was still predominant in his day. His paper not only exerted a powerful influence on the philosophy of the Meinong school, it also marked the beginning of the Gestalt tradition in psychology, later associated with the work of Wertheimer, Köhler and Koffka in Berlin. Includes papers by C. (...) Von Ehrenfels, Kurt Grelling and Paul Oppenheim and contributions by K. Mulligan, P.M. Simons and Barry Smith. (shrink)

An analogy between functional dependencies and implicational formulas of sentential logic has been discussed in the literature. We feel that a somewhat different connexion between dependency theory and sentential logic is suggested by the similarity between Armstrong's axioms for functional dependencies and Tarski's defining conditions for consequence relations, and we pursue aspects of this other analogy here for their theoretical interest. The analogy suggests, for example, a different semantic interpretation of consequence relations: instead of thinking ofB as a consequence of (...) a set of formulas {A1,...,A n} whenB is true on every assignment of truth-values on which eachA i is true, we can think of this relation as obtaining when every pair of truth-value assignments which give the same truth-values toA 1, the same truth-values toA 2,..., and the same truth-values toA n, also make the same assignment in respect ofB. We describe the former as the consequence relation inference-determined by the class of truth-value assignments (valuations) under consideration, and the latter as the consequence relation supervenience-determined by that class of assignments. Some comparisons will be made between these two notions. (shrink)

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 paper investigates formal logics for reasoning about determinacy and independence. Propositional Dependence Logic D and Propositional Independence Logic I are recently developed logical systems, based on team semantics, that provide a framework for such reasoning tasks. We introduce two new logics L_D and L_I, based on Kripke semantics, and propose them as alternatives for D and I, respectively. We analyse the relative expressive powers of these four logics and discuss the way these systems relate to natural language. We argue (...) that L_D and L_I naturally resolve a range of interpretational problems that arise in D and I.We also obtain sound and complete axiomatizations for L_D and L_I . (shrink)

Information exchange can be seen as a dynamic process of raising and resolving issues. The goal of this paper is to provide a logical framework to model and reason about this process. We develop an inquisitive dynamic epistemic logic , which enriches the standard framework of dynamic epistemic logic , incorporating insights from recent work on inquisitive semantics. At a static level, IDEL does not only allow us to model the information available to a set of agents, like standard epistemic (...) logic, but also the issues that the agents entertain. At a dynamic level, IDEL does not only allow us to model the effects of communicative actions that provide new information, like standard DEL, but also the effects of actions that raise new issues. Thus, IDEL provides the fundamental tools needed to analyze information exchange as a dynamic process of raising and resolving issues. (shrink)

We consider the modal logic of non-contingency in a general setting, without making special assumptions about the accessibility relation. The basic logic in this setting is axiomatized, and some of its extensions are discussed, with special attention to the expressive weakness of the language whose sole modal primitive is non-contingency , by comparison with the usual language based on necessity.

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. (shrink)

The formula A (it is noncontingent whether A) is true at a point in a Kripke model just in case all points accessible to that point agree on the truth-value of A. We can think of -based modal logic as a special case of what we call the general modal logic of agreement, interpreted with the aid of models supporting a ternary relation, S, say, with OA (which we write instead of A to emphasize the generalization involved) true at a (...) point w just in case for all points x, y, with Swxy, x and y agree on the truth-value of A. The noncontingency interpretation is the special case in which Swxy if and only if Rwx and Rwy, where R is a traditional binary accessibility relation. Another application, related to work of Lewis and von Kutschera, allows us to think of OA as saying that A is entirely about a certain subject matter. (shrink)

This paper investigates a generalized version of inquisitive semantics. A complete axiomatization of the associated logic is established, the connection with intuitionistic logic and several intermediate logics is explored, and the generalized version of inquisitive semantics is argued to have certain advantages over the system that was originally proposed by Groenendijk (2009) and Mascarenhas (2009).

In the first part of the paper it is proved that there exists a one–one mapping between a minimal contingential logic extended with a suitable axiom for a propositional constant τ, named KΔτw, and a logic of necessity ${K\square \tau{w}}$ whose language contains ${\square}$ and τ. The form of the proposed translation aims at giving a solution to a problem which was left open in a preceding paper. It is then shown that the presence of τ in the language of (...) KΔτw allows for the definition, in terms of the non-contingency operator Δ, not only of ${\square}$ but of a second necessity operator O. It is observed that this fact opens the road to an investigation of the τ-free multimodal fragments of KΔτw. (shrink)

The paper introduces a contingential language extended with a propositional constant τ axiomatized in a system named KΔτ , which receives a semantical analysis via relational models. A definition of the necessity operator in terms of Δ and τ allows proving (i) that KΔτ is equivalent to a modal system named K□τ (ii) that both KΔτ and K□τ are tableau-decidable and complete with respect to the defined relational semantics (iii) that the modal τ -free fragment of KΔτ is exactly the (...) deontic system KD. In §4 it is proved that the modal τ -free fragment of a system KΔτw weaker than KΔτ is exactly the minimal normal system K. (shrink)

We introduce an atomic formula ${\vec{y} \bot_{\vec{x}}\vec{z}}$ intuitively saying that the variables ${\vec{y}}$ are independent from the variables ${\vec{z}}$ if the variables ${\vec{x}}$ are kept constant. We contrast this with dependence logic ${\mathcal{D}}$ based on the atomic formula = ${(\vec{x}, \vec{y})}$ , actually equivalent to ${\vec{y} \bot_{\vec{x}}\vec{y}}$ , saying that the variables ${\vec{y}}$ are totally determined by the variables ${\vec{x}}$ . We show that ${\vec{y} \bot_{\vec{x}}\vec{z}}$ gives rise to a natural logic capable of formalizing basic intuitions about independence and dependence. (...) We show that ${\vec{y} \bot_{\vec{x}}\vec{z}}$ can be used to give partially ordered quantifiers and IF-logic an alternative interpretation without some of the shortcomings related to so called signaling that interpretations using = ${(\vec{x}, \vec{y})}$ have. (shrink)

Dependence logic and its many variants are new logics that aim at establishing a unified logical theory of dependence and independence underlying seemingly unrelated subjects. The area of dependence logic has developed rapidly in the past few years. We will give a short introduction to dependence logic and review some of the recent developments in the area.

The idea of a property's being supervenient on a class of properties is familiar from much philosophical literature. We give this idea a linguistic turn by converting it into the idea of a predicate symbol's being supervenient on a set of predicate symbols relative to a (first order) theory. What this means is that according to the theory, any individuals differing in respect to whether the given predicate applies to them also differ in respect to the application of at least (...) one of the predicates in the set. The latter relationship we show turns out to coincide with something antecedently familiar from work on definability: with what is called the piecewise (or modelwise) definability, in the theory in question, of the given predicate in terms of those in the set. (shrink)

The sophisticated philosophical literature on supervenience stands in need of supplementation by a treatment of more fundamental questions about what features this notion possesses solely in virtue of the form of the definition it is standardly given. We provide a discussion of these features without getting involved in the merits of particular supervenience claims advanced and contested in that literature.