A PARADOX IN ILLATIVE COMBINATORY LOGIC MW ... - Project Euclid

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H(HW+1 *) ϊ-H(Hn+llx ^ Gnx), so h T 2 *. v-H(Gn+1x). This completes the inductive proof of (1). Now let. (Y is the paradoxical combinator WS(BWB); see [3]), then.
467 Notre Dame Journal of Formal Logic Volume XI, Number 4, October 1970

A PARADOX IN ILLATIVE COMBINATORY LOGIC M. W. BUNDER

Curry, in [1] and [2], has shown the inconsistency of a system of illative combinatory logic containing the axiom: hH^X for all obs X, for k = 2 (and 1). ("HX" stands for "X is a proposition''.) He also stated that the inconsistency held for k > 2, this more general result is proved below. Assume the following: Al A2 A3 A4 A5 A7

HX, H8, H S l - X ^ . δ ^ S D X D g . D . X HX, Hg hX D . g D X . X, P X g H g . X hHX. !-H* + ι X/or fl/zj; X and k 5*0.

D

'3

// (-HX αwrf X t-Hg flzew h H ( P X f ) .

From Al, A2, A3 and A7 it follows (as is proved in [4]) that if T(Xi,.. .,XJ is any theorem of pure implicational intuitionistic propositional calculus for indeterminates Xi,..., Xw, then H Ϊ ! , HX 2 ,...,HX, h T ( X 1 , . . . , X j . This fact is used in several places below. Let

Ξ

G 0 [x].x ^ δ ,

and for n ^ 0 let Now H W+1 ^HH(G^) is proved by induction, thus: By A6 and A7 Hx hH(Gox). Received November 29, 1969

(1)

468

M. W. BUNDER

Now assume n+1

H x

hH(Gnx);

then by A7 H(H

W+1

n+ll

*) ϊ-H(H x

^ Gnx),

so 2

hT *

v-H(Gn+1x).

This completes the inductive proof of (1). Now let

(Y is the paradoxical combinator WS(BWB); see [3]), then I

Ξ

GkX.

But by (1) Hk+1X

hH(GkX),

so H*+1I i-HI, so by A5 and A4 for i ^ 1, f-H'X Thus also for i > 1,

hH(G, X).

Now for j ^ 1, X D G7 I h I D . H ; XD Gy.i-X, so by the propositional calculus, as above noted, I D GjX\-HjX -D. X-D Gy.iX Now for j ^ 1, I D G; XK\ΓD Gy-xX, and so for j ^ 1, I D

G/IHX D Gi-Y.

Now as X = GkX and I-HI, h I D GkX HHIDIID.IDE

as G0X=X^U. Now by the propositional calculus h l D , XΏ £ l : D . I D « ,

(2)

A PARADOX IN ILLATIVE COMBINATORY LOGIC

469

and thus using (2) f-HX3.Xz)«,

(3)

that is But also HH 2 X; so by A2 and A4 hH 2 X D GxX, that is hG 2 X. Similarly \-G3X,...

\-GkX

that is \-X; and by (3)

Now eliminating assumption A6, we have for any H, HH hfi. therefore H(H*H) \-H% and by A 5

Similarly

I-H*" 1 *,...

HH«, H«,

so hβ has been proved for any II. Of the assumptions used to derive this inconsistency, Al, A2 and A3 are ordinary propositional calculus results and A4 merely says that if X is true then it is a proposition. Thus it seems that we should reject either A5 or A7. If •HX.H8 ι-H(PXD)

(4)

is taken instead of A7, the paradox does not go througho However in some systems A7 is preferable to (4) and we have to reject A5.

470

M. W. BUNDER

REFERENCES [1] Curry, H. B., "Some advances in the combinatory theory of quantification,'' Proceedings of the National Academy of Sciences, USA, vol. 28 (1942), pp. 564569. [2] Curry, H. B., 4iThe combinatory foundations of mathematical logic," The Journal of Symbolic Logic, vol. 7 (1942), pp. 49-64. [3] Curry, H.B., andR.Feys, Combinatory Logic, North-Holland, Amsterdam (1958). [4] Bunder, M. W., Set Theory Based on Combinatory Logic, PhD Thesis, Amsterdam (1969).

University of New South Wales Wollongong, New South Wales, Australia