Infinite Logic and The Axiom of Determinacy
Many different versions of infinitary logic were proposed in the late 20th century. One reason that has been given for believing in the axiom of determinacy is that it can be written as follows (in a version of infinite logic):
OR
Note: Seq(S) is the set of all -sequences of S. The sentences here are infinitely long with a countably infinite list of quantifiers where the ellipses appear.
In an infinitary logic, this principle is therefore a natural generalization of the usual (de Morgan) rule for quantifiers that are true for finite formulas, such as OR 
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