By R. Parikh
Parikh R. (ed.) common sense colloquium. Symposium on good judgment, Boston, 1972-73 (LNM0453, Springer, 1975)(ISBN 3540071555)(1s)_Mln_
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Proof: are obtained from the a t o m i c formu- is called existential if it is of the form T B ~ (s•t> = -~ (S = U & t = v) ~ ~ M ~ ~[f] be the closed term ~ where s•t ~ of the typed ~- are introduced as a b b r e v i a t i o n s in the standard are d i s t i n c t v a r i a b l e s of type (~x((xs)t)) £ Specifically• The Y - q u a n t i f i e r s q u a n t i f y over a g i v e n type onlyo d e f i n e d in the obvious way. manner. s = t ~ for terms The formulae of ~4e Does the C o r o l l a r y to for the theory of functionals of finite type over a n o n e m p t y domain.
LEMMA Suppose by Lemma gl~g 2 for all v a r i a b l e s = @ , for all Proof: k = 0 . w(t(h2))X)g2(x) Set o n the Val(s~f) x3y . wsX)(gl) . of for 3 so t h a t k (S(hl))Xl(x)o Hence now U PV(t) ~ for a l l show that = g2(x) i0. Proof: Val(s~f) Obvious is w e l l defined~ from Lemma 9. and Val(s~f) Let The = Hence case x E FV(s) U FV(t) . [J U BV(s) We have h i = (gl)x~ = Val(t,f) , N FV(g2(y)) . z E FV(t) " . h 2 = (g2)xx . = ~- s ( g l ) ~ t ( g 2) . 2 LEMMA t . x I- ( X w ( s ( h l ) ) w ) g l ( x ) (x) [g(x)] = ~ w ~ FV(s) Let = (t(h2))g = = BV(t) n .
C. letters for formulae while we use Roman capitals (for nearly everything). 4. Transformation o f ~ -founded trees~ sharpening (*) in para. 3 by:(**) imposing (additional) requirements on mappings to R2-derivations (of A) when (*) is true. from Rl-derivations (of A) In the literature the distinc- tion between (*) and (**) is sometimes expressed by nonual form and normalization theorems (when cut free rules are called 'normal'). For finite Rl-derivations the distinction was subtle because the property of being an Rl-derivation is decidable end so on__eemapping~ 0~ is obtained trivially by simply running through all Q-derivations till one hits an R2-derivation with end formula A.