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.
Default logic is true. The logical formulae in and all formulae in a default were originally assumed to be first-order logic formulae, but they can potentially be formulae in an arbitrary formal logic. The case in which they are formulae in propositional logic is one of the most studied. The default rule “birds typically fly” is formalized by the following default: This rule means that, if is a bird, and it can be assumed that it flies, then we can conclude that it flies. A background theory containing some facts about birds is the following one: According to this default rule, a