Page:Russell, Whitehead - Principia Mathematica, vol. I, 1910.djvu/27

 To sum up, the three salient facts connected with the use of the variable are: (1) that a variable is ambiguous in its denotation and accordingly undefined: (2) that a variable preserves a recognizable identity in various occurrences throughout the same context, so that many variables can occur together in the same context each with its separate identity: and (3) that either the range of possible determinations of two variables my be the same, so that a possible determination of one variable is also a possible determination of the other, or the ranges of two variables may be different, so that, if a possible determination of one variable is given to the other, the resulting complete phrase is meaningless instead of becoming a complete unambiguous proposition (true or false) as would be the case if all variables in it had been given any suitable determinations.

The uses of various letters. Variables will be denoted by single letters, and so will certain constants; but a letter which has once been assigned to a constant by a definition must not afterwards be used to denote a variable. The small letters of the ordinary alphabet will all be used for variables, except $$\scriptstyle{p}$$ and $$\scriptstyle{s}$$ after *40, in which constant meanings are assigned to these two letters. The following capital letters will receive constant meanings: $\scriptstyle{B}$, $\scriptstyle{C}$, $\scriptstyle{D}$, $\scriptstyle{E}$, $\scriptstyle{F}$, $$\scriptstyle{I}$$ and $\scriptstyle{J}$. Among small Greek letters, we shall give constant meanings to $\scriptstyle{\epsilon}$, $\scriptstyle{\iota}$, $$\scriptstyle{\pi}$$ and (at a later stage) to $\scriptstyle{\eta}$, $$\scriptstyle{\theta}$$ and $\scriptstyle{\omega}$. Certain Greek capitals will from time to time be introduced for constants, but Greek capitals will not be used for variables. Of the remaining letters, $\scriptstyle{p}$, $\scriptstyle{q}$, $$\scriptstyle{r}$$ will be called propositional letters, and will stand for variable propositions (except that, from *40 onwards, $$\scriptstyle{p}$$ must not be used for a variable); $\scriptstyle{f}$, $\scriptstyle{g}$, $\scriptstyle{\phi}$, $\scriptstyle{\psi}$, $\scriptstyle{\chi}$, $$\scriptstyle{\theta}$$ and (until *33) $$\scriptstyle{F}$$ will be called functional letters, and will be used for variable functions.

The small Greek letters not already mentioned will be used for variables whose values are classes, and will be referred to simply as Greek letters. Ordinary capital letters not already mentioned will be used for variables whose values are relations, and will be referred to simply as capital letters. Ordinary small letters other than $\scriptstyle{p}$, $\scriptstyle{q}$, $\scriptstyle{r}$, $\scriptstyle{s}$, $\scriptstyle{f}$, $$\scriptstyle{g}$$ will be used for variables whose values are not known to be functions, classes, or relations; these letters will be referred to simply as small Latin letters.

After the early part of the work, variable propositions and variable functions will hardly ever occur. We shall then have three main kinds of variables: variable classes, denoted by small Greek letters; variable relations, denoted by capitals; and variables not given as necessarily classes or relations, which will be denoted by small Latin letters.

In addition to this usage of small Greek letters for variable classes, capital letters for variable relations, small Latin letters for variables of type wholly undetermined by the context (these arise from the possibility of