On the Expressive Power of IF-Logic with Classical Negation

Abstract

Abstract. It is well-known that Independence Friendly (IF) logic is equivalent to existential second-order logic (Σ 1 1 ) and, therefore, is not closed under classical negation. The boolean closure of IF sentences, called Extended IF-logic, on the other hand, corresponds to a proper fragment of ∆ 1 2 . In this paper we consider IF-logic extended with Hodges' flattening operator, which allows classical negation to occur also under the scope of IF quantifiers. We show that, nevertheless, the expressive power of this logic does not go beyond ∆ 1 2 . As part of the proof, we give a prenex normal form result and introduce a non-trivial syntactic fragment of full second-order logic that we show to be contained in ∆ 1 2 .

Independence friendly logic with classical negation via flattening is a second-order logic with weak dependencies

“…The latter can be seen as the closure by (classical) negation of the logic in which only one top-level Henkin quantifier can be used, which is known to be equivalent to IF. Some of the results contained in the present paper appeared in [10].…”

Classical Negation and Game-Theoretical Semantics

“…Our semantic games show that this claim does not categorically hold if relatively small modifications in game rules are allowed. Earlier a way of capturing classical negation using 3-player strategic games was found by Figueira, Gorín, and Grimson [6] (see Section 9.3 below). Hintikka could not object to our formulation of semantic games on the basis that we interpret + as acting on the strategy level.…”

“…[12] did not attempt to phrase the semantics of # game-theoretically, and neither did he comment on the expressive power of the language obtained by having # available in the IF-like logic he formulated. Both of these endeavors are undertaken by Figueira, Gorín, and Grimson [5], [6]. Write L.#/ for the set of formulas obtained from L IF -formulas by allowing arbitrary occurrences of # in the prefix.…”

“…The authors show that for L.#/-formulas, Hodges's compositional semantics and their game semantics coincide. In [6], the authors prove, utilizing their game-theoretically formulated semantics, that L.#/ Ä 1 2 . We may note that actually L 1 D L.#/: for every ' 2 L 1 there is ' 2 L.#/ such that M; ˆC II ' if and only if Mˆt ¹ º ' ; and for every 2 L.#/ there is ' 2 L 1 such that Mˆt ¹ º if and only if M; ˆC II ' .…”