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.
Multi-adjoint logic programming Multi-adjoint logic programming Multi-adjoint logic programming defines syntax and semantics of a logic programming program in such a way that the underliying maths justifying the results are a residuated lattice and/or MV-algebra. The definition of a multi-adjoint logic program is given, as usual in fuzzy logic programming, as a set of weighted rules and facts of a given formal language F. Notice that we are allowed to use different implications in our rules. Definition: A "multi-adjoint logic program" is a set P of rules of the form <("A" ←"i B"), δ> such that: 1. The "rule" (A ←i B) is