Methods to compute interesting consequences

In modern deductive analysis, the main tasks include the following: finding proof of a given statement using axioms and rules of inference and checking the correctness of a given consequence from certain premises. Little is currently known about inference problems with predetermined properties (problems with interesting consequences). There are no clear answers to the following questions: what properties are inherent in an interesting consequence and how to calculate an interesting consequence? To solve these problems, it is proposed to use the methods of n-tuple algebra (NTA) based on properties of the Cartesian product of sets. The objects of NTA are arbitrary n-ary relations, which can be interpreted as the formulas of mathematical logic. They are matrix-like structures which cells do not contain elements, but subsets of the corresponding attributes. In NTA, operations (addition, generalized intersection and generalized union) in the tuple algebra correspond to logical connectives of mathematical logic (negation, conjunction, disjunction), and the generalized inclusion relation corresponds to the derivability relation. The calculation of quantifier operations is performed using operations on attributes (adding a dummy attribute, which corresponds to the generalization rule in predicate calculus, and eliminating an attribute). For two of the four types of NTA structures, the elimination of attributes corresponds to computing the projection of a relation. To derive interesting consequences in NTA, a structure called the minimal consequence is proposed, which is equal to the generalized intersection of premises. Interesting consequences are calculated as projections of the minimal consequence. As a result of calculations and checks, consequences are obtained with a reduced or a given composition of variables, as well as with a reduced amount of notation.