Analysis vs Deduction

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V. I. Shalack

Abstract

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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Статьи

References

Интервью Владимира Воеводского (часть 1). URL: https://sspr.livejournal.com/620950.html (дата обращения: 24.02.2018).
Интервью Владимира Воеводского (часть 2). URL: http://baaltii1.livejournal.com/200269.html (дата обращения: 24.02.2018).
Марков А.А. Элементы математической логики. М.: Изд-во Моск. ун-та, 1984. 80 с.
Метельский Н.В. Дидактика математики: Общая методика и ее проблемы: Учеб. пособие для вузов. Минск: Изд-во БГУ, 1982. 256 с.
Смирнов В.А. Творчество, открытие и логические методы поиска доказательства // Логико-философские труды В.А. Смирнова. М.: Эдиториал УРСС, 2001. С. 438–447.
Чёрч А. Введение в математическую логику. М.: ИЛ, 1960. 486 с.
Шалак В.И. Дедуктивно-аналитический подход к планированию целей // Логико-философские штудии. 2016. Т. 13. № 2. С. 134–135. URL: http://ojs.philosophy.spbu.ru/index.php/lphs/article/view/418/423 (дата обращения: 24.02.2018).
Шалак В.И. Аналитический подход к решению задач // Логические исследования. 2017. № 23(1). С. 121–139. URL: https://iphras.ru/uplfile/logic/log23_1/121 (дата обращения: 24.02.2018).
Шалак В.И. Виды дедуктивных задач и их решение // 10-е Смирновские чтения по логике: Материалы международной научной конференции. М., 2017. С. 121–123.
Cellucci C. Why Proof? What is a Proof? // Deduction, Computation, Experiment. Exploring the Effectiveness of Proof. Berlin: Springer-Verlag, 2008. P. 1-27. URL: https://www.academia.edu/184307/Why_Proof_What_is_a_Proof (дата обращения: 24.02.2018).
Cellucci C. Rethinking Logic. Logic in Relation to Mathematics, Evolution, and Method. Berlin: Springer, 2013. 390 p.
Cellucci C. Does logic slowly pass away, or has it a future? URL: https://www.academia.edu/4433563/Does_Logic_Slowly_Pass_Away_or_Has_It_a_Future (дата обращения: 24.02.2018).
Cellucci C. Is Mathematics Problem Solving or Theorem Proving? URL: https://www.academia.edu/16448950/Is_Mathematics_Problem_Solving_or_Theorem_Proving (дата обращения: 24.02.2018).
Cellucci C. Philosophy of Mathematics: Making a Fresh Start // Studies in History and Philosophy of Science. 2013. Vol. 44. P. 32–42. URL: https://www.academia.edu/3153062/Philosophy_of_Mathematics_Making_a_Fresh_Start (дата обращения: 24.02.2018).
Cellucci C. Why Should the Logic of Discovery Be Revived? A Reappraisal. URL: https://www.academia.edu/7077297/Why_Should_the_Logic_of_Discovery_Be_Revived_A_Reappraisal (дата обращения: 24.02.2018).
Fitting M.C. First-Order Logic and Automated Theorem Proving. Berlin: Springer-Verlag, 1990. 242 p. (дата обращения: 24.02.2018).
Fitting M. Intuitionistic Logic, Model Theory and Forcing. Amsterdam: NorthHolland Publishing Company, 1969. 192 p.