Page:Russell, Whitehead - Principia Mathematica, vol. I, 1910.djvu/119

This is done, for example, when the law of identity is asserted in the form "$\scriptstyle {A}$ is $\scriptstyle {A}$." Here $\scriptstyle {A}$ is left undetermined, because, however $\scriptstyle {A}$ may be determined, the result will be true. Thus when we assert $\scriptstyle {\phi x}$, leaving $\scriptstyle {x}$ undetermined, we are asserting an ambiguous value of our function. This is only legitimate if, however the ambiguity may be determined, the result will be true. Thus take, as an illustration, the primitive proposition *1·2 below, namely

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

i.e. "'$\scriptstyle {p}$ or $\scriptstyle {p}$' implies $\scriptstyle {p}$." Here $\scriptstyle {p}$ may be any elementary proposition: by leaving $\scriptstyle {p}$ undetermined, we obtain an assertion which can be applied to any particular elementary proposition. Such assertions are like the particular enunciations in Euclid: when it is said "let $\scriptstyle {ABC}$ be an isosceles triangle; then the angles at the base will be equal," what is said applies to any isosceles triangle; it is stated concerning one triangle, but not concerning a definite one. All the assertions in the present work, with a very few exceptions, assert propositional functions, not definite propositions.

As a matter of fact, no constant elementary proposition will occur in the present work, or can occur in any work which employs only logical ideas. The ideas and propositions of logic are all general: an assertion (for example) which is true of Socrates but not of Plato, will not belong to logic^[1], and if an assertion which is true of both is to occur in logic, it must not be made concerning either, but concerning a variable $\scriptstyle {x}$. In order to obtain, in logic, a definite proposition instead of a propositional function, it is necessary to take some propositional function and assert that it is true always or sometimes, i.e. with all possible values of the variable or with some possible value. Thus, giving the name "individual" to whatever there is that is neither a proposition nor a function, the proposition "every individual is identical with itself" or the proposition "there are individuals" will be a proposition belonging to logic. But these propositions are not elementary.

(5) Negation. If $\scriptstyle {p}$ is any proposition, the proposition "not-$\scriptstyle {p}$," or "$\scriptstyle {p}$ is false," will be represented by "$\scriptstyle {\sim p}$." For the present, $\scriptstyle {p}$ must be an elementary proposition.

(6) Disjunction. If $\scriptstyle {p}$ and $\scriptstyle {q}$ are any propositions, the proposition "$\scriptstyle {p}$ or $\scriptstyle {q}$," i.e. "either $\scriptstyle {p}$ is true or $\scriptstyle {q}$ is true," where the alternatives are to be not mutually exclusive, will be represented by

"$\scriptstyle {p\lor q}$."

This is called the disjunction or the logical sum of $\scriptstyle {p}$ and $\scriptstyle {q}$. Thus "$\scriptstyle {\sim p\lor q}$" will mean "$\scriptstyle {p}$ is false or $\scriptstyle {q}$ is true"; $\scriptstyle {\sim (p\lor q)}$ will mean "it is false that either $\scriptstyle {p}$ or $\scriptstyle {q}$ is true," which is equivalent to "$\scriptstyle {p}$ and $\scriptstyle {q}$ are both false";

1. When we say that a proposition "belongs to logic," we mean that it can be expressed in terms of the primitive ideas of logic. We do not mean that logic applies to it, for that would of course be true of any proposition.