110 
non-exclusion of such totalities, have always 
infected logic and justified skepticism as to 
the ultimate soundness of all discourse, how- 
ever seemingly rigorous. (Such theoretic 
skepticism may persist anyhow, on other 
grounds.) Perhaps the most obvious example 
of an illegitimate totality is the so-called 
elass of all classes. Its illegitimacy may be 
shown as follows. If A is a class (say that 
of men) and # is a member of it, we say, H 
isan A. Now let W be the class of all classes 
such that no one of them is a member of itself. 
Then, whatever class x may be, to say that x 
is a W is equivalent to saying that x is not 
an «, and hence to say that W is a W is 
equivalent to saying that W is not a W! 
Such illegitimate totalities (and the fallacies 
they breed) are in general exceedingly sly, 
insinuating themselves under an endless va- 
riety of most specious disguises, and that, not 
only in the theory of classes but also in con- 
nection with every species of logical subject- 
matter, as propositions, relations and proposi- 
tional functions. As the propositional func- 
tion—any expression containing a real (as 
distinguished from an apparent) variable and 
yielding either non-sense or else a proposition 
whenever the variable is replaced by a con- 
stant term—is the basis of our authors’ work, 
their theory of logical types is fundamentally 
a theory of types of propositional functions. 
It can not be set forth here nor in fewer pages 
than the authors have devoted to it. Suffice 
it to say that the theory presents propositional 
functions as constituting a summitless hier- 
archy of types such that the functions of a 
given type make up a legitimate totality; and 
that, in the light of the theory, truth and 
falsehood present themselves each in the form 
of a systematic ambiguity, the quality of 
being true (or false) admitting of distinctions 
in respect of order, level above level, without 
a summit. When Epimenides, the Cretan, 
says that all statements of Cretans are false, 
and you reply that then his statement is false, 
the significance of “false” here presents two 
orders or levels; and logic must by its ma- 
chinery automatically prevent the possibility 
of confusing them. 
SCIENCE 
[N.S. Vou. XXXV. No. 890 
Next follows a chapter of 20 pages, which 
all philosophers, logicians and grammarians 
ought to study, a chapter treating of Incom- 
plete Symbols wherein by ingenious analysis 
it is shown that the ubiquitous expressions 
of the form “the so and so” (the “the” being 
singular, as “the author of Waverley,” “the 
sine of a,’ “the Athenian who drank hem- 
lock,” ete.) do not of themselves denote any- 
thing, though they have contextual signifi- 
cance essential to discourse, essential in par- 
ticular to the significance of identity, which, 
in the world of discourse, takes the form of 
“ais the so and so” and not the form of the 
triviality, @ is a. 
After the introduction of 88 pages, we reach 
the work proper (so far as it is contained in 
the present volume), namely, Part I.: Mathe- 
matical Logic. Here enunciation of primi- 
tives is followed by series after series of the- 
orems and demonstrations, marching through 
578 pages, all matter being clad in symbolic 
garb, except that the continuity is interrupted 
here and there by summaries and explanations 
in ordinary language. Logic it is called and 
logic it is, the logic of propositions and func- 
tions and classes and relations, by far the 
greatest (not merely the biggest) logic that 
our planet has produced, so much that is new 
in matter and in manner; but it is also math- 
ematics, a prolegomena to the science, yet 
itself mathematics in the most genuine sense, 
differing from other parts of the science only 
in the respects that it surpasses these in fun- 
damentality, generality and precision, and 
lacks traditionality. Few will read it, but all 
will feel its effect, for behind it is the urgence 
and push of a magnificent past: two thousand 
five hundred years of record and yet longer 
tradition of human endeavor to think aright. 
©. J. Keyser 
COLUMBIA UNIVERSITY 
A LETTER OF LAMARCK 
Letters of Lamarck are not often found. 
M. Landrieux, who has recently published a 
life of Lamarck, states that “one can count 
the number of his letters which have come 
