Questions & Quantification: A study of first order inquisitive logic
Gianluca Grilletti
Abstract:
This dissertation focuses on the study of inquisitive first order logic, a logical formalism encompassing questions in the presence of quantification, developed in order to employ questions in formal inferences and study their logical properties. In particular, we focus on developing tools and techniques to study the expressive power of inquisitive first order logic and the properties of its entailment. The dissertation can be divided in four parts, each considering a different approach to study the logic.
In the first part, consisting of Chapters 4 and 5, we adapt a tool from the field of model theory to inquisitive first order logic: Ehrenfeucht-Fraïssé games. We show that the technique of Ehrenfeucht-Fraïssé games adapts to this context and allows to detect when two inquisitive models are indistinguishable by formulas of a given complexity. The game developed is quite flexible and can be modified to capture properties other than logical equivalence, as for example the submodel relation. Using the game, we achieve a characterization of the cardinality quantifiers definable in inquisitive first order logic, generalizing the result for classical logic to this more expressive setting.
The second part, consisting of Chapter 6, takes another step in the model-theoretic direction and presents several ways to manipulate and combine models of first order inquisitive logic. The theory developed allows us to prove that two hallmarks of constructive logics hold for inquisitive first order logic: the Disjunction and Existence properties. The proof we give is semantical in nature: we develop several constructions to combine and transform inquisitive models, and use them to prove the disjunction and existence properties. Some of these constructions are inspired by operations on intuitionistic Kripke-frames (e.g., disjoint union) while others are based on constructions typical of classical predicate logic (e.g., models of terms). This approach allows us to prove also more general results: we define several classes of theories for which the corresponding consequence relations have the disjunction and/or the existence property.
In the third part, consisting of Chapters 7 and 8, we shift our attention on the axiomatization problem. As of now it is not known whether first order inquisitive logic is axiomatizable. We tackle a restricted version of the axiomatization problem, that is, we axiomatize fragments and variations of the logic. Chapter 7 focuses on the classical antecedent fragment, which can be intuitively characterized as the fragment where questions are not allowed in the antecedent of a conditional. This fragment is particularly interesting since it contains—modulo logical equivalence—all formulas corresponding to natural language sentences. We prove that the natural deduction system proposed in [Ciardelli, 2016, Section 4.6], restricted to the classical antecedent fragment, provides a sound and strongly complete axiomatization. Chapter 8 focuses on the finite-width inquisitive logics and on the bounded-width fragment. Finite-width inquisitive logics were introduced by Sano [2011] as a hierarchy closely related to inquisitive first order logic. Sano axiomatized one of these logics and left open two questions: whether the other elements of the hierarchy are axiomatizable, and whether first order inquisitive logic is the limit of this hierarchy. We give a positive answer to the former and a negative answer to the latter. Chapter 8 also treats the bounded-width fragment, characterized by the following property: if a formula of the fragment is not supported by an information state, then there exists a finite subset of the state which still does not support the formula. This rather peculiar property allows to derive several interesting results on the fragment (e.g., validities in the fragment are recursively enumerable, the restricted entailment is compact), building on the completeness result for the finite-width inquisitive logics.
The fourth part, consisting of Chapter 9, is an exploratory work not yet developed for the first order case, but only for the propositional case: we present an algebraic and a topological semantics for inquisitive propositional logics. Generalizing these semantic accounts to the first order case could prove to be a precious tool to study first order inquisitive logic from new perspectives, for example using the methods employed by Rasiowa and Sikorski [1950] or Görnemann [1971]. On the algebraic side, we introduce a new semantics based on Heyting algebras by restricting the valuations of propositional atoms only over regular elements. From this we obtain an algebraic semantics for inquisitive logic by restricting the semantics to the class of inquisitive Heyting algebras. On the topological side, we apply a duality result developed by Bezhanishvili and Holliday [2020] to characterize inquisitive algebras in terms of their dual topological UV-spaces. This allows to define a topological semantics for inquisitive logic which, as far as the author knows, is the first attempt to study inquisitive logic from a topological perspective.