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 atomism Russell thought that a moral of the revolutionary work in mathematics was this: equally revolutionary work in philosophy could occur, if we only make logic the essence of philosophizing. This aspiration lies at the origin, and motivates and runs through, logical atomism. Russell referred to his atomistic doctrine as contrary to the tier "of the people who more or less follow Hegel" (PLA 178). The first principle of logical atomism is that the World contains "facts". The facts are complex structures consisting of objects ("particulars"). A fact may be that an object has a property or that it stands in