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**1 - 3**of**3**### History of constructivism in the 20th century

"... In this survey of the history of constructivism, more space has been devoted to early developments (up till ca 1965) than to the work of the last few decades. Not only because most of the concepts and general insights have emerged before 1965, ..."

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In this survey of the history of constructivism, more space has been devoted to early developments (up till ca 1965) than to the work of the last few decades. Not only because most of the concepts and general insights have emerged before 1965,

### PROLOG EXTENSIONS BASED ON TABLEAU CALCULUS

"... The intention of this paper is to help bridging the gap between logic programming and theorem proving. It presents the design of a Gentzen type proof search procedure, based on classical tableau calculus, for knowledge bases consisting of arbitrary first order formulas. At each proof search step, wh ..."

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The intention of this paper is to help bridging the gap between logic programming and theorem proving. It presents the design of a Gentzen type proof search procedure, based on classical tableau calculus, for knowledge bases consisting of arbitrary first order formulas. At each proof search step, when a new formula is to be chosen from the knowledge base, the procedure chooses in such a way that the search space is small. When applied to a Horn clause knowledge base and an atomic goal, it performs the same proof search steps as any PROLOG interpreter would do. Hence, PROLOG can be viewed as a special Gentzen type procedure just as it is a special (namely, linear input) resolution procedure. 1.

### History of Constructivism in the 20th Century

"... notions, such as `constructive proof', `arbitrary number-theoretic function ' are rejected. Statements involving quantifiers are finitistically interpreted in terms of quantifier-free statements. Thus an existential statement 9xAx is regarded as a partial communication, to be supplemented ..."

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notions, such as `constructive proof', `arbitrary number-theoretic function ' are rejected. Statements involving quantifiers are finitistically interpreted in terms of quantifier-free statements. Thus an existential statement 9xAx is regarded as a partial communication, to be supplemented by providing an x which satisfies A. Establishing :8xAx finitistically means: providing a particular x such that Ax is false. In this century, T. Skolem 4 was the first to contribute substantially to finitist 4 Thoralf Skolem 1887--1963 History of constructivism in the 20th century 3 mathematics; he showed that a fair part of arithmetic could be developed in a calculus without bound variables, and with induction over quantifier-free expressions only. Introduction of functions by primitive recursion is freely allowed (Skolem 1923). Skolem does not present his results in a formal context, nor does he try to delimit precisely the extent of finitist reasoning. Since the idea of finitist reasoning ...