5.5 (00)



Every truth-function is a result of the successive application of the operation  (- - - - -V) ( xi ,....)  to elementary propositions.

This operation denies all the propositions in the right-hand bracket and I call it the negation of these propositions.

5.501    An expression in brackets whose terms are propositions I indicate - if the order of the terms in the bracket is indifferent - by a sign of the form "( xi-bar )". "( xi )" is a variable whose values are the terms of the expression in brackets, and the line over the variable indicates that it stands for all its values in the bracket.

(Thus if  xi  has the 3 values P, Q, R, then

( xi-bar ) = (P, Q, R) .)

The values of the variables must be determined.

The determination is the description of the propositions which the variable stands for.

How the description of the terms of the expression in brackets takes place is unessential.

We may distinguish 3 kinds of description: 1. Direct enumeration. In this case we can place simply its constant values instead of the variable. 2. Giving a function fx, whose values for all values of x are the propositions to be described. 3. Giving a formal law, according to which those propositions are constructed. In this case the terms of the expression in brackets are all the terms of a formal series.

5.502    Therefore I write instead of "(- - - - -V) ( xi ,....)" "N( xi-bar )".

N( xi-bar ) is the negation of all the values of the propositional variable  xi .

5.503    As it is obviously easy to express how propositions can be constructioned by means of this operation and how propositions are not to be constructed by means of it, this must be capable of exact expression.

5.51 - 5.57