# relation between inquisitive logic and logic as games?

In the very intriguing thesis "Questions in Logic" Ivano A. Ciardelli shows how to build a semantics of questions that reduces to Truth Conditional logic for factual statements where ¬¬p = p, but has an intuitionistic flavor for questions

proofs involving questions have an interesting constructive interpretation, reminiscent of the proofs-as-programs interpretation of intuitionistic logic: namely, a proof of an entailment Φ ⊨ ψ may be seen as encoding a logical resolution function f : Φ ⤳ ψ.

Inquisitive Logic replaces the satisfaction relation ⊨ between a world and a sentence with a support relation between sets of worlds and statements. The only new operator introduced is ⩖ such that s ⊨ η⩖τ read as the set of worlds s supports η⩖τ iff η is true in all worlds s or τ is true in all worlds s. From this one can define relations of support between questions and facts as in

In what year did Galileo discover Jupiter's Moon? ⊨ Galileo discovered Jupiter's Moons ⊨ Did Galileo Discover anything?

The first question has Galileo's discovery as presupposition, the last question is resolved by any information that supports Galileo's discovery, and the first question determines that last one, namely any answer to the first will be an answer to the last.

The difference between questions and statements has to do with the upper bound of the information space of a proposition. Factual statements have one upper bound, questions must have two or more. Eg:

(Leave the EU ⩖ ¬ Leave the EU) ≡ Leave the EU?

The truth conditional version of the above replaces ⩖ with ∨ is the tautology A ∨ ¬A which is always true since any world can only be one or the other, whereas the question can only be supported by sets of possibilities that are either only one or the other. Any subset of those two sets would support one of the erotetic disjuncts, including the empty set which supports anything.

The work in Inquisitive Semantics follows on from Hintikka's work on putting questions at the center of logic with his Independence Friendly Logic, and the book develops that with an extension of dependency logic in terms of questions.

Anyway I was wondering (wondering is also defined in terms of Kripke modal logics later on in the thesis) how Independence Friendly Logic then integrates with a game theoretic view of logic proposed by Hintikka, and developed by many others such as John Van Benthem in "Logic In Games", Abramski in "Game Semantics" which he suggests could be used as a new foundation for Homotopy Type Theory in the 2015 paper "Games for Dependent Types". One interesting notion that games bring, as pointed out in Hintikka's "Socratic Epistemology" is the notion of strategy: Hintikka thinks of the rules of logic as the rules of a game of chess: one cannot really be said to know how to play chess just by learning the rules: one also needs to understand strategy, and this requires one to be able to ask questions.

So would Inquisitive Semantics form a basis for a logic of games? Where does strategy come back into play?

• If anyone has enough points it may be worth adding the tag "erotetics" which is the word for the logic of questions to the tag list :-) – Henry Story Jan 18 at 17:25