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.
Exportation (logic) Exportation (logic) Exportation is a valid rule of replacement in propositional logic. The rule allows conditional statements having conjunctive antecedents to be replaced by statements having conditional consequents and vice versa in logical proofs. It is the rule that: formula_1 Where "formula_2" is a metalogical symbol representing "can be replaced in a proof with." The "exportation" rule may be written in sequent notation: where formula_4 is a metalogical symbol meaning that formula_5 is a syntactic equivalent of formula_6 in some logical system; or in rule form: where the rule is that wherever an instance of "formula_9" appears on a line