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 negation [not],
conjunction [and], disjunction
[or], conditional [if ...then],
universal quantification [for all]. The
symbol ` 
' [yields or, in the present
context, implies or entails] stands for
derivability or deducibility in the sense
of classical elementary logic; the symbol ` 
'
[is logically equivalent to] stands for mutual
derivability. The connectives , , ,
are understood to be operators, in the first
instance, on syntactical items called
statements. (It might be better to call them
sentences, in order to emphasize their formal or
morphological character, but it is less confusing if
Popper's usage is followed.) Two statements that are
not identical are distinct. Two statements that
are not logically equivalent are logically
distinct. Logical truths--statements that are
derivable from all statements, or from none--will,
laxly but conveniently, be called tautologies.
There are infinitely many distinct tautologies, but of
course no two are logically distinct.
(8) If X is a set of statements and 
,
we shall say that x is a [logical]
consequence of X. The set of consequences of
X is written 
. A [deductive]
theory 
is a set of statements closed under
the operation 
; that is to say, 
. A
theory is inconsistent if it contains every
statement; otherwise consistent. The connectives
and and the relation may be
extended in a natural way from statements to theories:
if & only if 
.
Whatever the terminology may suggest, it is of no
importance to this paper whether the consequence
operation is characterized syntactically (by
rules of derivation) or semantically (in terms of
models).
(9) If 
for some statement x, we
call [finitely] axiomatizable. If
is consistent, and has no consistent proper
extension 
, then is maximal (or
complete). If 
is both axiomatizable
and maximal, 
has no proper
consequences but tautologies (for if 
then 
). In this case the theory
, and by extension the statement
, are called irreducible
(Tarski 1936, section 4).
It follows at once that has only finitely many
distinct consequences if & only if 
where each of the 
is
irreducible. It is plain that cannot have a
finite content unless it is axiomatizable, and thus
identical to for some statement x.
(10) The proof that Percival gives (§45) of the infinitude of the content of any interesting axiomatizable theory t comes straight from note 18 of (Popper 1974, Popper 1976). Percival writes:
Suppose an infinite list of statements that are pair-wise contradictory and which individually do not entail t:Then the statement `t or a or both' follows from t. 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 `t or a or both', `t or b or both' etc., is interderivable. Thus the logical content of t must 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 t is not a
severe one. Popper himself notes that `[f]or most
t's, something like a: ``the number of planets is
0'', b: ``the number of planets is 1'', and so on,
would be adequate'
(1974, p. 19;
1976, pp. 26f). The
proof does not even require that each pair of elements
of the sequence consist of contraries; only that each
pair entail t. I do not know whether either of these
conditions is necessary for t to have infinitely many
logically distinct consequences, but there are closely
related conditions that are separately both necessary
and sufficient for this: (i) there exists a denumerable
sequence 
of pairwise incompatible
theories that individually do not imply t; (ii) there
exists a denumerable sequence of
theories that pairwise, but not individually, imply
t. Since (i) is stronger than (ii), we need prove
only the sufficiency of (ii) and the necessity of (i).
These results are of only supplementary significance in
the present context, and those who are not itching to
see proofs of them are invited to resume the main line
of argument in 2.4.
(13) The sufficiency of (ii) is established in much
the same way as in 2.2. Instead of
constructing a sequence of statements we construct a sequence of theories 
. (The disjunction of two theories
consists of their common consequences.) Now if any one
of these new theories is not axiomatizable, then t
must have infinite content; for if t had only finite
content, then its subtheories would have finite content
too. In other words, we may assume that each of can be axiomatized by a single
statement, and proceed as before.
(14) The proof of the necessity of (i) is a little
trickier, and can only be sketched. First recall
from 2.1 that t has only finitely many
logically distinct consequences if & only if it is
logically equivalent to the conjunction of a finite set
of irreducible statements. It follows, not quite
obviously, that t has finite content if & only if
has only finitely many maximal extensions,
all axiomatizable. In other words, it is necessary &
sufficient for t to have infinite content that
has either infinitely many axiomatizable
maximal extensions, or at least one unaxiomatizable
maximal extension. Now it is immediate from a theorem
of Mostowski
(1937, Theorem 8, p. 13;
reported on p. 370 of Tarski
1936) that if
a theory has even one unaxiomatizable maximal
extension, then it has infinitely many maximal
extensions. We may conclude that if t has infinite
content, then has infinitely many maximal
extensions. From this set of maximal extensions we may
extract a denumerable sequence 
of theories individually implying
. Since each such 
is consistent,
none of them implies t, and since each is maximal,
they are pairwise incompatible. In this way (i) is
proved.
(15) Popper admits that the result
of 2.2 is both `well known' and
`trivial' (though he means trite), but suggests that it
may appear rather more significant if it is phrased not
in terms of logical content, but in terms of the
closely related idea of informative content. He calls
the informative content of a theory the class of
statements that are incompatible with it; the class,
that is, of statements that it excludes or forbids
(Popper 1974, p. 18;
1976, p. 26). Now on the
one hand it is quite plain that, since implies x
if and only if is incompatible with , the
informative content of a theory is infinite whenever
its logical content is infinite. But on the other hand
we can see that amongst the elements of the informative
content of a theory will be many statements that the
inventor of the theory may not have had in mind when he
formulated it, and may indeed never come to appreciate.
Kepler's theory, to vary slightly the example given in
turn by Popper, by Bartley, and by Percival, excludes
Newton's theory, at least in the presence of the
assumption that the individual planets have non-zero
mass. Kepler's first law, which says that the sun
occupies one focus of the elliptical orbit of each
planet, is corrected by Newton's theory in at least two
ways: first, by introducing perturbations due to
gravitational attraction among the planets; and second
by insisting that, even in a one-planet system, it is
not the sun itself but the centre of mass of the
sun-planet complex that is at the focus of the
elliptical orbit. The informative content of a theory,
that is, is in some sense not present in its entirety
to the inventor of the theory, and exactly the same
therefore applies in the case of its logical content.
(16) Later in the section (in 2.11) and throughout section 3, I shall have cause to applaud this idea that the content of a theory is determined by what it rules out--an idea that goes back, as Percival notes, to the discussion of empirical content in §§31-35 of Logik der Forschung (Popper 1934). For the time being, if I may, I shall continue to investigate the significance of the theorem stated and proved in 2.2.
(17) Popper describes the situation in the words: `we never know what we are talking about' (1974, p. 19; 1976, p. 27). Bartley (1990) says: `we do not know what we are saying or ...what we are doing'. The point in each case is that if understanding a theory to the full requires understanding all its logical consequences, we cannot be said to understand to the full even our own creations. A similar evaluation was given by Ryle in his inaugural lecture at Oxford (1945, p. 7; 1971, p. 198): `Thus people can correctly be said to have only a partial grasp of most of the propositions they consider. They could usually be taken by surprise by certain of the remoter logical connexions of their most ordinary propositions.' As we shall see in 2.6, Ryle went on to qualify this judgement in an important way.
(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 t has infinitely
many consequences that are logically distinct from each
other. For it is even more straightforward to
establish that every theory, even one that states only
a tautology, has infinitely many syntactically distinct
consequences; that is, that there are infinitely many
distinct statements that are derivable from any logical
truth. Everyone who has studied logic knows (though
may not be able to prove) that there exist infinitely
many syntactically distinct tautologies, indeed
infinitely many distinct forms of tautologies. It
would be plainly an exaggerated idealization to suggest
that any logician, however skilled, could recognise all
these distinct tautologies. But we ought to jib at the
idea that the understanding of 
requires any
ability to recognise all its equivalents in
propositional logic (let alone all its equivalents in
elementary predicate logic) as logical equivalents.
Indeed, no one actually has the psychological prowess
(or time) to recognise all tautologies of the simplest
form . Once we reach instances of this scheme
involving several million variables and many billions
of pairs of parentheses we are beyond what is
accessible even to the keenest brain. And it cannot be
thought that Popper and Bartley imagine that there is a
sharp and significant difference here between the
transparent relation of logical equivalence and the
opaque relation of logical implication. The mere
possession of infinitely many equivalents, or of
infinitely many consequences, though having
psychological implications of the most banal kind, does
not in itself make a theory unfathomable or
ununderstandable.
(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 i greater than 1 a statement, which we call
(the point of this notation will
become clear in 2.10), to the effect
that there do not exist exactly i objects. But it is
surely an absurd conceit to maintain that this theory
(plainly false as it is) is beyond our full
comprehension. Ryle is worth quoting again: `though
people's understanding of the propositions that they
use is in this sense [the sense of grasping all logical
connexions] imperfect, there is another sense in which
their understanding of some of them may be nearly or
quite complete'
(1945, p. 7;
1971, p. 198). Indeed, if
we restrict ourselves to the calculus of elementary
logic with identity and no other predicates or
relations, we can without too much difficulty give a
characterization of all the consequences of
(which does not mean that we can
recognise presumptive consequences of great length).
It might be thought that considerations of complexity
or effectiveness could enable us to draw a useful
distinction here. For within elementary logic with
identity the theory
is a decidable theory
(Tarski et al. 1953, p. 19); there is a
mechanical procedure for determining of any statement
whether or not it is a consequence of the theory.
Decidable theories, it might be conjectured, are
understandable (even if we cannot recognise all their
consequences), but undecidable theories are not.
Percival briefly alludes to Gödel's theorem
concerning the incompleteness of consistent recursively
axiomatizable theories of arithmetic (§49), from which
follows the undecidability of arithmetic, but he does
not contrast undecidable theories with decidable ones.
But the trouble with this suggestion is that even
elementary logic is undecidable (Church), indeed
undecidable in the highest degree; a solution to the
decision problem is also a solution to the halting
problem for Turing machines (this is the gist of the
proof of Church's theorem, due to Büchi, presented in
Chapter 10 of Boolos & Jeffrey
1974). On the other hand,
elementary Euclidean geometry is decidable
(Tarski 1948). Should we identify full
intelligibility with decidability, we would therefore
find ourselves back in the untenable position of saying
that the tautology is beyond our
understanding, and having to admit at the same time
that Euclidean geometry is not. But nothing in this
paragraph should be taken to deny that some more
delicate deployment of ideas from recursion theory
might provide real illumination of the problem.
(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 t, even though finitely axiomatizable, may be beyond us.
(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 2.2 above. If we look again at that
proof, we shall see that although no one element of the
sequence implies any other,
any two elements of it are together logically
equivalent to t, and therefore together imply all the
others. For since a and b are incompatible by
hypothesis, we have (using the distributive law):
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 t is not equivalent to any infinite independent subset, under the conditions already propounded, it does include, within its content, such a set. This shows that there is one sense (though one that I shall claim to be unimportant) in which we can correctly assert that t does have infinitely many different things to say. Those who want to concentrate on the main problem of the paper, and those who can't be bothered with proofs, are once again permitted to move on, either to 2.10 or, if they are desperate, directly to section 3.
(26) The result of Popper's that we need is this: if
the set 
of all true statements is not
axiomatizable, and t is axiomatizable and false, then
its truth content 
is not axiomatizable.
Phrased more generally, this asserts that if 
is an unaxiomatizable maximal theory that does not
imply t, then 
is an
unaxiomatizable subtheory of t. It may be shown that
if t has infinite content then there must exist such
an unaxiomatizable that does not imply t. It
follows that is not
axiomatizable. Now by a theorem of Tarski (proved
informally on p. 362 of
1935), every
theory is logically equivalent to an independent set,
and hence must be equivalent to
an infinite independent set (for otherwise, as already
noted, it would be finitely axiomatizable).
(27) To complete the proof we need to establish the
result attributed to Popper, and to establish also that
if t has infinite content then at least one
unaxiomatizable maximal theory does not imply
it.
(28) First suppose that does not imply t.
Since it is maximal, it implies . Now if
were axiomatizable, so would be its
conjunction with . But
meaning that too would be axiomatizable. This proves
Popper's corollary.
(29) If t has infinite content then, as noted in
2.3, has an infinite number
of maximal extensions. That these cannot all be
axiomatizable is part of Theorem 8 of (Mostowski
1937; Tarski
1936,
p. 370). Here is the simple proof. If a theory
(whether axiomatizable or not) has infinitely many
axiomatizable maximal extensions 
, then it is consistent with each finite
subset of 
, and
hence (by the principle of finitude) 
is consistent.
By Lindenbaum's theorem, has a maximal extension
that is not identical with any element of
(for maximal
theories are pairwise incompatible). Thus
has an unaxiomatizable extension , which cannot
also be an extension of t.
(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 t, then t includes amongst its consequences an infinite independent set. A simple and revealing example is provided by Kepler's three laws, augmented by a finite set of initial conditions sufficiently copious for the prediction of the positions of the planets at all future times. It is plain that within the content of this theory we can find an infinite set of statements that, taken in isolation, are logically independent of each other; statements of the positions of Venus at different times suffice for this, since it is only in the presence of universal laws that there is any logical connection between the position of Venus at one time and its position at any other time. But we should not be eager to conclude that there is any important sense in which Kepler's laws say infinitely many distinct things. On the contrary, one of the virtues of logical systematization and axiomatization is that it enables us to replace scattered sets of results by unifying principles. There is something suspiciously retrograde about the claim that Kepler's laws have an infinity of things to say (just as there is something retrograde about the claim that a theory may be replaced by its Ramsey sentence). We must note in any case that to obtain the full force of Kepler's laws (plus initial conditions) we should need to add to the infinite set of predictions concerning Venus (or any other infinite independent subset of the content of Kepler's laws) some statement (or finite set of statements) that renders all but finitely many of those predictions redundant.
(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 k irreducibles generates a theory with
distinct consequences; and an axiomatizable
theory that is logically equivalent to the conjunction
of denumerably many irreducible statements is, by
finitude, equivalent to the conjunction of some finite
subset of them, so that we return to the previous case.
A simple example is supplied by the calculus described
in 2.6, elementary logic with identity
as the only relation: for each positive i the
statement is irreducible, but the
conjunction of all these statements yields a theory
(`the number of objects is not finite', or
`the universe is infinite') that is not finitely
axiomatizable.
(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 2.4 was called informative
content. If we identify the content of a theory not
with `the class ...of statements that it excludes
or forbids' but with the class of maximal theories (or,
if you like, models or possible worlds) that it
excludes or forbids, then most of the difficulties
paraded above disappear. Some theories may exclude
only finitely many maximal theories; if such theories
exist, they will be just the same as those with finite
contents--in other words, they will be equivalent to
the conjunction of a finite number of irreducibles. But
usually, and in some calculi always, non-tautological
theories will succeed in excluding infinitely many
maximal theories. The main difficulty lies in
explaining in what sense maximal theories can be
thought of as independent of each other; that is, in
showing that if an axiomatizable theory has infinite
content this is not simply a result of duplication.
Plainly the sense required is not simple logical
independence, since any maximal theory is implied by
the conjunction of two others. And the hunch that
maximal theories can never duplicate each other, or get
in each other's way, and that any set of maximal
theories can constitute a content, is unfortunately
false. For example, in the calculus just mentioned, no
theory can exclude the maximal theory unless
it also excludes one (in fact, almost all) of the
(this was in effect proved at the end
of 2.8 above). Despite these worries, it
is quite easy to defend the view that an axiomatizable
theory can make an infinite number of independent
exclusions, and this suffices for the claim that it has
genuinely infinite content. Whether it entitles us to
claim that the theory is infinitely applicable as well
is quite a different matter. In section 3 it
will be suggested that it does not so entitle us.
(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 by implication in the premises, and being directly asserted in them'. In other words, an axiomatizable theory t does directly (if not transparently) assert in finitely many words everything that its infinitely many consequences take infinitely many words to assert. Mill expresses puzzlement that `a science, like geometry, can be all ``wrapt up'' in a few definitions and axioms' (loc. cit.). But we should not allow ourselves to be taken in here. Although virtually all theories `wrap up' infinitely many thoughts in the sense that we can find infinitely many thoughts within them, it is a capital mistake to suppose that a theory's content is synthesized from logically more primitive (weaker) components. As we have seen, in many cases there are no weakest components. (Aristotle's treatment of Zeno's paradox of Achilles and the tortoise invites comparison.) I am not of course defending the view that the understanding of a rich scientific theory is a straightforward business, and that some acquaintance with the theory's consequences is not essential to its understanding. As I have already noted, I incline to the view that understanding a theory fundamentally means understanding the problem situation it addresses, and how well it addresses it. It is possible to go further and to recognise that understanding may be enhanced when it is realized that the theory solves, or is unable to solve, some unexpected, some newly emerged problem. I am quite happy to admit that newly identified consequences may lead to a sharp improvement in the understanding of a theory. My purpose here is only to question the doctrine that much can be explained by the infinitude of a theory's content alone. It is such a flimsy matter that we could hardly expect it to yield substantial returns.
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