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.
Analytic philosophy sought to understand language (and hence philosophical problems) by using formal logic to formalize the way in which philosophical statements are made. Wittgenstein developed a comprehensive system of logical atomism in his "Tractatus Logico-Philosophicus" (, 1921). He thereby argued that the universe is the totality of actual states of affairs and that these states of affairs can be expressed by the language of first-order predicate logic. Thus a "picture" of the universe can be construed by means of expressing atomic facts in the form of atomic propositions, and linking them using logical operators. During the late 1920s, to 1940s, a