First-Order Inquisitive Pair Logic Katsuhiko Sano Department of Humanistic Informatics, Graduate School of Letters Kyoto University / JSPS katsuhiko.sano@gmail.com Abstract. We introduce two different calculi for a first-order extension of inquis- itive pair semantics (Groenendijk 2008): Hilbert-style calculus and Tree-sequent calculus. These are first-order generalizations of (Mascarenhas 2009) and (Sano 2009), respectively. First, we show the strong completeness of our Hilbert-style calculus via canonical models. Second, we establish the completeness and sound- ness of our Tree-sequent calculus. As a corollary of the results, we semantically establish that our Tree-sequent calculus enjoys a cut-elimination theorem. 1 Introduction Groenendijk [1] first introduced the inquisitive pair semantics for a language of propo- sitional logic to capture both classical and inquisitive meanings of a sentence. For exam- ple, the classical meaning of p ∨ q is |p ∨ q| and the inquisitive meaning of it is {|p|, |q|}, where | A| is the set of all truth functions making A true. In the first logical study for inquisitive pair semantics [2], Mascarenhas revealed that the corresponding inquisitive pair logic is an axiomatic extension of intuitionistic logic (however, it is not closed under uniform substitutions) and established the completeness of it. Independently, fol- lowing the idea of [3], the author gave a complete and cut-free Gentzen-style sequent calculus for inquisitive pair logic [4]. After these studies, Ciardelli and Roelofsen [5] generalized inquisitive pair semantics within the propositional level and revealed that their generalized inquisitive logic has various beautiful logical properties. Disjunction ∨ allows us to formalize an English sentence containing ‘or’. However, in order to handle the sentences containing quantifications as well as ‘which’, ‘who’, etc., we need a first-order extension of inquisitive semantics. Ciardelli [6] studied how to give a recursive definition of inquisitive meaning in a first-order setting. As far as the author knows, however, there is no complete axiomatization of first-order inquisitive logic, though there was a preliminary study toward this direction [7, Ch.6]. This paper contributes to this point. In this paper, we focus on a first-order extension of the original inquisitive pair semantics and give two different complete calculi for a first-order in- quisitive pair logic: Hilbert-style calculus and Gentzen-style sequent calculus. We can regard these as first-order generalizations of [2] and [4], respectively. There are various ways of considering first-order extensions of intuitionistic logic for Kripke semantics: e.g. by expanding the domain or keeping it constant. Following [7, Ch.6], this paper also concerns the constant-domain semantics, which means that we adopt CD: ∀ x. (A ∨ B( x)) → (A ∨∀ x. B( x)) ( x is not free in A) as our logical axiom. In the first part of this paper, we establish the correspondence between the first-order M. Banerjee and A. Seth (Eds.): ICLA 2011, LNAI 6521, pp. 147–161, 2011. © Springer-Verlag Berlin Heidelberg 2011