By Scott Menard

The point of interest during this moment variation is back on logistic regression versions for person point facts, yet combination or grouped info also are thought of. The booklet contains distinctive discussions of goodness of healthy, indices of predictive potency, and standardized logistic regression coefficients, and examples utilizing SAS and SPSS are incorporated. extra distinctive attention of grouped in place of case-wise info through the e-book up to date dialogue of the houses and acceptable use of goodness of healthy measures, R-square analogues, and indices of predictive potency dialogue of the misuse of odds ratios to symbolize probability ratios, and of over-dispersion and under-dispersion for grouped information up to date insurance of unordered and ordered polytomous logistic regression models.

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PROOF. This is similar to the proof of Theorem 2* preceded by the proof of a result which corresponds to Lemma 10*. Suppose X E S, and that X is provable in L i, then TX . Z)i and is provable in Ltz by Theorem 6. Suppose now that X E S, and that TX . =:> ai is provable in Ltz. /TX'. /X' and therefore X is provable in L; Hence, for i = 1, 2, if X E Si then X is provable in L, if and only if TX . z. z is primitive recursive in those of Lj , L z can be obtained, using Theorem 6, by trivial modification of the corresponding part of the proof of Theorem 2*.

56 JAAKKO HINTIKKA 5. The existence of distributive normal forms The possibility of converting every formula F (of first-order logic without identity) with given parameters (P. l)-(P. 3) into the first distributive normal form with the same parameters (or with certain fixed larger ones) can be proved by induction on the depth d of F. A basis for induction is given us by the propositional normal forms (d = 0). In the general case F is a truth-function of formulae of the following two forms: (a) A;(al' ...

A RELA TIVIZA TION PROCEDURE FOR PROPOSITION AL CALCULI 29 where R; is , Pil ... Pih, R~ (25) are all the vafs which occur in R;, and P;j (1 ::;; j ::;; hJ and P; are obtained from Pij (1 ::;; j ::;; h;) and Pi by replacing A r by A r + I for each r, 1 ::;; r ::;; Si. The special restriction imposed on the form of the rules of K is automatically satisfied in view of the satisfaction of the AI' ... , As, corresponding conditions imposed on the form of the axiom schemes and rules of J. We can now prove the following results: (a) A formula is provable in K if and only if the corresponding formula scheme is provable in K and vice versa.