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? in just the same way as he argues the principle of bivalence is. To grasp why, consider why truth tables work for classical logic: first, it must be the case that the variable parts of the proposition are either true or false: if they could be other values, or fail to have truth values at all, then the truth table analysis of logical connectives would not exhaust the possible ways these could be applied. For example intuitionistic logic respects the classical truth tables, but not the laws of classical logic, because intuitionistic logic allows propositions to be other than true