Infinitary logic logical axiom, an element of T, or is deduced from previous statements using a rule of inference. As before, all rules of inference in finitary logic can be used, together with an additional one: The logical axiom schemata specific to infinitary logic are presented below. Global schemata variables: formula_3 and formula_18 such that formula_23. The last two axiom schemata require the axiom of choice because certain sets must be well orderable. The last axiom schema is strictly speaking unnecessary as Chang's distributivity laws imply it, however it is included as a natural way to allow natural weakenings to the logic.
Is Logic Empirical? idea of a propositional logic with rules radically different from Boolean logic in itself was not new. Indeed a sort of analogy had been established in the mid-nineteen thirties by Garrett Birkhoff and John von Neumann between a non-classical propositional logic and some aspects of the measurement process in quantum mechanics. Putnam and the physicist David Finkelstein proposed that there was more to this correspondence than a loose analogy: that in fact there was a logical system whose semantics was given by a lattice of projection operators on a Hilbert space. This, actually, was the correct logic for reasoning about