THE LOGICAL STATUS OF MENTAL FACTORS 103

(b) Bipolar or Two-signed factors, i.e. factors which can
vary in opposite or antagonistic directions, and whose satura-
tion coefficients may therefore be either positive or negative.

(2) Particular factors, each common to a certain group of traits
only, and hence usually termed * Growp-factors ' : they have some-
times been called £ special factors,' ' overlapping specific factors *
(' overlapping,5 because any single trait may contain more than one
of them), or * general factors of limited range'—phrases which all
rather blur the real nature of the distinction.

B. Individual or Unique factors, i.e. those influencing one test
or trait alone, viz. :

(3) Singular factors, each peculiar to a single trait, and usually
called £ Specific factors ' (sometimes also c individual' or * unique '
factors) ; when the characteristics they cover are regarded as
irrelevant to the main inquiry, these factors, like the following, are
frequently described as ' errors,' and then, in contrast to the follow-
ing, are designated * constant' or * systematic errors.'

(4) Accidental factors, each peculiar to the particular occasion
on which the particular trait was measured, and therefore sometimes
called * factors of error' or of c unreliability' (the latter term in
factor-analysis merely means inconsistency). Here the errors are
the results, not of some gross and traceable bias, but of a very
large number of very small causes. Hence the minor fluctuations
for which they are responsible show the random distribution
characteristic of * chance,' and the factors themselves are con-
sequently often called * random errors ' or * chance factors.'

We may sum up these preliminary distinctions in what I
have called the four-factor theorem. " The measurement
of any individual for any one of a given set of traits may be
regarded as a function of four kinds of components : namely,
those characteristic of (i) all the traits, (ii) some of the traits,
(iii) the particular trait in question whenever it is measured,
(iv) the particular trait in question as measured on this
particular occasion." This I regard as a fundamental
logical postulate from which all factor theories must neces-
sarily start. If for the sake of clearness we prefer to con-
dense it into a symbolic equation (assuming with most fac-
torists that the ' function? may be expressed as a weighted
sum), we may write—