Alternating (In)Dependence-Friendly Logic

Abstract

Hintikka and Sandu originally proposed Independence Friendly Logic (▪) as a first-order logic of imperfect information to describe game-theoretic phenomena underlying the semantics of natural language. The logic allows for expressing independence constraints among quantified variables, in a similar vein to Henkin quantifiers, and has a nice game-theoretic semantics in terms of imperfect information games. However, the ▪ semantics exhibits some limitations, at least from a purely logical perspective. It treats the players asymmetrically, considering only one of the two players as having imperfect information when evaluating truth, resp., falsity, of a sentence. In addition, truth and falsity of sentences coincide with the existence of a uniform winning strategy for one of the two players in the semantic imperfect information game. As a consequence, ▪ does admit undetermined sentences, which are neither true nor false, thus failing the law of excluded middle. These idiosyncrasies limit its expressive power to the existential fragment of Second Order Logic (▪). In this paper, we investigate an extension of ▪, called Alternating Dependence/Independence Friendly Logic (▪), tailored to overcome these limitations. To this end, we introduce a novel compositional semantics, generalising the one based on trumps proposed by Hodges for ▪. The new semantics (i) allows for meaningfully restricting both players at the same time, (ii) enjoys the property of game-theoretic determinacy, (iii) recovers the law of excluded middle for sentences, and (iv) grants ▪ the full descriptive power of ▪. We also provide an equivalent Herbrand-Skolem semantics and a game-theoretic semantics for the prenex fragment of ▪, the latter being defined in terms of a determined infinite-duration game that precisely captures the other two semantics on finite structures.