Analysis vs Deduction


V. I. Shalack


In the paper, we consider four types of problems that naturally arise in connection with the definition of a logical inference: 1) verifying the proof of: $\varGamma \langle A_{1},..,A_{n}\rangle A $;; 2) search for interesting consequences: $ \varGamma \langle ...\rangle ?$; 3) search for the proof: $ \varGamma \langle .?.\rangle A$; 4) search for hypotheses: $ ?\langle ...\rangle A $. Modern logic focuses on the problem of finding the proof of statements. G _odel’s restrictive theorems have a direct relation to it. At the same time in real practice, the task of search for hypotheses is much more common. The main part of this work is devoted to the investigation of this problem. A target proposition $A$ is given, and it is required to find the set of premises $\varGamma$ from which it is logically deducible. The choice of suitable premises $\varGamma$ occurs on the basis of the logical analysis of proposition $A$. We distinguish six different grounds for the selection of these premises: 1) acceptance of explicit definitions for predicate and functional symbols; 2) acceptance of axiomatic definitions for predicate and functional symbols; 3) acceptance of previously proved theorems; 4) acceptance of empirically true statements; 5) acceptance of statements describing the result of some action; 6) acceptance of plausible hypotheses that may be relevant to the problem being solved. In this paper we construct a calculus that formalizes the problem of an analytic search for the justification of a thesis. Two metatheoremes are proved, from which it follows that the constructed calculus really allows us to solve this type of problems.DOI: 10.21146/2074-1472-2018-24-1-26-45






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