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.
Multiple-conclusion logic the informal, non-mathematical sense) and favors truth over falsity (or assertion over denial). Multiple-conclusion logic A multiple-conclusion logic is one in which logical consequence is a relation, formula_1, between two sets of sentences (or propositions). formula_2 is typically interpreted as meaning that whenever each element of formula_3 is true, some element of formula_4 is true; and whenever each element of formula_4 is false, some element of formula_3 is false. This form of logic was developed in the 1970s by D. J. Shoesmith and Timothy Smiley but has not been widely adopted. Some logicians favor a multiple-conclusion consequence relation over the