IF logic is characterized by branching quantifiers.

Also see branching quantifiers and the plural quantifiers of George Boolos and others.

In recent decades, he has worked mainly on game semantics, and on independence-friendly logic, known for its "branching quantifiers" which he believes do better justice to our intuitions about quantifiers than does conventional first-order logic.

Conversely, any dependence logic sentence is equivalent to some sentence in the logic of branching quantifiers, since all existential second order sentences are expressible in branching quantifier logic.

Foundational considerations of game semantics have been more emphasised by Jaakko Hintikka and Gabriel Sandu, especially for Independence-friendly logic (IF logic, more recently Information-friendly logic), a logic with branching quantifiers.

Observing that is not closed under negation, Barwise also proposed a practical test to determine whether natural language sentences really involve branching quantifiers, namely to test whether their natural-language negation involves universal quantification over a set variable (a sentence).

Hintikka in a 1973 paper advanced the hypothesis that some sentences in natural languages are best understood in terms of branching quantifiers, for example: "some relative of each villager and some relative of each townsman hate each other" is supposed to be interpreted, according to Hintikka, as:

Dependence logic is a logic of imperfect information, like branching quantifier logic or independence-friendly logic: in other words, its game theoretic semantics can be obtained from that of first-order logic by restricting the availability of information to the players, thus allowing for non-linearly ordered patterns of dependence and independence between variables.