Independence-friendly logic

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Independence-friendly logic (IF logic), proposed by Jaakko Hintikka and Gabriel Sandu in 1989, aims at being a more natural and intuitive alternative to classical first-order logic (FOL). IF logic is characterized by branching quantifiers. It is more expressive than FOL because it allows one to express independence relations between quantified variables.

For example, the formula ∀a ∀b ∃c/b ∃d/a φ(a,b,c,d) ("x/y" should be read as "x independent of y") cannot be expressed in FOL. This is because c depends only on a and d depends only on b. First-order logic cannot express these independences by any linear reordering of the quantifiers. In part, IF logic was motivated by game semantics for games with imperfect information.

IF logic is translation equivalent with existential second-order logic (\Sigma^1_1) and also with Väänänen's dependence logic and with first-order logic extended with Henkin quantifiers. Although it shares a number of metalogical properties with first-order logic, there are some differences, including lack of closure under negation and higher complexity for deciding the validity of formulas. Extended IF logic addresses the closure problem, but it sacrifices game semantics in the process, and it properly belongs to higher fragment of second-order logic ( \Delta_2^1).

Hintikka's proposal that IF logic and its extended version be used as foundations of mathematics has been met with skepticism by other mathematicians, including Väänänen and Solomon Feferman.

Semantics[edit]

Since Tarskian semantics does not allow indeterminate truth values, it cannot be used for IF logic. Hintikka further argues that the standard semantics of FOL cannot accommodate IF logic because the principle of compositionality fails in the latter. Wilfrid Hodges (1997) gives a compositional semantics for it in part by having the truth clauses for IF formulas quantify over sets of assignments rather than just assignments (as the usual truth clauses do).

The game-theoretic semantics for FOL treats a FOL formula as a game of perfect information, whose players are Verifier and Falsifier. The same holds for the standard semantics of IF logic, except that the games are of imperfect information.

Independence relations between the quantified variables are modelled in the game tree as indistinguishability relations between game states with respect to a certain player. In other words, the players are not certain where they are in the tree (this ignorance simulates simultaneous play). The formula is evaluated as true if there Verifier has a winning strategy, false if Falsifier has a winning strategy, and indeterminate otherwise.

A winning strategy is informally defined as a strategy that is guaranteed to win the game, regardless of how the other players play. It can be given a completely rigorous, formal definition.

Extended IF logic[edit]

IF logic is not closed under classical negation. The boolean closure of IF logic is known as extended IF logic and it is equivalent to a proper fragment of \Delta_2^1 (Figueira et al. 2011). Hintikka (1996, p. 196) claimed that "virtually all of classical mathematics can in principle be done in extended IF first-order logic".

Properties and critique[edit]

A number of properties of IF logic follow from logical equivalence with \Sigma^1_1 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 \Sigma^m_n 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 \Sigma_1^0 (r.e.) \Pi_2
Non-validity \Pi_1^0 (co-r.e.) \Sigma_2
Consistency \Pi_1^0 \Pi_1^0
Inconsistency \Sigma_1^0 \Sigma_1^0

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 \Pi_1^1 do not admit a game-theoretic interpretation.

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