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It is a central feature of the systematics of I and E rules of intuitionistic logic, that an I step followed by an E step whose major premiss is just the conclusion of the I step represents a 'detour' and can be omitted: the premisses of the I step already 'contain' the conclusion of the E step in a certain sense. In Prawitz (1965) this fact is called the 'inversion principle' and used as the basis of the normalization procedures developed there. For example, the sequence of inference steps + + + ß 0.

Mt: * is the Cartesian produet of the sets t E T} and lETMt The following definition of the matrix eonsequenee ]M: 2F ~ 2F is due to bos and Suszko (1959). , ~) is the set of all homomorphisms of the algebra! into M. The next theorem ean be found in Woj cieki (1973). (I. I) If a matrix 1M2 is an epimorphie image of a matrix 1M I then 1M I = 1M 2 • T. PRUCNAL 34 Let H be any implicational intermediate logic. Then any H-supersystem Y determines a binary relation ;y on F by x; z iff x Y z, z ~ ~ x E Y.

J ZU J 0, 1, 2 , ... and zn , (p. ) ~ «p. ) ~ po)' where j > i > 0, 1~ ~ J 1~ 2 2 u- 1 x 2 ~ (x 3 ~ ... ~ (xn ~ (x 3 ~ (x 4 ~ ... ~ (x n ~ po) ... ). 1) For every strongly compact algebra! E PIA and for every n ~ 3: if zn E i(0) then card(A) < n, where card(A) is the cardinal number of the set A. We say that an implicational intermediate logic H is tabular if and only if H = 1(0) for some finite non-degenerate algebra! E PIA. 38 T. 2) For every implicational intermediate logic H the following conditions are equivalent: (cl) H is tabular, , or f some> n _ 3• (c2) Z n € H Proof.