On the Expressive Power of IF-Logic with Classical Negation

It is well-known that Independence Friendly (IF) logic is equivalent to existential second-order logic (Sigma(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 Delta(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 Delta(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 Delta(1)(2).