Page:Russell, Whitehead - Principia Mathematica, vol. I, 1910.djvu/28

"systematic ambiguity," explained later in the explanations of the theory of types), the reader need only remember that all letters represent variables, unless they have been defined as constants in some previous place in the book. In general the structure of the context determines the scope of the variables contained in it; but the special indication of the nature of the variables employed, as here proposed, saves considerable labour of thought.

The fundamental functions of propositions. An aggregation of propositions, considered as wholes not necessarily unambiguously determined, into a single proposition more complex than its constituents, is a function with propositions as arguments. The general idea of such an aggregation of propositions, or of variables representing propositions, will not be employed in this work. But there are four special cases which are of fundamental importance, since all the aggregations of subordinate propositions into one complex proposition which occur in the sequel are formed out of them step by step.

They are (1) the Contradictory Function, (2) the Logical Sum, or Disjunctive Function, (3) the Logical Product, or Conjunctive Function, (4) the Implicative Function. These functions in the sense in which they are required in this work are not all independent; and if two of them are taken as primitive undefined ideas, the other two can be defined in terms of them. It is to some extent—though not entirely—arbitrary as to which functions are taken as primitive. Simplicity of primitive ideas and symmetry of treatment seem to be gained by taking the first two functions as primitive ideas.

The Contradictory Function with argument $\scriptstyle {p}$, where $\scriptstyle {p}$ is any proposition, is the proposition which is the contradictory of $\scriptstyle {p}$, that is, the proposition asserting that $\scriptstyle {p}$ is not true. This is denoted by $\scriptstyle {\sim p}$. Thus $\scriptstyle {\sim p}$ is the contradictory function with $\scriptstyle {p}$ as argument and means the negation of the proposition $\scriptstyle {p}$. It will also be referred to as the proposition not-$\scriptstyle {p}$. Thus $\scriptstyle {\sim p}$ means not-$\scriptstyle {p}$, which means the negation of $\scriptstyle {p}$.

The Logical Sum is a propositional function with two arguments $\scriptstyle {p}$ and $\scriptstyle {q}$, and is the proposition asserting $\scriptstyle {p}$ or $\scriptstyle {q}$ disjunctively, that is, asserting that at least one of the two $\scriptstyle {p}$ and $\scriptstyle {q}$ is true. This is denoted by $\scriptstyle {p\lor q}$. Thus $\scriptstyle {p\lor q}$ is the logical sum with $\scriptstyle {p}$ and $\scriptstyle {q}$ as arguments. It is also called the logical sum of $\scriptstyle {p}$ and $\scriptstyle {q}$. Accordingly $\scriptstyle {p\lor q}$ means that at least $\scriptstyle {p}$ or $\scriptstyle {q}$ is true, not excluding the case in which both are true.

The Logical Product is a propositional function with two arguments $\scriptstyle {p}$ and $\scriptstyle {q}$, and is the proposition asserting $\scriptstyle {p}$ and $\scriptstyle {q}$ conjunctively, that is, asserting that both $\scriptstyle {p}$ and $\scriptstyle {q}$ are true. This is denoted by $\scriptstyle {p\cdot q}$, or—in order to make the dots act as brackets in a way to be explained immediately—by $\scriptstyle {p:q}$, or by $\scriptstyle {p:.q}$, or by $\scriptstyle {p::q}$. Thus $\scriptstyle {p\cdot q}$ is the logical product with