Questions in Logic
Ivano A. Ciardelli
Abstract:
This dissertation pursues two tightly interwoven goals: to bring out the relevance of questions for the field of logic, and to establish a solid theory of the logic of questions within a classical logical setting. These enterprises feed into each other: on the one hand, the development of our formal systems is motivated by our considerations concerning the role to be played by questions; on the other hand, it is via the development of concrete, workable logical systems that the potential of questions in logic is made clear and tangible.
We begin by showing that, if we move from the standard relation of truth at a possible world to a relation of support at an information state, we obtain a semantic foundation which allows us to interpret statements and questions in a uniform way. This move leads to a substantial generalization of the classical notion of entailment, which encompasses not only the standard relation of consequence, but also the relation of dependency—a relation that plays an important role in many contexts, from physics to computer science. We show that, once logic is extended to questions, dependency emerges as contextual question entailment.
In Chapter 2, our approach is made concrete in the simplest possible setting, that of propositional logic. We describe how classical propositional logic can be enriched with questions, discuss the features of this logic, and show how propositional dependencies may be captured as cases of question entailment.
In Chapter 3 a natural deduction system for our propositional logic is provided. We show that proofs involving questions in this system have an interesting computational interpretation, reminiscent of the proofs-as-programs interpretation of intuitionistic logic: namely, whenever a proof witnesses a certain dependency, it actually encodes a method for computing this dependency. Finally, we abstract away from the details of the given proof system, and focus on the role played by questions in inferences. Essentially, we find that a question can be used as a placeholder for a generic piece of information of a certain type, much like individual constants are sometimes used in first-order logical proofs as placeholders for generic individuals. By manipulating such placeholders it is possible to provide formal proofs of the validity of certain dependencies.
In Chapter 4 we take our approach to the more expressive setting of first-order logic. We describe a conservative extension of classical first-order logic with questions, and discuss how a broad range of interesting questions becomes expressible in this system, touching upon issues such as reference and identity. We identify two fragments of the language which jointly cover the most salient classes of first-order questions, and for each of them we provide a simple axiomatization.
In Chapter 5, we discuss in detail the similarities and differences between our approach to dependency and the one adopted in the framework of dependence logic, which has seen considerable development in recent years. We conclude that the fundamental difference lies in the fact that in dependence logic, dependency is construed as a relation between variables, while in this thesis, it is construed as a relation between questions—none other than the relation of entailment. We argue that viewing dependency as question entailment presents some important benefits: first, it allows us to recognize and handle a broader range of dependencies than those considered in dependence logic. Moreover, by uncovering dependency as a facet of entailment, the question-based approach does not only lead to a neat conceptual picture, but also allows us to handle this relation by means of familiar logical tools, both semantically and proof-theoretically.
We then turn to the setting of modal logic. In Chapter 6, we show how Kripke modalities can be generalized to the context of a logic which includes questions. As an application, this makes it possible to generalize the knowledge modality of epistemic logic to embed both statements and questions, allowing for a uniform analysis of sentences like “A knows that B is home”, “A knows whether B is home” and, in the first-order case, “A knows where B is”. Technically, our main result is a uniform axiomatization result for the inquisitive counterpart of any given canonical modal logic. Moreover, in this chapter we propose and defend a modal account of dependence statements, i.e., statements such as “whether Alice will come depends on whether she finishes her homework”. According to this account, a dependence statement can be formalized in our modal logic as a modal conditional among questions. Finally, we provide an axiomatization for the logic that arises from taking this conditional as our primitive modal operator.
In Chapter 7, we explore a richer view on modal operators, which is suggested by our enriched semantics. We replace Kripke models with inquisitive modal models, which equip each world not just with an information state—a set of successors—but with an inquisitive state, encoding both information and issues. As an application, we investigate inquisitive epistemic logic, an enrichment of epistemic logic in which we can reason about agents who do not only have certain information, but also entertain certain issues, both individually and as a group. We provide completeness results both for the basic inquisitive modal logic, and for a range of modal logics that result from imposing various interesting frame conditions. We show that, while inquisitive modalities are more expressive than standard Kripke modalities, they retain a very well-behaved mathematical theory, and they are characterized by simple logical features.
Finally, in Chapter 8 we dynamify inquisitive epistemic logic, by generalizing the standard account of public announcements in epistemic logic. The resulting logic allows us to reason about the way in which an inquisitive-epistemic situation evolves not only when new information is provided by announcing a statement, but also when a new issue is raised by asking a question. A remarkable feature of this logic is that we do not need two separate actions for announcing and asking: one action of public utterance suffices; it is the meaning of the sentence being uttered that determines whether the effect of the utterance is to provide new information, or to raise new issues. We establish a complete axiomatization of this dynamic logic by means of reduction rules that allow us to transform every formula of this logic into a formula of inquisitive epistemic logic.