WHAT IS LOGICAL EMPIRICISM?
67
Every significant statement must therefore have empirically determ–
inable consequences. Apart from them, the statement has no mean–
ing. The pragmatic theory of meaning could not, however, adequately
account for the meaning of mathematical statements. Analytic studies
in mathematics tended to undermine the empiricistic interpretation
for which mathematical statements were abstract, but nevertheless
empirically descriptive, assertions about matters of fact.
Empiricism has traditionally relied on a psychological analysis of
knowledge for support of this view, and so tended to base mathe–
matical propositions, as well as empirical statements, on
th~theory
of
perception. It became more and more evident, however, that psycho–
logical theories of knowledge were not competent to solve this problem.
And the work of Peano and Frege in the foundations of mathematics
directed attention to the need for a new analysis of mathematical and
empirical science,-an analysis which would study science not psy–
chologically but in its formal or logical aspect.
Considered historically this program for logical analysis of science
was a revival of the projects of Raymond Lully and of Leibnitz for a
systematization of the language of science whereby it could be em–
ployed as a universal calculus. The mechanical operation of such
calculus would,
if
the system were comprehensive enough, be identical
with all reasoning. A radical difference, however, separates this ration–
alistic analysis from the logical empiricist analysis of scientific language
For the former the operations of the calculus can in themselves yield
knowledge concerning matters of fact; for the latter such operations
are purely formal and devoid of reference to the empirical world.
This leads in logical empiricism to a distinction between two
kinds of meaning: formal and empirical. Both kinds of meaning are
operational, that is to say, meanings-whether formal or empirical–
lie in operatiomi.l consequences. The operations involved in pure
mathematics and logic are formal. Those entering into the manipula–
tion and observation of experimental ·apparatus are empirical.
More generally, formal operations are those which pertain only to
the symbolic, linguistic functions of things; other operations are em–
pirical. By identifying meanings with operational consequences,
logical empiricism, like instrumentalism, denies that there are im–
manent directly apprehended ,meanings. No expression in isolation
can have meaning. It derives its meaning from its determinable con–
sequences -within a system. A statement whose operational conse–
quences are entirely formal has no meaning with reference to the
empirical world. Mathematical or logical statements are of this type.
Empirical meanings are derived from empirical operations. To
every proposition which makes assertions about anything in the em-
