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.
Connexive logic Connexive logic Connexive logic names one class of alternative, or non-classical, logics designed to exclude the so-called paradoxes of material implication. (Other logical theories with the same agenda include relevance logic, also known as relevant logic.) The characteristic that separates connexive logic from other non-classical logics is its rejection of Aristotle's Thesis, i.e. the formula, as a logical truth. Aristotle's Thesis asserts that no statement follows from its own denial. Stronger connexive logics also accept Boethius' Thesis, which states that if a statement implies one thing, it does not imply its opposite. Connexive logic is arguably one of the oldest