Peano–Russell notation

In mathematical logic, Peano–Russell notation was Bertrand Russell's application of Giuseppe Peano's logical notation to the logical notions of Frege and was used in the writing of Principia Mathematica in collaboration with Alfred North Whitehead:

"The notation adopted in the present work is based upon that of Peano, and the following explanations are to some extent modelled on those which he prefixes to his Formulario Mathematico." (Chapter I: Preliminary Explanations of Ideas and Notations, page 4)

In the notation, variables are ambiguous in denotation, preserve a recognizable identity appearing in various places in logical statements within a given context, and have a range of possible determination between any two variables which is the same or different. When the possible determination is the same for both variables, then one implies the other; otherwise, the possible determination of one given to the other produces a meaningless phrase. The alphabetic symbol set for variables includes the lower and upper case Roman letters as well as many from the Greek alphabet.

The four fundamental functions are the contradictory function, the logical sum, the logical product, and the implicative function.

The contradictory function applied to a proposition returns its negation.

The logical sum applied to two propositions returns their disjunction.

The logical product applied to two propositions returns the truth-value of both propositions being simultaneously true.

The implicative function applied to two ordered propositions returns the truth value of the first implying the second proposition.

Equivalence is written as ${\displaystyle p\equiv q}$, standing for ${\displaystyle p\supset q\cdot q\supset p}$.

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