By Morse Anthony P.

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1 DEFl N I T I O N S . (St tx UX v X ( 2 =X A UX)) (substitute z for x in ux E st zx ux) 24 0. Language and Inference DEFlNIT10 NAL SCHEMAS. 58 ‘ (St Zp U’XX‘ E vXvX’(Z = (4) A U‘XX’)) ’. 1 We accept as a definition each expression which can be obtained by replacing ‘p’ by a march whose terminal symbol is ‘x”’ in ‘ (St Zp U””’X’’ vXVX’vX”(2= (p) A U”#X’X”)) ’. etc. Thus among our theorems are : ‘ (St Z X fJ U’XfJ ‘ (St 2 X +tJ U’XtJ I VXvtJ(2 = ( X VXVfJ(Z I = (X tJ) A U’XtJ)) ’, +fJ) A U’XtJ))’.

0. Language and Inference 12 I t is then natural to expect that (EnE 4 n = 6 = E m E 4 m). Also it seems clear that ‘ E x E y x’ is a formula. Inasmuch as ‘ (Y AxY) ’ is a formula in which ‘y’ is free it seems inescapable that -+ ‘ ( x -+ A x x ) ’ is a formula, but not, we hope, a theorem. Thus, since ‘y’ is free in ‘ Z x E y x’ we feel compelled to admit that ‘Ex Ex x’ is also a formula. 28 were simplified in the way we momentarily have in mind, then we would be unable to interpret the formula ‘Ex E x x ’ .

If A is the expression ‘ ( X u X’ n X” u x “ ) ’ then ‘ n ’ is of prime importance in A, ‘ x ’ and ‘ x ” ’ are left in A, ‘x” and ‘ x ” ’ are right in A , and the complicate of A is ‘( ( x u x’) n (x” u x ” ) ) ’. However, if A is the expression ‘(x n X ’ u x” n X I ) ’ then ‘ u ’ is of prime importance in A and the complicate of A is ‘((x n x’) u (x” n x ” ) ) ’. Theory of Notation 21 If A is the expression '(X A< then 'A<< X' A < < X" A < X'" A < < X"")' ' is of prime importance in A and the complicate of A is A<< '((XA