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.
Logical disjunction to but different from the logical "or". For example, "Please ring me or send an email" likely means "do one or the other, but not both". On the other hand, "Her grades are so good that either she's very bright or she studies hard" does not exclude the possibility of both. In other words, in ordinary language "or" (even if used with "either") can mean either the inclusive "or" [inclusive-]or the exclusive "or." Logical disjunction In logic and mathematics, or is the truth-functional operator of (inclusive) disjunction, also known as alternation; the "or" of a set of operands is true