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.
Paraconsistent logic disjunction. As one can verify, LP preserves most other inference patterns that one would expect to be valid, such as De Morgan's laws and the usual introduction and elimination rules for negation, conjunction, and disjunction. Surprisingly, the logical truths (or tautologies) of LP are precisely those of classical propositional logic. (LP and classical logic differ only in the "inferences" they deem valid.) Relaxing the requirement that every formula be either true or false yields the weaker paraconsistent logic commonly known as first-degree entailment (FDE). Unlike LP, FDE contains no logical truths. It must be emphasized that LP is but one