The British Society for the Philosophy of Science
On Popper's Definitions of Verisimilitude
Author(s): Pavel Tichý
Source: The British Journal for the Philosophy of Science, Vol. 25, No. 2 (Jun., 1974), pp.
155-160
Published by: Oxford University Press on behalf of The British Society for the Philosophy of Science
Stable URL: http://www.jstor.org/stable/686819
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Brit. J. Phil. Sci. 25 (I974), 155-188 Printedin GreatBritain
155
Discussions
ON POPPER'S DEFINITIONS OF VERISIMILITUDE1
Introduction.
z Preliminaries.
z Popper'sLogicalDefinitionof Verisimilitude.
3
4
Popper's Probabilistic Definition of Verisimilitude.
Conclusion.
Introduction.
Sir Karl Popper'sepistemologicalposition is best characterisedas an optimistic
scepticism. It is a scepticism since it affirmsthat no non-trivial theory can be
justified and that more likely than not all the theories we entertainand use are
false. The position is optimistic in contending that in science we nevertheless
make progress:that we have a way of improvingon our false theories. Progress,
however, hardly ever consists in supplantinga false theory by a true one. As a
rule, the new theoryis alsofalse but somehowless so than its antecedent.Popper's
epistemologythus calls for a discriminatingapproachto false theories: he has to
assume that of two false theories, one can be preferableto the other in being
'closer to the truth' or 'more like the truth'.
In an attemptto legitimise this sort of talk Popper has proposedtwo rigorous
definitionsof verisimilitude,I shall call them logical and probabilistic.The aim
of this note is to show that for simple logical reasons,both are totally inadequate.
In Section x are given Popper's definitions of several auxiliarynotions. The
logical definitionof verisimilitudeis consideredin Section z. It is demonstrated
that on this definition a false theory can never enjoy more verisimilitudethan
anotherfalse theory. The probabilisticdefinitionis dealt with in Section 3. An
example of two theories A and B is given such that A is patently closer to the
truth than B, yet on Popper'sdefinitionA has strictlyless verisimilitudethan B.
x Preliminaries.
Consider a languagehaving (as is usual) a finite numberof primitivedescriptive
constants. Any finite set of (closed) sentences of the languagewill be called a
theory. In what follows, A, B, C,... are understood to be arbitrarytheories.
Cn(A) is the set of theorems of A, i.e., the set of logical consequences of A.
Furthermore,let T and F be the set of true and false sentences of the language
respectively.Popper has proposedthe following definitions.2
1 An earlier version of this paper was presented to the Philosophy Seminar of the University
of Otago in March 1973. The author benefited from conversations with Sir Karl Popper,
Alan Musgrave, and John Harris, and adopted a terminological suggestion made by
David Miller.
2The latest formulations of these definitions can be found in Professor Popper's [1972].
In what follows, all page references are to this book.
156 Pavel Tichj
Definition i.i.x The truth content A, of A is Cn(A) n T.
Definition1.2.2 The relativecontentA, B of A given B is Cn(A U B)--Cn(B).
DefinitionI.3.3 The falsity contentAF of A is the relative content of A given
AT, i.e., A, A T.
Definitions 1.1, 1.2, and 1.3 yield
Proposition 1.4. A, = Cn(A) n F.4
Proof. A, = A, A, = Cn(A U AT)-Cn(AT)
= Cn(A)-AT
(by 1.3 and
I.2)
(since A U A, = A and
Cn(AT) = A,)
= Cn(A)-(Cn(A) fr T)
(by i.r)
= Cn(A) n F
(since T= F).
z Popper'sLogicalDefinitionof Verisimilitude.
Popper never explicitly states but obviously presupposes:
just in case one of
Definition2.1. A, and B, (or A, and B.) are comparable
them is a (proper or improper) subclass of the other.
Now we can state Popper's logical definitionof verisimilitude:
Definition2.2.5 A has less verisimilitudethan B just in case (a) AT and A, are
respectivelycomparablewith BT and BF, and (b) either AT c B, and A, ? B,
or B q:A, and B, c A,.
Definitions 2.I and 2.2 yield immediately
Proposition2.3. A has less verisimilitudethan B just in case either AT
and BF,
AF or AT,
a
B,
B, and B, c A,.
Definition 2.2 is inadequateas explicationof verisimilitudein view of
Proposition2.4. If B is false then A does not have less verisimilitudethan B.
Proof. Since B is false, there is a false sentence, say f, in Cn(B). First assume
is how the concept of truth content is defined on p. 330. On p. 48 we are given a
slightly different definition, whereby the truth content of A is rather (C(A) nr T)-L,
where L is the set of tautologies or logically valid sentences. But I take this to be a mere
slip, since some statements on the same page are in conflict with this definition. At all
events, the difference is marginal and does not affect our ensuing considerations.
2 This is how the concept of relative content is defined on p. 332. On p. 49 the relative
content of A given B is characterised as the class of all sentences deducible from A with
the help of B. This might be construed as suggesting that the relative content of A given
B is simply Cn(A U B). However, from several subsequent remarks it transpires that this
is not what is intended.
3 See pp. 49, 51 and 332.
content of A (as the class of false
4 The proposition shows that the definition of the falsity
consequences of A), which is considered and rejected on p. 48 is in effect logically
equivalent to the definition actually proposed at the bottom of p. 49.
5 See p. 52. The signs 9 and c stand for set inclusion and proper set inclusion respectively.
1 This
On Popper'sDefinitionsof Verisimilitude 157
A, c BT. Then there is a sentence, say b, in BT-AT. But then (f . b) e B,. On
the other hand, (f. b) 0 A ,, since otherwise, by 1.4 and I.x, b e AT, in contradiction to the choice of b. Thus B, t A,. Now assumeBF c A,. Then there is
a sentence, say a, in AF-BF. But then (f = a) e A,. On the other hand,
(f = a) A p, since otherwise, by I.x and 1.4, a e AT, in contradiction to the
choice of a. Thus AT - BT. The Propositionnow follows by 2.3.
To illustrate Proposition 2.4, let A consist of the sole sentence 'It is now
between 9.40 and 9.48' and let B consist of the sole sentence 'It is now between
9.45 and 9-48', where 'between' is understood to exclude the two bounds.
Suppose that the actual time is 9-48.1 Then B is false. Moreover, AT c B,.
Yet A does not have less verisimilitudethan B on Definition 2.2. For clearly the
(only) member of B is in BF but not in Ap, thus BF t AF.2
3. Popper'sProbabilisticDefinitionof Verisimilitude.
Where A and B are theories, let p(A) be the logical probabilityof A and p(A, B)
the relativelogical probabilityof A given B. Popper has proposedthe following
definitions.
Definition3.I.3 The measurectT(A) of the truthcontentof A is I--p(A ).
Definition3.2.3 The measurectF(A) of thefalsity contentof A is I--p(A, AT).
Popper's probabilisticexplicationof truthlikenessis then in terms of ct, and
ct,. Popper offers, in fact, two alternativeexplications.They will be spoken of as
verisimilitude, and verisimilitude2.The definitionsare as follows.
Definition3.3.4 The verisimilitude1
vs,(A) of A is ctT(A)--ctF(A).
Definition3.4.4 The verisimilitude,vs2(A) of A is
(ctT(A)-ctp(A))/(2-ctT(A)-ctF(A)).
Both concepts are drasticallyat variancewith the intuitive notion of closeness
to the truth. Preparatoryto a justificationof this claim I shall introduceseveral
notationalconventions and prove an auxiliaryproposition.
Let a, b,...,
t,...
be arbitrary sentences of the language in question. In
what follows, a symbol standingfor a sentence will also be used to denote the set
whose only element is that sentence.
Proposition 3.5. If T = Cn(t) then aT = Cn(a v t).
Proof. Assume T = Cn(t) and consider an arbitrary sentence b. By 1.i,
1 See p. 56.
2
In private conversation Professor Popper suggested to the author that things might be
remedied if we forgot all about falsity contents and simplified Definition 2.2 to the
following: A has less verisimilitude than B just in case AT C BT. It is easy to show,
however, that on this definition a false theory A has less verisimilitude than another false
theory B just in case Cn(A) C Cn(B), i.e., just in case A is a logical consequence of B.
If this definition was adequate it would be child's play to increase the verisimilitude of
any false theory A: it would suffice to add to A an arbitrary sentence which does not
follow from it.
In a personal letter David Miller has informed the author that he independently
1 See pp.
obtained the results of Section 2.
51, 337.
4 See p. 334.
158
Pavel Tich3
be a iff b e Cn(a) n Cn(t). But by propositionallogic, b e Cn(a) n Cn(t) iff
b e Cn(a v t).
The inadequacyof 3.3 and 3.4 will now be demonstratedon a simple example.
Consider a rudimentaryweather-languageL containing no predicates and
only three primitivesentences, 'it is raining','it is windy' and 'it is warm'.Let us
abbreviatethem respectivelyas 'p', 'q', and 'r'. Moreover,assume all the three
sentences are, as a matter of fact, true. Then, writing t for p. q. r, we have
T== Cn(t).
The eight sentences p. q. r, p. q.
r,...,
p.
. q.
-
r will be spoken
,
of as constituents.The constituents are,pmutually,incompatible,
jointly exhaustive
and of equallogicalstrength.Hence the logical probabilityof each is I/8. As well
known, every consistent sentence a of L is logically equivalentto a disjunction
of constituentsthe (disjunctive)normalformof a. The followingfour propositions
clearly hold of any sentences a and b of L:
3.6. a is compatible with b just in case the normal forms of a and b have a
constituent in common.
3.7. a is true just in case a is compatiblewith t.
3.8. If a is incompatiblewith b then p(a v b) = p(a)+p(b).
3.9. The relative probabilityp(a, b) of a given b is p(a . b)/p(b).
Now it is easy to prove
Proposition 3.10. If a is false then ctT(a) = (7/8)-p(a)
and
ctF(a) = I-[p(a)l(p(a)+1/8)].
Proof. Let a be false. We have:
ctT(a) = I--p(aT)
I--p(a v t)
= I -[p(a)+p(t)]
(by 3.z)
(by 3.5)
(by 3.7 and 3.8),
ctF(a)
(by 3.2)
(by 3-5)
(by 3.9)
I--p(a, aT)
= I-p(a, a v t)
(a v t))/p(a v t)]
- I -[p(a.
= I -[p(a)l(p(a)+p(t))]
(by 3-7, 3.8, and propositionallogic).
But p(t) = 1/8. Which completes the proof.
From 3.zo, 3.1, and 3.2 it immediatelyfollows that the values of vs1 and vs, at
false sentences of L depend solely on the logical probabilitiesof the sentences.A
little reflectionrevealsthat this fact alone makesvs, and vs, unfit to explicatethe
intuitive notion of proximityto the truth. Since surely we want it to be possible
for one false theory to be closer to the truth than anotherfalse theory despite the
two theorieshavingthe same logicalprobability.If Popper'sproposalswere right
then in order to decide which one of two false theories is closer to the truth, no
factualknowledgewould be requiredover and above the knowledgethat the two
theories are indeed false. Which is clearlyabsurd.
To illustratethis point, let us consider a couple of examples.
and vs2at some false senThe following table gives the values of ct,, ct,,
vsl,
tences of L:
On Popper'sDefinitionsof Verisimilitude I59
qp.
q
p.q.
-r
-p. -
q. er
6/8
6/8
5/8
1/2
1/3
1/2
2/8
7/8
2/8
vS1
1/3
21/25
1/3
vs2
Now imagine that Jones and Smith, two prisonerssharing a windowless and
air-conditionedcell, are using L to discuss the weather.Jonestakes the view that
it is a dry, still day, with a low temperature.In other words, Jones's conjectureis
- p . -. q. - r. Smith disagrees.Although he also thinks that the temperature
is low, he (rightly) insists that it is raining and windy. In other words, Smith's
theory is p. q . ~ r.
It seems hardlydeniablethat Smith is by far nearerto the truththan Jones. He
is admittedlywrong on temperature,but he is dead right as far as rain and wind
are concerned.Jones,on the other hand, is wrong on three counts. He could not,
in fact, be fartherfrom the truth than he is (without contradictinghimself).
Thus one would expect Smith's theory to exceed Jones's in measure of truth
content and in verisimilitude.One would also expect Jones's theory to exceed
Smith's in measureof falsity content. Yet, as seen in the above table, each of the
functions ctT, ctF, vs1,and vs2takes the same value at Jones'stheory as it does at
Smith's.
But Popper'sfunctionsvs1and vs2not only fail to discriminatebetweentheories
which, like the two above, are vastly unlike in proximityto the truth. In many
cases the functions accord strictly greater verisimilitudeto a theory which is
patently fartherfrom the truth than anothertheory.
Let us alterslightly the above example. Imaginethat while Smith sticks to his
theoryp. q. - r, Jones has weakenedhis claim to - p. . q. Jones's theory is
now marginallybetter than before: while previouslyJones was positivelywrong
on temperature,this time he withholds judgement. But Jones's new theory is
surely not better enough to match, let alone exceed, Smith's in closeness to the
truth. Jones'sis still one of the lousiest and Smith's one of the best false theories,
as false theoriesgo. Smith is only wrong on one count, whereasJones is wrong on
two. One would certainly expect Jones's theory to exceed Smith's in falsity
content. Yet, the ct, of Jones'stheory is strictlyless than the ct, of Smith's. One
would also expect Smith's theory to have greater verisimilitude than Jones's.
of Smith's theory is strictlyless than the vs1of Jones's and similarly
Yet, the
for vs2. vsI
ctT
ct,
4 Conclusion.
To do justice to the intuitive notion of truthlikenessone must clearly make it
possiblefor a false theory to be closerto the truth than anotherfalse theory of the
same logical probability.For a simple languagewhich, like L, is based on propositionallogic only, this is easily done. The 'distance'between two constituents
can be naturallydefined as the number of primitive sentences negatedin one of
the constituentsbut not in the other. The verisimilitudeof an arbitrarysentence
a can then be defined as the arithmeticalmean of the distancesbetween the true
constituent t and the constituentsappearingin the disjunctivenormalform of a.
It is easily seen that such a definitionmeets all intuitive requirements.
i6o Pavel Tich?
Things, of course, get vastly more complicatedwhen we turn to the more
typical kind of theories, i.e., theories formulatedin a first-orderlanguage. But
the idea underlyingthe above definitionof 'distance'can be carriedover to firstorder theories if one employs, in lieu of disjunctivenormal forms, Hintikka's
distributivenormalforms for first-orderformulas.This, however,is a topic for a
separatearticle.
PAVEL TICHY
Universityof Otago,
Dunedin,New Zealand
REFERENCE
POPPER, K. R. [1972]:
Objective Knowledge.
POPPER'S DEFINITIONS OF 'VERISIMILITUDE"
I One of the majorproblemsPopperhas attackedis that of findingand making
intelligible a coherent view of critical common-senserealismwhich agrees with
the practices of science. He sees science as progressingand finding ever better
theories. And since for him the only significantprogresswould be that of getting
closerto the (absolute)truth, he is obviouslyled to the minorproblemof explaining what it would mean (at least in principle)to say that one theory is closer to
the truth than another, especiallyin the case when both theories are false. His
attemptsat explicatingthis particularconceptusuallyappearin his writingsunder
the heading of 'verisimilitude'.
For any two theoriesA and B let us write 'A <, B' if intuitivelyB is closer to
the truth than A. (This is not to imply that this intuitive concept has a unique
sense and won't somedaybe found to be ambiguous.But whetherthis is the case
is part of the problem.)Popper has given essentiallytwo differentformal definitions of '<T', which Miller2calls the qualitativeand the quantitativedefinitions.
I will denotethe firstby '< '. Miller3and Tichy4have independentlyshown that
neither of the formal definitions is faithful to Popper's intuitive notion of
verisimilitude.In particularthey provethat if A and B arefalse theories,then on
Popper's qualitativedefinitionneither can be closer to the truth than the other:
(I)
A 4,
B and B c 1 A.
The main purpose of this article is (i) to consider a mathematicallymore
generalproblem,that of comparingtheories relativeto an arbitrarycomparison
theory, (ii) to show the philosophicalrelevanceof this generalisationand (iii) to
shed light on the Miller-Tichy result (i) by obtaining some general results of
which (i) is a special case.
2 When we say that science finds ever better theories or that one theory is
better than another,implicit in such a statementis the assumptionthat there is
some criterion of comparison such as elegance, ease of calculation, degree of
falsifiability, agreement with a portion of currently accepted background knowverisiSI am deeply indebted to Pavel Tichy for bringing to my attention the problem of
2
militude and his negative results. I am also indebted to P. Tichf, D. Miller and A.
Musgrave for criticism of earlier drafts of this article.
Miller [1974].
4 Tich'
3 Ibid.
[i974].