Get Logic, Methodology and Philosophy of Science, Volume 9: PDF

By Brian Skyrms, Dag Westerståhl, Dag Prawitz

ISBN-10: 0444893415

ISBN-13: 9780444893413

Those lawsuits hide quite a lot of diversified concerns within the box of common sense, method and philosophy of technology.

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Additional info for Logic, Methodology and Philosophy of Science, Volume 9: Proceedings of the Ninth International Congress of Logic, Methodology, and Philosophy of Science

Example text

Bo(b, ~, Wit~B2(g2(v~, b, c~, c-*), . . , Wit~Bt(gl(v~, b, ~ , ~ for functions g 2 , . . , g / which are -< win-primitive recursive in E~_I. The difficulty is that these functions take b as an argument, but b is not free in the endsequent so we can not just set fi = gi. ,b~) such that -~Bo(bo,... ,b~) and ( U ~ c ( ~ 1)o equals bo. , Wit~Ak (wk, - - ~ B a (~, Wit~2 (f2(C, ~ , ~ , . . , Wit~Bt(fe(C, c], 51 and f 2 , . . , gk are and since (3v)C is in E~-I. LOP(-< win, II~_l): Suppose the last inference is OZ ~ no, A I (OL, c~, A2, .

If 7 -~ c~+ czz then 3' < c~+ ca6 for some 5 -~/3 so A(3') holds by J(5). Since 3` was arbitrary, J(/~) follows. That completes the proof of (a). The proof of (b) consists of a partial formalization of the Main Lemma 10 in the theory IA0 + TI(< cJm+l, En-2). First an important lemma is necessary: LEMMA 12 Let m > 2 and n > 2. IAo + TI(-z, ~m+l, En-2) can En-detine precisely the -~ Wm-primitive recursive in En-I functions. Proof By the just established part (a) of Theorem 11, every Endefinable function of I A 0 + TI(-'

Ge must produce a witness for the corresponding B 1 , . . , Be. D. Lemma 10 and Theorems 7, 8 and 9 The above proof did not consider the case where the last inference of the proof is an induction inference: since induction is restricted to A0-formulas and the witness formula for a A0-formula is just the formula itself, that 52 case is completely trivial. However, IE~ is, by Proposition 2 a consequence of IA0 + TI(-< Wm, En-1) and it must, a priori, be possible to handle IE~ induction inferences by the witness function method as above.

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Logic, Methodology and Philosophy of Science, Volume 9: Proceedings of the Ninth International Congress of Logic, Methodology, and Philosophy of Science by Brian Skyrms, Dag Westerståhl, Dag Prawitz


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