Properties and Critique
A number of properties of IF logic follow from logical equivalence with and bring it closer to first-order logic including a compactness theorem, a Löwenheim–Skolem theorem, and a Craig interpolation theorem. (Väänänen, 2007, p. 86). However, Väänänen (2001) proved that the set of Gödel numbers of valid sentences of IF logic with at least one binary predicate symbol (set denoted by ValIF) is recursively isomorphic with the corresponding set of Gödel numbers of valid (full) second-order sentences in a vocabulary that contains one binary predicate symbol (set denoted by Val2). Furthermore Väänänen showed that Val2 is the complete Π2-definable set of integers, and that it is Val2 not in for any finite m and n. Väänänen (2007, pp. 136-139) summarizes the complexity results as follows:
| Problem | first-order logic | IF/depence/ESO logic |
|---|---|---|
| Decision | (r.e.) | |
| Non-validity | (co-r.e.) | |
| Consistency | ||
| Inconsistency |
Feferman (2006) cites Väänänen's 2001 result to argue (contra Hintikka) that while satisfiability might be a first-order matter, the question of whether there is a winning strategy for Verifier over all structures in general "lands us squarely in full second order logic" (emphasis Feferman's). Feferman also attacked the claimed usefulness of the extended IF logic, because the sentences in do not admit a game-theoretic interpretation.
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