Page:Russell, Whitehead - Principia Mathematica, vol. I, 1910.djvu/118

96 in sensation, will be elementary. Any combination of given elementary propositions by means of negation, disjunction or conjunction (see below) will be elementary. In the primitive propositions of the present number, and therefore in the deductions from these primitive propositions in *2—*5, the letters $\scriptstyle{p}$, $\scriptstyle{q}$, $\scriptstyle{r}$, $$\scriptstyle{s}$$ will be used to denote elementary propositions.

(2) Elementary propositional functions. By an "elementary propositional function" we shall mean an expression containing an undetermined constituent, i.e. a variable, or several such constituents, and such that, when the undetermined constituent or constituents are determined, i.e. when values are assigned to the variable or variables, the resulting value of the expression in question is an elementary proposition. Thus if $$\scriptstyle{p}$$ is an undetermined elementary proposition, "not-$\scriptstyle{p}$" is an elementary propositional function.

We shall show in *9 how to extend the results of this and the following numbers (*1—*5) to propositions which are not elementary.

(3) Assertion. Any proposition may be either asserted or merely considered. If I say "Caesar died," I assert the proposition "Caesar died," if I say "'Caesar died' is a proposition," I make a different assertion, and "Caesar died" is no longer asserted, but merely considered. Similarly in a hypothetical proposition, e.g. "if $\scriptstyle{a=b}$, then $\scriptstyle{b=a}$," we have two unasserted propositions, namely "$\scriptstyle{a=b}$" and "$\scriptstyle{b=a}$," while what is asserted is that the first of these implies the second. In language, we indicate when a proposition is merely considered by "if so-and-so" or "that so-and-so" or merely by inverted commas. In symbols, if $$\scriptstyle{p}$$ is a proposition, $$\scriptstyle{p}$$ by itself will stand for the unasserted proposition, while the asserted proposition will be designated by The sign "$\scriptstyle{\vdash}$" is called the assertion-sign ; it may be read "it is true that" (although philosophically this is not exactly what it means). The dots after the assertion-sign indicate its range; that is to say, everything following is asserted until we reach either an equal number of dots preceding a sign of implication or the end of the sentence. Thus "$\scriptstyle{\vdash:p.\supset.q}$" means "it is true that $$\scriptstyle{p}$$ implies $\scriptstyle{q}$," whereas "$\scriptstyle{\vdash.p.\supset\vdash.q}$" means "$\scriptstyle{p}$ is true; therefore $$\scriptstyle{q}$$ is true ." The first of these does not necessarily involve the truth either of $$\scriptstyle{p}$$ or of $\scriptstyle{q}$, while the second involves the truth of both.

(4) Assertion of a propositional function. Besides the assertion of definite propositions, we need what we shall call "assertion of a propositional function." The general notion of asserting any propositional function is not used until *9, but we use at once the notion of asserting various special elementary propositional functions. Let $$\scriptstyle{\phi x}$$ be a propositional function whose argument is $\scriptstyle{x}$; then we may assert $$\scriptstyle{\phi x}$$ without assigning a value to $\scriptstyle{x}$.