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Download Essays on Mathematical and Philosophical Logic: Proceedings by G. Kreisel (auth.), Jaakko Hintikka, Ilkka Niiniluoto, Esa PDF

By G. Kreisel (auth.), Jaakko Hintikka, Ilkka Niiniluoto, Esa Saarinen (eds.)

The Fourth Scandinavian common sense Symposium and the 1st Soviet-Finnish common sense convention have been held in JyvaskyIa, Finland, June 29-July 6, 1976. The meetings have been equipped through a committee which consisted of the editors of the current quantity. The meetings have been supported financially through the Ministry of schooling of Finland, through the Academy of Finland, and by means of the department of common sense, method, and Philosophy of technology of the overseas Union of historical past of technological know-how. The Philosophical Society of Finland and the Jyvaskyla summer time competition gave priceless assist in a variety of practicalities. 35 papers by way of authors representing 10 international locations have been awarded on the conferences. of these papers 24 seem right here. THE EDITORS v desk OF CONTENTS PREFACE v half 1/ evidence concept GEORG KREISEL / a few proof from the speculation of Proofs and a few Fictions from basic evidence concept three DAG PRAWITZ / Proofs and the that means and Completeness of the Logical Constants 25 v. A. SMIRNOV / conception of Quantification and tff-calculi forty-one LARS SVENONIUS/Two forms of Extensions of Primitive Recursive mathematics forty nine DIRK VAN DALEN and R. STATMAN / Equality within the Presence of Apartness ninety five half II / INFINITARY LANGUAGES VEIKKO RANTALA / Game-Theoretical Semantics and Back-and- Forth 119 MAARET KAR TTUNEN / Infinitary Languages N oo~.

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Additional resources for Essays on Mathematical and Philosophical Logic: Proceedings of the Fourth Scandinavian Logic Symposium and of the First Soviet-Finnish Logic Conference, Jyväskylä, Finland, June 29–July 6, 1976

Sample text

10. II. 3rpVC(Tl \- C VC(T1 \- C == rp( C» == rp(C» Tl C T2 and Tl =f T2 VC(T2 \- C == rp(C» 3C(T2 \- C and T)CIT) T2 \- C T1lfC T2 \- rp( C) T \- rp(C) T s:: Tl Tl \- rp(C) Tl \- C assumption 3E ; 1 condition 2,3 3 5 5 4,6 T2 is a conservative extension of T; 8 condition 9,10 2,11 12. Contradiction. 13. Consequently, Tl is not an inessential extension of T. Now we shall use this lemma as a proof that HlffC is not an inessential extension of He. Let HlffC + be an extension of HlffC at the expense of an addition to the axiom schema 44 V.

Hintikka, I. Niiniluoto, and E. ), Essays on Mathematical and Philosophical Logic, 49-94. All Rights Reserved. Copyright © 1978 by D. Reidel Publishing Company, Dordrecht, Holland. •. , l k - 1 IS a numerical term (formula, function term). ) (iii) When X is a function term, and t a numerical term, then {x}(t) is a numerical term. (iv) If A and B are formulas, then so are -,A, (A V B), (A 1\ B), (A _ B), and (A ~ B). 3. l' 0 is freely generated from a set of 'initial constants ' (enumerated below) by the partially defined operations (i) - (vi) to be described.

Thus, instead of description fffxA(x, Yl> ... , Yn) it is possible to use a constant with indexes ayt , • • • , Yn' In this case different letters correspond to different descriptions. This method is developed in the book by Slupecki and Borkowski. This is another method of fixation of a description. Thus the system HfffC is equivalent to the system of natural deduction with the direct rule of elimination of the existential quantifier. Hence the system of natural deduction with the direct rule of elimination of the existential quantifier is not equivalent to the standard calculus of predicates, as is often stated.

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