The Critical Rationalist Vol. 03 No. 01 ISSN: 1393-3809 17-Apr-1998

3 The Application of Scientific Theories

1 Introduction

* (7)* Throughout this section I shall make use of a
number of items of standard logical notation, together
with some ideas that are less familiar. The signs
, , , , are
respectively the signs of

* (8)* If

* (9)* If for some statement

* (10)* The proof that Percival gives (§45) of the infinitude of
the content of any interesting axiomatizable theory

Suppose an infinite list of statements that are pair-wise contradictory and which individually do not entailt: Then the statement `toraor both' follows fromt. The same holds for each and every one of the statements in the infinite list. Since the statements in the list are pair-wise contradictory one can infer that none of the statements `toraor both', `torbor both' etc., is interderivable. Thus the logical content oftmust be infinite.

* (11)* More explicitly: if
then . But by hypothesis . Hence , contrary to
supposition. We may conclude that no element of the
list implies any other, and
hence that no two are logically equivalent.

* (12)* This result is rather general, since the
requirement that there exist a denumerable sequence of statements so related to

* (13)* The sufficiency of (ii) is established in much
the same way as in

* (14)* The proof of the necessity of (i) is a little
trickier, and can only be sketched. First recall
from

* (15)* Popper admits that the result
of

* (16)* Later in the section (in

* (17)* Popper describes the situation in the words:
`

* (18)* Now I will admit that I am not averse to the
main line of thought here, especially not to the idea
that we often discover in our theories consequences
that we never suspected. The history of remarkable
theorems in Euclidean geometry, one of the most
extensively studied of all mathematical theories, is
evidence enough that our knowledge has an uncanny
ability to surprise us; I need only cite my favourite
theorem in Euclidean geometry, Morley's theorem (which
says that the points of intersection of adjacent
trisectors of adjacent angles of a triangle always form
an equilateral triangle), or some of the theorems
collected in (Evelyn et al. 1974).
But though sympathetic to the general idea that we do
not know half of what we know, I think that a few
unsympathetic comments deserve a hearing.

* (19)* In the first place, note that the interpretation given to
the result is dangerously strong. For since most of, or even all,
the non-tautological consequences of a theory themselves have
infinitely many logically distinct consequences--all those, in
fact, that are not equivalent to the conjunction of some finite
number of irreducible statements--we are forced to acknowledge
that we do not understand properly most, or even any, of the
non-tautological consequences of any theory that we hold. If
understanding its consequences is what is important to
understanding a theory, then we do not really understand theories
at all; it is not just that our understanding is limited--it is
unbegotten. I for one therefore want to hold on to the
alternative idea, also endorsed by Popper (and by many others,
such as Collingwood), that the real path to understanding a

theory is by way of understanding its response to the problem situation that provoked it. Bartley (1990, p. 34) lumps together these different varieties of understanding, but they deserve to be kept cleanly apart.

* (20)* The second point to be made is that Popper and
Bartley can hardly be drawing attention only to the
frailty of our subjective apprehension of the items of
objective knowledge that we (and others) have
constructed. There must be more to what they are
insisting on than that we are not logically omniscient,
that we are unable to recognise all the consequences of
what we say. If that were really all that was meant,
then it would be hard to see why the proof should
bother to establish the (admittedly simple) point that
under suitable conditions a theory

* (21)* This, it seems to me, continues to hold even
when we move away from the degenerate case of logical
truth and consider theories that do indeed satisfy
Popper's theorem. The statement , for example, says that there exists exactly
one object. In elementary logic it has infinitely many
distinct consequences, including for each natural
number

* (22)* Whether our theories are fully understandable,
and in what sense they have infinite content, are
separate and, I have suggested, independent issues. For
the remainder of this section I shall content myself
with probing further into the latter problem, the
technical one. I suspect that what led Popper (and, in
his footsteps, Bartley) to think that the issue of the
ununderstandability of our theories amounts to more
than the incontrovertible psychological fact that we
can be surprised by some of their consequences was
something like this. The fact that most statements
have infinitely many distinct consequences can easily
be conflated with the claim that--as we might put it
informally--they have infinitely many different things
to say; that their contents consist of infinitely many
separate nuggets of information, each distinct from and
independent of the others. There is, that is to say, an
objective sense in which a theory

* (23)* I would certainly be prepared to consider this
as a relevant difference. But unfortunately no
such thing has been demonstrated by the proof
in

The only sense, that is, in which the different
elements of the sequence say
different things is that no one says the same as
another. But taken together, any two say exactly what
all the others say. Moreover there is no possibility
that a theory *t* such as the one with which we started
should have an equivalent formulation in terms of an
infinite sequence of statements each of which is
genuinely independent of all the others. For suppose
that *t* were equivalent to the infinite set of independent statements. By
the principle of finitude (often called compactness),
if *t* is derivable from this set, as we are assuming,
it is derivable from some finite subset, say . But then:

and the set is not independent after all.

* (24)*
Any independent axiomatization of an unaxiomatizable theory is
infinite, and any independent axiomatization of an axiomatizable
theory is finite. The difference is that the word `independent'
can be dropped from the first assertion, but not from the second.
Unaxiomatizable theories never look finite, but axiomatizable
theories sometimes look infinite. My thesis is that this is
something of an optical illusion, a logical hologram, an infinite-
dimensional Necker hypercube; what is really a single thing is
made to assume simultaneously an infinity of different guises.
But once we sort out the dependences among them, the finiteness
is restored.

* (25)* Yet the idea that infinite content implies the
existence of infinitely many independent nuggets of
information can be pressed a little further. Using a
corollary stated (but not explicitly proved) by
Popper (1966, p. 349), it is
possible to establish the somewhat unexpected
proposition that, although

* (26)* The result of Popper's that we need is this: if
the set of all true statements is not
axiomatizable, and

* (27)* To complete the proof we need to establish the
result attributed to Popper, and to establish also that
if

* (28)* First suppose that does not imply

meaning that too would be axiomatizable. This proves Popper's corollary.

* (29)* If

* (30)* It may be concluded that when Popper's original
assumption holds, so that there exists an infinite set
of statements that pairwise are contradictory and
individually do not entail

* (31)* Lying behind the view that scientific theories
are infinitely varied in their consequences there is, I
suspect, an atomistic view of content: the idea that
there exist minimal independent morsels of information
from which the contents of all more informative
statements and theories are compounded by finite or
infinite conjunction. The reader is warned not to be
misled by the use of the word `bit' in information
theory to express what sounds like exactly this idea.
It is not the same idea. Bits are not minimal, except
in the logically insignificant sense of being
representable by expressions of minimal length. Indeed,
the atomistic thesis as it stands is untenable. For
the only possible candidates for the role of atomic
contents would be irreducible statements, and no
axiomatizable theory with infinite content can be built
entirely from irreducibles. This is easily shown. For
a set of

* (32)* The objection may be strengthened by noting
that in many calculi there exist no irreducible
theories at all. This follows from the result of
Mostowski already cited. An example is provided by
ordinary classical sentential calculus with denumerably
many sentence letters. In such calculi, of course, all
non-tautological theories have infinite content.

* (33)* If an atomistic approach of content is possible
at all, it will only be, I think, through a move away
from logical content to a construe related to
what in

* (34)* I have been trying without success to find
something defensible in the view that the infinitude of
a theory's content has more than psychological
significance; that there is an objective sense in which
it is true that an axiomatizable theory must say more
than we can ever appreciate. In his discussion of the
syllogism Mill (1843, Book II, Chapter III,
section II) rightly
dismisses any attempt `to attach any serious scientific
value to such a mere salvo as the distinction drawn
between being involved

3 The Application of Scientific Theories

1 Introduction

The Critical Rationalist Vol. 03 No. 01 ISSN: 1393-3809 17-Apr-1998

Copyright © 1998 All Rights Reserved.

TCR Issue Timestamp: Fri Apr 17 07:52:54 GMT 1998