Logic

Foundations of Logic and Linguistics: Problems and Their by Wolfgang Rautenberg (auth.), Georg Dorn, Professor Paul

By Wolfgang Rautenberg (auth.), Georg Dorn, Professor Paul Weingartner (eds.)

This quantity includes a variety of papers that have been contributed to the seventh overseas Congress of common sense, method and Philosophy of technology, which was once held in Salzburg from the eleventh - sixteenth July, 1983. there have been 14 sections during this congress: 1. facts idea and foundations of arithmetic 2. version conception and its applica ti on three. recursion concept and conception of computation four. axiomatic set concept five. philosophical common sense 6. basic technique of technology 7. foundations of likelihood and induction eight. foundations and philosophy of the actual sciences nine. foundati ons and phi 1 osophy of biology 10. foundations and philosophy of psychology foundations and philosophy eleven. of the social sciences 12. foundati ons and philosophy of linguistics thirteen. background of common sense, method and philosophy of technology 14. basic ideas of the ethics of technological know-how In each one part, 3 or 4 invited addresses got, to be able to be released within the Congress lawsuits (Ruth Barcan Marcus, Georg J. W. Dorn and Paul Weingartner, eds. : common sense, Metho­ dology and Philosophy of technology VII. complaints of the 7th foreign Congress of good judgment, method and Philosophy of v PREFACE technology, Salzburg, 1983. - Amsterdam, ny, Oxford: North-Holland Publishing 'Company, 1985. ) each part except for part 14 additionally contained contributed papers.

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It is a central feature of the systematics of I and E rules of intuitionistic logic, that an I step followed by an E step whose major premiss is just the conclusion of the I step represents a 'detour' and can be omitted: the premisses of the I step already 'contain' the conclusion of the E step in a certain sense. In Prawitz (1965) this fact is called the 'inversion principle' and used as the basis of the normalization procedures developed there. For example, the sequence of inference steps + + + ß 0.

Mt: * is the Cartesian produet of the sets t E T} and lETMt The following definition of the matrix eonsequenee ]M: 2F ~ 2F is due to bos and Suszko (1959). , ~) is the set of all homomorphisms of the algebra! into M. The next theorem ean be found in Woj cieki (1973). (I. I) If a matrix 1M2 is an epimorphie image of a matrix 1M I then 1M I = 1M 2 • T. PRUCNAL 34 Let H be any implicational intermediate logic. Then any H-supersystem Y determines a binary relation ;y on F by x; z iff x Y z, z ~ ~ x E Y.

J ZU J 0, 1, 2 , ... and zn , (p. ) ~ «p. ) ~ po)' where j > i > 0, 1~ ~ J 1~ 2 2 u- 1 x 2 ~ (x 3 ~ ... ~ (xn ~ (x 3 ~ (x 4 ~ ... ~ (x n ~ po) ... ). 1) For every strongly compact algebra! E PIA and for every n ~ 3: if zn E i(0) then card(A) < n, where card(A) is the cardinal number of the set A. We say that an implicational intermediate logic H is tabular if and only if H = 1(0) for some finite non-degenerate algebra! E PIA. 38 T. 2) For every implicational intermediate logic H the following conditions are equivalent: (cl) H is tabular, , or f some> n _ 3• (c2) Z n € H Proof.

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