Page:Russell, Whitehead - Principia Mathematica, vol. I, 1910.djvu/67

which $\scriptstyle {\phi x}$ has first truth; thus we may define "$\scriptstyle {\{(\exists x).\phi x\}}$ has second truth" as meaning "some value for $\scriptstyle {\phi {\hat {x}}}$ has first truth," i.e. "$\scriptstyle {(\exists x).(\phi x}$ has first truth)." Similar remarks apply to falsehood. Thus "$\scriptstyle {\{(x).\phi x\}}$ has second falsehood" will mean "some value for $\scriptstyle {\phi {\hat {x}}}$ has first falsehood," i.e. "$\scriptstyle {(\exists x).(\phi x}$ has first falsehood)," while "$\scriptstyle {\{(\exists x).\phi x\}}$ has second falsehood" will mean "all values for $\scriptstyle {\phi {\hat {x}}}$ have first falsehood," i.e. "$\scriptstyle {(x).(\phi x}$ has first falsehood)." Thus the sort of falsehood that can belong to a general proposition is different from the sort that can belong to a particular proposition.

Applying these considerations to the proposition "$\scriptstyle {(p).p}$ is false," we see that the kind of falsehood in question must be specified. If, for example, first falsehood is meant, the function "$\scriptstyle {\hat {p}}$ has first falsehood" is only significant when $\scriptstyle {p}$ is the sort of proposition which has first falsehood or first truth. Hence "$\scriptstyle {(p).p}$ is false" will be replaced by a statement which is equivalent to "all propositions having either first truth or first falsehood have first falsehood." This proposition has second falsehood, and is not a possible argument to the function "$\scriptstyle {\hat {p}}$ has first falsehood." Thus the apparent exception to the principle that "$\scriptstyle {\phi \{(x).\phi x\}}$" must be meaningless disappears.

Similar considerations will enable us to deal with "not-$\scriptstyle {p}$" and with "$\scriptstyle {p}$ or $\scriptstyle {q}$." It might seem as if these were functions in which any proposition might appear as argument. But this is due to a systematic ambiguity in the meanings of "not" and "or," by which they adapt themselves to propositions of any order. To explain fully how this occurs, it will be well to begin with a definition of the simplest kind of truth and falsehood.

The universe consists of objects having various qualities and standing in various relations. Some of the objects which occur in the universe are complex. When an object is complex, it consists of interrelated parts. Let us consider a complex object composed of two parts $\scriptstyle {a}$ and $\scriptstyle {b}$ standing to each other in the relation $\scriptstyle {R}$. The complex object "$\scriptstyle {a}$-in-the-relation-$\scriptstyle {R}$-to-$\scriptstyle {b}$" may be capable of being perceived; when perceived, it is perceived as one object. Attention may show that it is complex; we then judge that $\scriptstyle {a}$ and $\scriptstyle {b}$ stand in the relation $\scriptstyle {R}$. Such a judgment, being derived from perception by mere attention, may be called a "judgment of perception." This judgment of perception, considered as an actual occurrence, is a relation of four terms, namely $\scriptstyle {a}$ and $\scriptstyle {b}$ and $\scriptstyle {R}$ and the percipient. The perception, on the contrary, is a relation of two terms, namely "$\scriptstyle {a}$-in-the-relation-$\scriptstyle {R}$-to-$\scriptstyle {b}$," and the percipient. Since an object of perception cannot be nothing, we cannot perceive "$\scriptstyle {a}$-in-the-relation-$\scriptstyle {R}$-to-$\scriptstyle {b}$" unless $\scriptstyle {a}$ is in the relation $\scriptstyle {R}$ to $\scriptstyle {b}$. Hence a judgment of perception, according to the above definition, must be true. This does not mean that, in a judgment which appears to us to be one of perception, we are sure of not being in error, since we may err in thinking that our judgment has really been derived merely by analysis of