This paper argues that questions have an important role to to play in logic, both semantically and proof-theoretically. Semantically, we show that by generalizing the classical notion of entailment to questions, we can capture not only the standard relation of logical consequence, which holds between pieces of information, but also the relation of logical dependency, which holds between information types. Proof-theoretically, we show that questions may be used in inferences as placeholders for arbitrary information of a given type; by manipulating (...) such placeholders, we may construct formal proofs of dependencies. Finally, we show that such proofs have a specific kind of constructive content: they do not just witness the existence of a certain dependency, but actually encode a method for transforming information of the types described by the assumptions into information of the type described by the conclusion. (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)

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).

We take a fresh look at the logics of informational dependence and independence of Hintikka and Sandu and Väänänen, and their compositional semantics due to Hodges. We show how Hodges’ semantics can be seen as a special case of a general construction, which provides a context for a useful completeness theorem with respect to a wider class of models. We shed some new light on each aspect of the logic. We show that the natural propositional logic carried by the semantics (...) is the logic of Bunched Implications due to Pym and O’Hearn, which combines intuitionistic and multiplicative connectives. This introduces several new connectives not previously considered in logics of informational dependence, but which we show play a very natural rôle, most notably intuitionistic implication. As regards the quantifiers, we show that their interpretation in the Hodges semantics is forced, in that they are the image under the general construction of the usual Tarski semantics; this implies that they are adjoints to substitution, and hence uniquely determined. As for the dependence predicate, we show that this is definable from a simpler predicate, of constancy or dependence on nothing. This makes essential use of the intuitionistic implication. The Armstrong axioms for functional dependence are then recovered as a standard set of axioms for intuitionistic implication. We also prove a full abstraction result in the style of Hodges, in which the intuitionistic implication plays a very natural rôle. (shrink)

Intuitionistic dependence logic was introduced by Abramsky and Väänänen [1] as a variant of dependence logic under a general construction of Hodges’ (trump) team semantics. It was proven that there is a translation from intuitionistic dependence logic sentences into second order logic sentences. In this paper, we prove that the other direction is also true, therefore intuitionistic dependence logic is equivalent to second order logic on the level of sentences.

This paper introduces and explores a conservative extension of inquisitive logic. In particular, weak negation is added to the standard propositional language of inquisitive semantics, and it is shown that, although we lose some general semantic properties of the original framework, such an enrichment enables us to model some previously inexpressible speech acts such as weak denial and ‘might’-assertions. As a result, a new modal logic emerges. For this logic, a Fitch-style system of natural deduction is formulated. The main result (...) of this paper is a theorem establishing the completeness of the system with respect to inquisitive semantics with weak negation. At the conclusion of the paper, the possibility of extending the framework to the level of first order logic is briefly discussed. (shrink)

This paper introduces a generalized version of inquisitive semantics, denoted as GIS, and concentrates especially on the role of disjunction in this general framework. Two alternative semantic conditions for disjunction are compared: the first one corresponds to the so-called tensor operator of dependence logic, and the second one is the standard condition for inquisitive disjunction. It is shown that GIS is intimately related to intuitionistic logic and its Kripke semantics. Using this framework, it is shown that the main results concerning (...) inquisitive semantics, especially the axiomatization of inquisitive logic, can be viewed as particular cases of more general phenomena. In this connection, a class of non-standard superintuitionistic logics is introduced and studied. These logics share many interesting features with inquisitive logic, which is the strongest logic of this class. (shrink)

We prove the following surprising property of Heyting's intuitionistic propositional calculus, IpC. Consider the collection of formulas, φ, built up from propositional variables (p,q,r,...) and falsity $(\perp)$ using conjunction $(\wedge)$ , disjunction (∨) and implication (→). Write $\vdash\phi$ to indicate that such a formula is intuitionistically valid. We show that for each variable p and formula φ there exists a formula Apφ (effectively computable from φ), containing only variables not equal to p which occur in φ, and such that for (...) all formulas ψ not involving $p, \vdash \psi \rightarrow A_p\phi$ if and only if $\vdash \psi \rightarrow \phi$ . Consequently quantification over propositional variables can be modelled in IpC, and there is an interpretation of the second order propositional calculus, IpC2, in IpC which restricts to the identity on first order propositions. An immediate corollary is the strengthening of the usual interpolation theorem for IpC to the statement that there are least and greatest interpolant formulas for any given pair of formulas. The result also has a number of interesting consequences for the algebraic counterpart of IpC, the theory of Heyting algebras. In particular we show that a model of IpC2 can be constructed whose algebra of truth-values is equal to any given Heyting algebra. (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)