Between Logic and Heuristic :: Calculus Mathematics Essays

Between Logic and Heuristic* ABSTRACT: This article aims to construct a new type of logical calculi-logical heuristic calculus which contains the means of reducing complete search. Such a heuristic component of calculus is reached with the help of meta-level means. The principal means for reducing search is structural information about information about contrary literals of formula. There are two major approaches to studying the process of reasoning ( «problem-solving »). On one hand, it is necessary to discover and investigate correct modes of reasoning in which the property of  «truth » is preserved. This task which can be formulated as the question  «what is a correct reasoning (proof)? » is considered in Logic. In order to decide this problem, Logic is based upon the concept of  «logical form ». There is a special syntactical method to deal with this concept—the method of construction of a logical calculus. In this respect, the calculus in question is a  «black box » which guarantees the  «true » conclusion under the  «true » premisses. Thus, Logic (logical form) gives the answer for the question about correct reasoning—  «the correct reasoning is a proof ». But logical syntax, as a  «black box »Ã¢â‚¬â€ calculus, isn’t interested in the real process of derivation building, in studying the question about methods of proof-search, in studying and construction a more manageable and efficient machin ery of  «truth » preserving. Availability of any method of exhaustive (complete) search, e.g.  «British museum algorithm », is quite enough for Logic (logical form). On the other hand, the process of  «problem-solving » can be investigated in the light of the following question:  «how is it possible to build a piece of correct reasoning? ». This task is considered in Heuristic. Heuristic investigates general principles and methods of  «problem-solving ». Computer Heuristic (computer heuristic method) is a system of rules (a rule) for essential reducing the complete search, i.e. heuristic methods are opposed to exhaustive search methods. The area of intersection of Logic and Computer Heuristic is proof-seach theory (PST), which investigates possible methods of  «problem-solving » ( «how is it possible to build a proof? ») in some calculus. PST deals with the heuristic component of proof systems. More precisely, the aim of proof-search theory can be defined as follows:  «discovering, on the basis of a calculus and an entity in the calculus in question of the structure of a possible derivation of this entity, a derivation which is interesting in some respect ».(1) Thus, proof-search theory can be included into the logical pragmatic (look at Fig.