Page:Russell, Whitehead - Principia Mathematica, vol. I, 1910.djvu/40

the two $\scriptstyle {x}$'s have no connection with each other. Since only one dot follows the $\scriptstyle {x}$ in brackets, the scope of the first $\scriptstyle {x}$ is limited to the "$\scriptstyle {\phi x}$" immediately following the $\scriptstyle {x}$ in brackets. It usually conduces to clearness to write rather than

${\begin{aligned}&\scriptstyle {(x).\phi x.\supset .\psi y}\\&\scriptstyle {(x).\phi x.\supset .\psi x,}\\\end{aligned}}$

since the use of different letters emphasises the absence of connection between the two variables; but there is no logical necessity to use different letters, and it is sometimes convenient to use the same letter. Ambiguous assertion and the real variable. Any value "$\scriptstyle {\phi x}$" of the function $\scriptstyle {\phi {\hat {x}}}$ can be asserted. Such an assertion of an ambiguous member of the values of $\scriptstyle {\phi {\hat {x}}}$ is symbolised by

"$\scriptstyle {\vdash .\phi x}$."

Ambiguous assertion of this kind is a primitive idea, which cannot be defined in terms of the assertion of propositions. This primitive idea is the one which embodies the use of the variable. Apart from ambiguous assertion, the consideration of "$\scriptstyle {\phi x}$," which is an ambiguous member of the values of $\scriptstyle {\phi {\hat {x}}}$, would be of little consequence. When we are considering or asserting "$\scriptstyle {\phi x}$," the variable $\scriptstyle {x}$ is called a "real variable." Take, for example, the law of excluded middle in the form which it has in traditional formal logic:

"$\scriptstyle {a}$ is either $\scriptstyle {b}$ or not $\scriptstyle {b}$."

Here $\scriptstyle {a}$ and $\scriptstyle {b}$ are real variables: as they vary, different propositions are expressed, though all of them are true. While $\scriptstyle {a}$ and $\scriptstyle {b}$ are undetermined, as in the above enunciation, no one definite proposition is asserted, but what is asserted is any value of the propositional function in question. This can only be legitimately asserted if, whatever value may be chosen, that value is true, i.e.' if all the values are true. Thus the above form of the law of excluded middle is equivalent to

"($\scriptstyle {a}$, $\scriptstyle {b}$).$\scriptstyle {a}$ is either $\scriptstyle {b}$ or not $\scriptstyle {b}$,"

i.e. to "it is always true that $\scriptstyle {a}$ is either $\scriptstyle {b}$ or not $\scriptstyle {b}$." But these two, though equivalent, are not identical, and we shall find it necessary to keep them distinguished. When we assert something containing a real variable, as in e.g.

"$\scriptstyle {\vdash .x=x}$,"

we are asserting any value of a propositional function. When we assert something containing an apparent variable, as in or

${\begin{aligned}&\scriptstyle {^{\prime \prime }\vdash .(x).x=x^{\prime \prime }}\\&\scriptstyle {^{\prime \prime }\vdash .(\exists x).x=x,^{\prime \prime }}\\\end{aligned}}$

we are asserting, in the first case all values, in the second case some value (undetermined), of the propositional function in question. It is plain that