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 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 ( ).
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.
Read more about Independence-friendly Logic: Semantics, Extended IF Logic, Properties and Critique
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