New PDF release: Solvable Cases of the Decision Problem

By W., Ackerman

A massive exposition of the periods of statements for which the choice challenge is solvable.

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Simpler proofs have been given by Th. Skolem [l] and by H. Behmann [l]. The method used by all of them consists in transforming a wff into an equivalent one from which the predicate variables have been eliminated. The three authors could make use of the preparatory work of E. Schroder in the first and second volume of his “Algebra der Logik”. A representation of the method used by H. Behmann is to be found in D. Hilbert and P. Bernays [I], I, pp. 146ff). More recent modifications of the methods (see W.

Under these conditions (EG)(x)%,(x,Gxy) and ( X ) ( E H ) % ~H ( Xy ,) , as well as 39 ELIMINATION THEOREMS (G)(Ez)%,(x,Gzy) and (Ex)(H)'2lV(z, H y ) , are equivalent wfls. An analogous theorem holds for wfls %,(x, Gyx). By application of this theorem a bound dyadic predicate variable is changed to a monadic one. Similar theorems concern the replacement of a bound triadic predicate variable by a dyadic predicate variable, and so forth. If (EG)(z)%Jx,Gxy) is true by an assignment in a domain D, then there is a dyadic propositional function Y such that for any a , %,(a, P ( a , y)) is true by this assignment.

Yk) are true in D. ) a,, Y1, This completes our induction and the proof of our lemma. , Gk are all monadic ones. , Yk are defined. We divide D, into several classes. Two elements a and b of D, are said to belong to have the same values; the same class if for all i, Y&) and Y$(b) otherwise they belong to different classes. The number of classes of D,is at most 2k. We now construct a subdomain 0,of D, consisting of just one element of each class. To every element of D, we let correspond the element of 0,belonging to the same class.

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Solvable Cases of the Decision Problem by W., Ackerman


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