popper-logic-scientific-discovery

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some structural components of a theory of experience
be an empirical way of defining a concept. It is customary to speak of ‘ostensive
definitions’. This means that a definite empirical meaning is assigned
to a concept by correlating it with certain objects belonging to the real
world. It is then regarded as a symbol of those objects. But it should
have been clear that only individual names or concepts can be fixed by
ostensively referring to ‘real objects’—say, by pointing to a certain
thing and uttering a name, or by attaching to it a label bearing a name,
etc. Yet the concepts which are to be used in the axiomatic system
should be universal names, which cannot be defined by empirical
indications, pointing, etc. They can be defined if at all only explicitly,
with the help of other universal names; otherwise they can only be left
undefined. That some universal names should remain undefined is
therefore quite unavoidable; and herein lies the difficulty. For these
undefined concepts can always be used in the non-empirical sense (i),
i.e. as if they were implicitly defined concepts. Yet this use must inevitably
destroy the empirical character of the system. This difficulty, I
believe, can only be overcome by means of a methodological decision.
I shall, accordingly, adopt a rule not to use undefined concepts as if
they were implicitly defined. (This point will be dealt with below in
section 20.)
Here I may perhaps add that it is usually possible for the primitive
concepts of an axiomatic system such as geometry to be correlated
with, or interpreted by, the concepts of another system, e.g. physics.
This possibility is particularly important when, in the course of the
evolution of a science, one system of statements is being explained by
means of a new—a more general—system of hypotheses which permits
the deduction not only of statements belonging to the first system,
but also of statements belonging to other systems. In such cases it may
be possible to define the fundamental concepts of the new system with
the help of concepts which were originally used in some of the old
systems.
18 LEVELS OF UNIVERSALITY. THE MODUS TOLLENS
We may distinguish, within a theoretical system, statements belonging
to various levels of universality. The statements on the highest level of
universality are the axioms; statements on the lower levels can be