Peano–Russell notation

Fundamental functions of propositions

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

Contradictory function

The contradictory function applied to a proposition returns its negation.

${\displaystyle \sim p}$

Logical sum

The logical sum applied to two propositions returns their disjunction.

${\displaystyle p\vee q}$

Logical product

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

${\displaystyle p\cdot q}$

Implicative function

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

${\displaystyle p\supset q}$

More complex functions of propositions

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

Assertion is same as the making of a statement between two full stops.

${\displaystyle ⊢p}$

An asserted proposition is either true or an error on the part of the writer.

Inference is equivalent to the rule modus ponens, where ${\displaystyle p\cdot p\supset q.\supset q}$

In addition to the logical product, dots are also used to show groupings of functions of propositions. In the above example, the dot before the final implication function symbol groups all of the previous functions on that line together as the antecedent to the final consequent.

The notation includes definitions as complex functions of propositions, using the equals sign "=" to separate the defined term from its symbolic definition, ending with the letters "Df".