ment, there is no corresponding constituent in the meaning of the statement. We may continue, if
we like, to say that numbers are eternal, immutable, and so on, but we must add that they are
logical fictions.
There is a further point. Concerning sound and colour, Plato says "both together are two, and each
of them is one." We have considered the two; now we must consider the one. There is here a
mistake very analogous to that concerning existence. The predicate "one" is not applicable to
things, but only to unit classes. We can say "the earth has one satellite," but it is a syntactical error
to say "the moon is one." For what can such an assertion mean? You may just as well say "the
moon is many," since it has many parts. To say "the earth has one satellite" is to give a property of
the concept "earth's satellite," namely the following property:
"There is a c such that 'x is a satellite of the earth' is true when, and only when, x is c."
This is an astronomical truth; but if, for "a satellite of the earth," you substitute "the moon" or any
other proper name, the result is either meaningless or a mere tautology. "One," therefore, is a
property of certain concepts, just as "ten" is a property of the concept "my finger." But to argue
"the earth has one satellite, namely the moon, therefore the moon is one" is as bad as to argue "the
Apostles were twelve; Peter was an apostle; therefore Peter was twelve," which would be valid if
for "twelve" we substituted "white."
The above considerations have shown that, while there is a formal kind of knowledge, namely
logic and mathematics, which is not derived from perception, Plato's arguments as regards all
other knowledge are fallacious. This does not, of course, prove that his conclusion is false; it
proves only that he has given no valid reason for supposing it true.
(2) I come now to the position of Protagoras, that man is the measure of all things, or, as it is
interpreted, that each man is the measure of all things. Here it is essential to decide the level upon
which the discussion is to proceed. It is obvious that, to begin with, we must distinguish between
percepts and inferences. Among percepts, each man is inevitably confined to his own; what he
knows of the percepts of others he knows by inference from his own percepts in hearing and
reading. The percepts of dreamers and madmen, as