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.
Resolution (logic) premises hold, then the conclusion formula_19 is true. Resolution rule can be generalized to first-order logic to: where formula_21 is a most general unifier of formula_22 and formula_23 and formula_24 and formula_25 have no common variables. The clauses formula_26 and formula_27 can apply this rule with formula_28 as unifier. Here x is a variable and b is a constant. Here we see that In first order logic, resolution condenses the traditional syllogisms of logical inference down to a single rule. To understand how resolution works, consider the following example syllogism of term logic: Or, more generally: To recast the reasoning