The Critical Rationalist Vol. 02 No. 02 ISSN: 1393-3809 05-Jun-1997
References
Popperian Ideas on Progress and Rationality in Science
(First presented at the American Philosophical Association Eastern Division Meeting 1996, at the Symposium on Scientific Rationality and Progress in Memory of Paul Feyerabend and Karl Popper.)
(1) I got to know Paul Feyerabend well in 1961, when I was in Berkeley. (I also met Tom Kuhn then.) I loved his racy, carefree, and often surprisingly erudite talk and letters. But after Against Method we got rather stuck into confrontational postures (see for instance our argy-bargy in Progress and Rationality in Science, Radnitzky &\ Andersson 1978). Karl Popper I first encountered back in 1947. He had a decisive formative influence on my intellectual life. I remain a tremendous admirer of the ideas about science, especially about the growth and progress of science, which he published during his heyday, from 1934 down to about 1960. But I am critical of certain tendencies in his later philosophy, one of which I turn to now.
(2) For Popper, science is like music or poetry in that it requires inventiveness, there being no recipe or method for creating it, and also unlike music and poetry in that there are criteria of progress for it but not for them. What are these criteria? Popperians are generally agreed that corroborations have a dominant role in determining whether one scientific theory is better than another; but there have been large disagreements as to why, or in what sense, the better corroborated of two theories is the better theory. This is the main question I shall address.
(3) Through the 1960s and early 1970s Popper
believed that he had got hold of an ordinal scale for
truth-likeness or verisimilitude. Its lower bound was
given by the contradiction, and its upper bound by the
set T of all true statements. In place of this latter
concept he sometimes operated with the idea of the
target theory 
towards which a sequence of
scientific theories 
, 
, 
, each of
which may be false, is progressing (see for instance
his 1976, p. 155).
(4)
When is a theory more truth-like than theory
? Popper's original answer was deceptively simple
and persuasive. A theory's content is the totality of
its logical consequences; its true consequences
constitute its truth-content and its false
consequences, if any, its falsity-content. And Popper
proposed the following three requirements for theory
to have greater verisimilitude than theory :
first, 's truth-content includes 's; second,
's falsity-content, which may be zero, is included
in 's; third, at least one of these inclusions is
strict. These requirements are trivially met when
is true and strictly entails . But his idea of
increasing verisimilitude should apply to false
theories. For it was a main plank of his
anti-inductivist view of science that, at least since
Anaximander's explanation of the earth's supposed
stability in terms of there being no sufficient reason
for it to move in any particular direction because it
is equidistant from the heavenly bodies around it,
science has been advancing explanatory theories that
typically turned out to be false, and were often aimed
at erroneous explananda.
(5) He broached these ideas in 1960, at the first LMPS meeting, in Stanford. They were published in Chapter 10 of Conjectures and Refutations (1963). As is well known, a decade later a fatal defect in them was discovered independently by David Miller (1974b) and the late Pavel Tichy (1974). Soon afterwards Herbert Keuth (1976) and Hermann Vetter (1977) independently discovered another discouraging result, and I will begin with this.
(6) I used to believe that I had made one small contribution to the theory of verisimilitude. In my Science and Scepticism (1984) I made much use of the idea of two statements being incongruent counterparts of one another; that is, their consequences, though different, are in one-to-one correspondence. And I took it for granted that for one incongruent counterpart to be closer to the truth than the other, the ratio of true to false among its consequences must be higher than it is among those of the other. I was unaware that Keuth and Vetter had shown that for any false statement, the number of its true consequences equals the number of its false consequences. I tried to put matters right in Watkins (1987).
(7) Miller (1994, pp. 209f) has recently come up with a neat new way of obtaining this result as well as his and Tichy's previous result. Let T be a false theory and let f be a proposition in its falsity-content. Now consider the bi-conditional f-iff-x, where x is some proposition; f-iff-x is true when x is false and false when x is true. To every x in T's truth-content there will correspond an f-iff-x in T's falsity-content, and to every x in T's falsity-content there will correspond an x-iff-f in T's truth-content. This reproduces the Keuth-Vetter result.
(8) I may mention that my idea concerning the
comparative verisimilitude of incongruent counterparts
would have avoided this result if I had restricted the
comparison to their empirical content, as understood in
Science and Scepticism where it is equated with
the class of the theory's singular predictive
implications, or SPIs, as I call them. These are simply
the negations of what Popper called its potential
falsifiers. Thus if, to take the simplest case, the
theory were (x)(Fx 
Gx) where F and
G are observational predicates, then Fa 
Ga is a potential falsifier and Fa
Ga is the corresponding SPI. As I
characterized them, SPIs generally have a pebbly
character: if you have two separate pebbles you don't
create a new pebble by sticking them together, not even
if you use super-glue; and nor do you create a new SPI
by sticking together two SPIs, at least if their
predicates do not overlap. Thus if f is one SPI and
x is another, f-iff-x is not a SPI. And if T
and T' are incongruent counterparts with their SPIs
in one-to-one correspondence, it is entirely possible
that the ratio of true to false among the SPIs of T
is higher than it is among those of T'. Miller once
used the analogy of invited guests arriving trailing
gatecrashers behind them to depict the defect in
Popper's definition of verisimilitude
(Miller 1974a). Restricting the
consequences that are taken into account to SPIs should
exclude `gatecrashers'.
(9) Now to Miller's new way of obtaining his and
Tichy's old result. The question is whether, given a
sequence , ...of theories which, though
false, are progressing towards the true target theory
, could have more verisimilitude than
in Popper's (1963) sense. Assume
that and satisfy Popper's first and second
requirements: 's truth-content includes 's
and 's falsity-content includes 's. Can they
satisfy his third requirement, that at least one of the
inclusions is strict? Let f now be a statement in
's falsity-content; by the second requirement f
will also be in . Let x be any consequence of
. Suppose first that x is false. Then by the
second requirement, again, this false x must be in
. Suppose next that x is true; then f-iff-x
is in 's falsity-content and again must be in
. But since already contains both f and
f-iff-x, it will also contain this true x. Thus
every x, whether true or false, in is also in
; has no excess content, and hence no excess
truth-content, over . Popper's third requirement
cannot be met.
(10) Popper's critics did not attach much importance to his concept of verisimilitude in its heyday; but after its debacle in 1974, news of which spread rapidly, some of them came to see it as the central plank of his philosophy of science, the whole system being brought down by its collapse. Popper's estimate of its importance changed in the opposite way. When he first introduced this idea he had indeed made it a, if not the, central plank of his philosophy of science, saying that we simply cannot do without something like it (1963 p. 232) and equating scientific progress with increasing verisimilitude. After the debacle his estimate flipped over. This shows up clearly in Realism and the Aim of Science (Popper 1982). In the main text, work on which had come to a stop in 1962, he had said that to describe one theory as better than, or superior to, another is to claim that it comes nearer to the truth (p. 25); and he even suggested that the problem of induction can be solved by replacing truth with `better approximation to the truth' (p. 67). But in his new Introduction, dated 1982, he remarked parenthetically that the idea of verisimilitude was not an essential part of his theory (p. xxxvii).
(11) Because attention has mainly focused on defects in Popper's definition of verisimilitude, not much attention has been paid to the role he was giving to verisimilitude in his later philosophy of science. That is what I now turn to. Various attempts have been made to replace his definition by a viable one, for instance Kuipers (1982, 1987a), Niiniluoto (1987), Oddie (1986), and Schurz & Weingartner (1987). I have not attempted to assess their comparative merits, partly because of the escalating technicality of the discussion. But I now assume for argument's sake that an adequate definition exists; and I now ask: how satisfactory would Popper's equation of the aim of science with ever-increasing verisimilitude have been if the latter had been adequately defined.
(12) What was Popper taking as the aim for science before he hit on the idea of verisimilitude? Well, from Logik der Forschung (1934) to `The Aim of Science' (1957) he was saying that science aims at ever better explanations, and that choices between competing explanatory theories are controlled by corroborations. He said some very interesting things about how, in the best case, a theory should stand to its predecessor(s) to constitute a better explanation. The bottom line was that it should be more corroborable, and go on to become better corroborated.
(13) Such a theory will not be certainly true or
probably true, but it will be possibly true in the
sense that it is well tested and, as yet, unfalsified.
So I proposed in Science and Scepticism that so
far as truth is concerned science aims at theories that
are possibly true. This brought Alan Musgrave's wrath
upon me: science aims at truth, not at possible
truth (see e.g. his 1989,
pp. 302-3). Miller agrees with Musgrave here. Now I
of course accept that the idea of truth is a regulative
ideal for science: truth is what science aspires after.
But to aspire after X is not equal to aiming at X. A
schoolboy who dreams of being a military hero does not
yet have military aims. If one is to aim at X, and
pursue one's aim rationally, one needs to be able to
monitor the success or failure of one's attempts to
achieve X. Are Popperians entitled to claim that one
could do so if X were simply truth? Here is a
simplified version of what, for us, would be a paradigm
of scientific progress. Within some problem-situation
a powerful new theory is advanced. It is tested
and for a time it only wins corroborations. But then a
more corroborable theory is advanced. Crucial
experiments between it and are performed, and
they go in favour of . The splendid has
fallen in battle. Later, the pattern is repeated, with
being refuted and superseded by the more
corroborable 
. Was science fulfilling the aim of
truth in this admirable progression? Not with ,
which turned out to be false, nor with which
suffered the same fate. Perhaps this aim was fulfilled
with ? Well, we may learn that it was not but we'll
never learn that it was.
(14)
But if X were ever better corroborated theories,
science could monitor very well the success or failure
of its attempts to achieve X. In our idealized
progression the latest theory's pattern of
corroborations dominates its predecessors', with every
test-result that corroborates them also corroborating
it, while some test-results which corroborate it are
either neutral to earlier theories, being in an area
where they are silent, or discorroborate them. Of
course, corroboration situations may be messy; but in
this clearcut situation Popper's pre-1960s vintage
methodology said that it is rational provisionally to
accept as the currently best corroborated theory in
its field.
(15) This was the cue, back in those
pre-verisimilitude days, for critics of this
purportedly non-inductivist philosophy to ask
why the best corroborated theory is the best
theory. As Feyerabend might have put it, what's so
great about corroboration? If Popperian corroboration
is free of any tincture of inductivism, if a
corroboration-appraisal merely compares the past
performances of the theories under test, of what
interest is it, except to historians, to know that
is better corroborated than ? Perhaps
will perform better in the future.
(16) In Science and Scepticism I attempted to answer this question without bringing in verisimilitude and in a way that would preserve the non-inductive character of this philosophy of science. I will come back to that later. In the meanwhile let us ask what use Popper made, in the 1960s and 70s, of his new theory of verisimilitude in his account of progress and rationality in science. The short answer is that he used it to turn what many saw as a pessimistic philosophy, in which the truth is permanently hidden, into an optimistic philosophy in which we can know, or at least have reason to believe, that we are making progress with respect to truth. Corroboration, he said, though not a measure, is an indicator of verisimilitude (1972 p. 103, his emphasis). An old inductivist view was that if, say, twenty pairs of experimental values are found to fit a simple linear function y = f(x), then this function very probably represents a law of nature since it would otherwise be extremely improbable that just these values would have obtained. Hermann Weyl criticised this on the ground that all sorts of other mathematical functions could be defined which these values will equally satisfy; and in The Logic of Scientific Discovery Popper went along with that (1959 pp. 139-140, and *n1). But after the introduction of verisimilitude his position changed. He now declared that a claim like, `It is highly improbable that Einstein's theory would make predictions that are precise and correct unless it were true' becomes valid when `true' is replaced by `high degree of verisimilitude' (1974 p. 1192). He also wrote: `If two competing theories have been criticized and tested as thoroughly as we could manage, with the result that the degree of corroboration of one of them is greater than that of the other, we will, in general, have reason to believe that the first is a better approximation to the truth than the second' (Popper 1982, p. 58, his emphasis). In short, corroboration-appraisals provide some justification for the corresponding verisimilitude-appraisals.
(17) It seems clear that an inductive element has
been let in here. Consider the following scenario.
and respectively entail conflicting
singular predictive implications SPI 
and SPI 
;
these are relevant to some urgent technological
problem, say to prevent a nuclear catastrophe. There
has as yet been no test on the two theories with
respect to predictions of these kinds, but the theories
have been severely tested in other places, and
has emerged as unambiguously better corroborated than
. If that gives us reason to believe that is
nearer the truth than , then we surely have some
reason to prefer SPI to SPI . Let us use
`reliable', in analogy with `desirable' and
`preferable', to mean `worthy of being relied upon'.
Popper once asked rhetorically: `But do I not really
draw inductive conclusions from past performance to
future performance?' (1982 p. 66).
The answer seems clear. To proceed from evidence about
past instances to a categorical conclusion about the
next instance is of course an inductive inference; and
Popper insisted that it would still be inductive if it
proceeded only to a probabilistic conclusion. To which
we may add that it would still be inductive if it
proceeded to such a conclusion in several steps. In the
present case, we proceed from evidence as to how
and have performed under test in the past via a
corroboration-appraisal to a verisimilitude-appraisal
and thence to a conclusion about their relative
reliability in the future. Inductivists may exclaim,
`So he's one of us after all! Good for him.' But I find
it sad when a philosophy of science whose proud claim
was to have dispensed with induction covertly reneges
on that claim.
(18) Is there any way in which Popperian ideas of rational choice between competing scientific theories, and of theoretical progress in science, can be upheld without a resort to some form of inductivism? I think there is. Popper had the valuable idea, which I exploited in Science and Scepticism, of separating justification of preferences for statements (hypotheses, theories) from justifications of the statements themselves. I will suggest that he misused this distinction but that it can be used to solve our problem.
(19)
Justificationists usually see justification as allowing
of degrees, with `verification', `proof', etc. reserved
for limiting cases; they would say that to justify a
statement you need arguments which go some way towards
establishing it as true, or at least as close to the
truth; actually to establish its truth, if that were
possible, would justify it conclusively. And with
respect to competing hypotheses they would say that to
justify hypothesis a against hypothesis b you need
arguments for a being more probable, or perhaps more
truthlike, than b. It seems obvious that the
justification of a preference for a over b, whether
these are statements or entities of some other kind,
must be with respect to some property (or complex of
properties), call it 
. You are free to prefer
lemonade to champagne; but if you seek to
justify your unusual preference you must bring
in some property, say keeping a clear head, with
respect to which lemonade tends to perform better than
champagne. If the foregoing elucidations are accepted,
it follows that to justify statement a is the very
same thing as to justify a preference for a with
respect to truth. One may justify a preference for a
with respect to without justifying a only if
is not truth, or some relative of truth such as
verisimilitude.
(20) In Realism and the Aim of Science (pp. 19-20) Popper distinguished between three problems: that of adjudicating between competing scientific theories, that of justifying scientific theories, and that of showing one scientific theory to be preferable to another. Whereas justificationist philosophers subordinate adjudication and preference to justification, Popper claimed to take an attitude to the problem of justification that was `as unambiguously negative as that of any irrationalist or sceptic'. His idea was to leave out justification and solve the problem of adjudication via the problem of preference. But now comes a big letdown: he added that by a theory being preferable to another he meant that we have reasons to think it a closer approximation to the truth (p. 20, his emphasis). At another place, after saying that there is a world of difference between justifying a preference and justifying a theory, he immediately added: `To justify a theory is to show that it is true. But we may justify a preference, even for a false theory, if we can show that of all the competing theories it appears to come nearer to the truth than any of the others' (1994 p. 138). His equation of justification with conclusive justification is a bit rich, coming from someone who spent much time attacking probabilistic versions of justificationism, like those of Keynes, Reichenbach and Carnap, and who insisted that the problem of justifying inductive conclusions remains the same if one puts `probable' in front of `conclusions' (1972 p. 4). Once it is accepted that justification allows of degrees, it seems clear that to justify a preference for a theory with respect to verisimilitude would tend to justify the theory itself. His later philosophy was tainted by justificationism as well as by inductivism.
(21)
A quick word now about the view of scientific progress
and rationality taken in my Science and Scepticism.
Call a property 
of a theory contingent or inherent
according to whether or not possession of depends on
factors external to the theory. Thus the consistency of
a consistent and falsified theory is an inherent
property, while its falsity is a contingent property.
And a theory is inherently -er than another if degrees
of -ness depend only on features of the theories and
are independent of factors external to them. Thus
may be inherently simpler, say in a "paucity of
parameters" sense, than . But if and
are neither tautological nor self-contradictory
but regular scientific theories, then 's being
closer to the truth than would be a contingent
property.
(22)
At one extreme, a proposed aim for science might erect
one contingent property, such as being verified, into
the sole desideratum. The aim proposed in Science
and Scepticism goes as far as it can in the other
direction. All but one of the properties of theories
which it holds up as desiderata are inherent
properties: its B-pole, as I call it, says that science
should aim at theories that are ever deeper, more
unified, more predictively powerful and exact. The
great advantage of such inherent properties in the
present context is, of course, that they are outside
the reach of inductive scepticism. Presented with two
theories suitably formulated in a language for whose
predicates there is an agreed partition into
observational and theoretical, logical examination
should enable one to tell which is -er when is one
of these properties. It turned out that the above four
properties can be collapsed into two: deeper, and
wider. Suppose that theory has been found to be
deeper and wider than , its only serious rival. In
that case it will typically happen both that some of
's testable content diverges, if only slightly,
from 's and also that has excess testable
content over . This brings me to this aim's
A-pole. For to be accepted as an advance over
there needs to have been at least one test on it
in areas of divergent or excess content, and its
pattern of corroborations should dominate 's in
the way mentioned earlier, with every test-result that
corroborates also corroborating it, and some
test-results which corroborate it either neutral to or
discorroborating .
(23)
I claimed this aim to be the optimum aim for science;
it contains what can be retained of the Bacon-Descartes
dream for science when this is de-utopianized and
rendered feasible. Like Popper's philosophy of science,
this revised version gives corroborations a decisive
role. But its way of justifying a preference for the
best corroborated theory in its field is not like the
way he started talking in the 1960s and early 1970s; it
does not treat corroborations as indicators of
verisimilitude, thereby surreptitiously turning a
justification for a preference for a theory into a
justification of the theory. In suitable cases it
justifies a preference for the better corroborated
over on the ground that is better
than with respect to the optimum aim of science,
where the latter does not require the thus preferred
theories to be certainly true, or in some sense
probably true, but only possibly true in the old
Popperian sense of having been severely tested and
surviving so far.
References
Popperian Ideas on Progress and Rationality in Science
The Critical Rationalist Vol. 02 No. 02 ISSN: 1393-3809 05-Jun-1997
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TCR Issue Timestamp: Fri Jun 6 16:34:01 GMT 1997