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 equivalence Logical equivalence In logic, statements formula_1 and formula_2 are logically equivalent if they have the same logical content. That is, if they have the same truth value in every model (Mendelson 1979:56). The logical equivalence of formula_1 and formula_2 is sometimes expressed as formula_5, formula_6, or formula_7. However, these symbols are also used for material equivalence. Proper interpretation depends on the context. Logical equivalence is different from material equivalence, although the two concepts are closely related. Logical equivalences involving conditional statements： Logical equivalences involving biconditionals： The following statements are logically equivalent: Syntactically, (1) and (2) are derivable from each other