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.
Interpretation (logic) valid or tautologous. An "interpretation of a theory" is the relationship between a theory and some subject matter when there is a many-to-one correspondence between certain elementary statements of the theory, and certain statements related to the subject matter. If every elementary statement in the theory has a correspondent it is called a "full interpretation", otherwise it is called a "partial interpretation". The formal language for propositional logic consists of formulas built up from propositional symbols (also called sentential symbols, sentential variables, and propositional variables) and logical connectives. The only non-logical symbols in a formal language for propositional logic are