Page:Russell, Whitehead - Principia Mathematica, vol. I, 1910.djvu/33

If this is to be asserted, we must put four dots after the assertion-sign, thus:

"$\scriptstyle {\vdash ::p\lor q.\supset :.p.\lor .q\supset r:\supset .p\lor r}$."

(This proposition is proved in the body of the work; it is *2·75.) If we wish to assert (what is equivalent to the above) the proposition: "if either $\scriptstyle {p}$ or $\scriptstyle {q}$ is true, and either $\scriptstyle {p}$ or '$\scriptstyle {q}$ implies $\scriptstyle {r}$' is true, then either $\scriptstyle {p}$ or $\scriptstyle {r}$ is true," we write

"$\scriptstyle {\vdash :.p\lor q:p.\lor q\supset r:\supset .p\lor r}$."

Here the first pair of dots indicates a logical product, while the second pair does not. Thus the scope of the second pair of dots passes over the first pair, and back until we reach the three dots after the assertion-sign.

Other uses of dots follow the same principles, and will be explained as they are introduced. In reading a proposition, the dots should be noticed first, as they show its structure. In a proposition containing several signs of implication or equivalence, the one with the greatest number of dots before or after it is the principal one: everything that goes before this one is stated by the proposition to imply or be equivalent to everything that comes after it.

Definitions. A definition is a declaration that a certain newly-introduced symbol or combination of symbols is to mean the same as a certain other combination of symbols of which the meaning is already known. Or, if the defining combination of symbols is one which only acquires meaning when combined in a suitable manner with other symbols^[1], what is meant is that any combination of symbols in which the newly-defined symbol or combination of symbols occurs is to have that meaning (if any) which results from substituting the defining combination of symbols for the newly-defined symbol or combination of symbols wherever the latter occurs. We will give the names of definiendum and definiens respectively to what is defined and to that which it is defined as meaning. We express a definition by putting the definiendum to the left and the definiens to the right, with the sign "$\scriptstyle {=}$" between, and the letters "$\scriptstyle {\text{Df}}$" to the right of the definiens. It is to be understood that the sign "$\scriptstyle {=}$" and the letters "$\scriptstyle {\text{Df}}$" are to be regarded as together forming one symbol. The sign "$\scriptstyle {=}$" without the letters "$\scriptstyle {\text{Df}}$" will have a different meaning, to be explained shortly.

An example of a definition is

$\scriptstyle {p\supset q.=.\sim p\lor q~{\text{Df.}}}$

It is to be observed that a definition is, strictly speaking, no part of the subject in which it occurs. For a definition is concerned wholly with the symbols, not with what they symbolise. Moreover it is not true or false, being the expression of a volition, not of a proposition. (For this reason,

1. This case will be fully considered in Chapter III of the Introduction. It need not further concern us at present.