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By Anthony P. Morse

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C is of order n if and only if C has a subject in which precisely n binariates appear. 56 D E F I N I T I O N A L SCHEMA. We accept as a definition each expression which can be obtained by replacing ‘E’ by an expression of class 0, ‘A’ by an expression of class 1, ‘ V ’ by an expression of class 2, and ‘sup’ by an expression of class 3 in any one of the following expressions: ‘(Ex ;gx yx = Exbx A yx))’ ‘( A x ;EX yx = A x(0 E EX yx))’ ‘( v X ;EX YX = v X(0 €EX A IX))’ ‘(sup x vx = sup x ; (x = x) VX)’.

AGREEMENT. s is a subject of A if and only if s is such a verbless expression, whose terminal symbol is a binariate, that either A is s or A can be obtained from ‘xyz’by replacing ‘x’by s, ‘y’by a verbal nexus, and ‘z’by an expression. 54 Clearly an expression can have at most one subject. AGREEMENT. C is of order n if and only if C has a subject in which precisely n binariates appear. 56 D E F I N I T I O N A L SCHEMA. We accept as a definition each expression which can be obtained by replacing ‘E’ by an expression of class 0, ‘A’ by an expression of class 1, ‘ V ’ by an expression of class 2, and ‘sup’ by an expression of class 3 in any one of the following expressions: ‘(Ex ;gx yx = Exbx A yx))’ ‘( A x ;EX yx = A x(0 E EX yx))’ ‘( v X ;EX YX = v X(0 €EX A IX))’ ‘(sup x vx = sup x ; (x = x) VX)’.

We agree that: S is parenthetic if and only if S is ‘( )’ or S can be obtained from ‘(x)’ by replacing ‘x’ by an expression in which no parenthesis appears; S evolves T if and only if T can be obtained from S by replacing a symbol which is not a parenthesis by a parenthetic 6 0. Language and Inference expression; S is parenthetical if and only if S is a framed expression which can be built by successive evolvement from some expression in which no parenthesis appears. Thus if A and B are parenthetical expressisons and C is obtained from A by replacing a symbol which is not a parenthesis by B, then C is parenthetical.

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