Page:Russell, Whitehead - Principia Mathematica, vol. I, 1910.djvu/91

Chapter III.

Incomplete Symbols.

(1) Descriptions. By an "incomplete" symbol we mean a symbol which is not supposed to have any meaning in isolation, but is only defined in certain contexts. In ordinary mathematics, for example, $\scriptstyle {\frac {d}{dx}}$ and $\scriptstyle {\int _{a}^{b}}$ are incomplete symbols: something has to be supplied before we have anything significant. Such symbols have what may be called a "definition in use." Thus if we put

$\scriptstyle {\nabla ^{2}={\frac {\partial ^{2}}{\partial x^{2}}}+{\frac {\partial ^{2}}{\partial y^{2}}}+{\frac {\partial ^{2}}{\partial z^{2}}}\quad {\text{Df}}}$,

we define the use of $\scriptstyle {\nabla ^{2}}$, but $\scriptstyle {\nabla ^{2}}$ by itself remains without meaning. This distinguishes such symbols from what (in a generalized sense) we may call proper names: "Socrates," for example, stands for a certain man, and therefore has a meaning by itself, without the need of any context. If we supply a context, as in "Socrates is mortal," these words express a fact of which Socrates himself is a constituent: there is a certain object, namely Socrates, which does have the property of mortality, and this object is a constituent of the complex fact which we assert when we say "Socrates is mortal." But in other cases, this simple analysis fails us. Suppose we say: "The round square does not exist." It seems plain that this is a true proposition, yet we cannot regard it as denying the existence of a certain object called "the round square." For if there were such an object, it would exist: we cannot first assume that there is a certain object, and then proceed to deny that there is such an object. Whenever the grammatical subject of a proposition can be supposed not to exist without rendering the proposition meaningless, it is plain that the grammatical subject is not a proper name, i.e. not a name directly representing some object. Thus in all such cases, the proposition must be capable of being so analysed that what was the grammatical subject shall have disappeared. Thus when we say "the round square does not exist," we may, as a first attempt at such analysis, substitute "it is false that there is an object $\scriptstyle {x}$ which is both round and square." Generally, when "the so-and-so" is said not to exist, we have a proposition of the form^[1]

"$\scriptstyle {\sim \mathbf {E!} (}$℩$\scriptstyle {x)(\phi x)}$,"

i.e.

$\scriptstyle {\sim \{(\exists c):\phi x.\equiv _{x}.x=c\}}$,

1. Cf. pp. 31, 32.