Page:Russell, Whitehead - Principia Mathematica, vol. I, 1910.djvu/32

($\scriptstyle {\exists x,y}$) or [$\scriptstyle {(\iota x)(\phi x)}$] or analogous expressions^[1]. Group III consists of dots which stand between propositions in order to indicate a logical product. Group I is of greater force than Group II, and Group II than Group III. The scope of the bracket indicated by any collection of dots extends backwards or forwards beyond any smaller number of dots, or any equal number from a group of less force, until we reach either the end of the asserted proposition or a greater number of dots or an equal number belonging to a group of equal or superior force. Dots indicating a logical product have a scope which works both backwards and forwards; other dots only work away from the adjacent sign of disjunction, implication, or equivalence, or forward from the adjacent symbol of one of the other kinds enumerated in Group II.

Some examples will serve to illustrate the use.of dots.

"$\scriptstyle {p\lor q.\supset .q\lor p}$" means the proposition "'$\scriptstyle {p}$ or $\scriptstyle {q}$' implies '$\scriptstyle {q}$ or $\scriptstyle {p}$.'" When we assert this proposition, instead of merely considering it, we write

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

where the two dots after the assertion-sign show that what is asserted is the whole of what follows the assertion-sign, since there are not as many as two dots anywhere else. If we had written "$\scriptstyle {p\lor q.\supset .q\lor p}$," that would mean the proposition "either $\scriptstyle {p}$ is true, or $\scriptstyle {q}$ implies '$\scriptstyle {q}$ or $\scriptstyle {p}$.'" If we wished to assert this, we should have to put three dots after the assertion-sign. If we had written "$\scriptstyle {p\lor q.\supset .q:\lor :p}$," that would mean the proposition "either '$\scriptstyle {p}$ or $\scriptstyle {q}$' implies $\scriptstyle {q}$, or $\scriptstyle {p}$ is true." The forms "$\scriptstyle {p.\lor .q.\supset .q\lor p}$" and "$\scriptstyle {p\lor q.\supset .q.\lor .p}$" have no meaning. "$\scriptstyle {p\supset q.\supset :q\supset r.\supset .p\supset r}$" will mean "if $\scriptstyle {p}$ implies $\scriptstyle {q}$, then if $\scriptstyle {q}$ implies $\scriptstyle {r}$, $\scriptstyle {p}$ implies $\scriptstyle {r}$." If we wish to assert this (which is true) we write

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

Again "$\scriptstyle {p\supset q.\supset .q\supset r:\supset .p\supset r}$" will mean "if '$\scriptstyle {p}$ implies $\scriptstyle {q}$' implies '$\scriptstyle {q}$ implies $\scriptstyle {r}$,' then $\scriptstyle {p}$ implies $\scriptstyle {r}$." This is in general untrue. (Observe that "$\scriptstyle {p\supset q}$" is sometimes most conveniently read as "$\scriptstyle {p}$ implies $\scriptstyle {q}$," and sometimes as "if $\scriptstyle {p}$, then $\scriptstyle {q}$.") "$\scriptstyle {p\supset q.q\supset r.\supset .p\supset r}$" will mean "if $\scriptstyle {p}$ implies $\scriptstyle {q}$, and $\scriptstyle {q}$ implies $\scriptstyle {r}$, then $\scriptstyle {p}$ implies $\scriptstyle {r}$." In this formula, the first dot indicates a logical product; hence the scope of the second dot extends backwards to the beginning of the proposition. "$\scriptstyle {p\supset q:q\supset r.\supset .\supset .p\supset r}$" will mean "$\scriptstyle {p}$ implies $\scriptstyle {q}$; and if $\scriptstyle {q}$ implies $\scriptstyle {r}$, then $\scriptstyle {p}$ implies $\scriptstyle {r}$." (This is not true in general.) Here the two dots indicate a logical product; since two dots do not occur anywhere else, the scope of these two dots extends backwards to the beginning of the proposition, and forwards to the end. "$\scriptstyle {p\lor q.\supset :.p.\lor .q\supset r:\supset .p\lor r}$" will mean "if either $\scriptstyle {p}$ or $\scriptstyle {q}$ is true, then if either $\scriptstyle {p}$ or '$\scriptstyle {q}$ implies $\scriptstyle {r}$' is true, it follows that either $\scriptstyle {p}$ or $\scriptstyle {r}$ is true."

1. The meaning of these expressions will be explained later, and examples of the use of dots in connection with them will be given on pp. 17, 18.