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

It is well-known that Independence Friendly (IF) logic is equivalent to existential second-order logic (Σ1) 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 ∆2. In this article we consider SL(↓), IF-logic extended with Hodges’ flattening operator ↓, which allows to define a classical negation. Furthermore, this negation, in Hodges’ style, may occur also under the scope of IF quantifiers. SL(↓) contains Extended IF-logic and hence it is at least as expressive as the Boolean closure of Σ1. We prove that SL(↓) corresponds to a weak syntactic fragment of SO which we show to be strictly contained in ∆2. The separation is derived almost trivially from the fact that Σn defines its own truth-predicate. We finally show that SL(↓) is equivalent to the logic of Henkin quantifiers, which shows, we argue, that Hodges’ notion of negation is adequate.