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On Popper's Definitions of Verisimilitude Author(S): Pavel Tichý Source: the British Journal for the Philosophy of Science, Vol

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On Popper's Definitions of Verisimilitude Author(S): Pavel Tichý Source: the British Journal for the Philosophy of Science, Vol 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 Accessed: 28/10/2010 16:12 Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. 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Sci. 25 (I974), 155-188 Printedin GreatBritain 155 Discussions ON POPPER'S DEFINITIONS OF VERISIMILITUDE1 Introduction. z Preliminaries. z Popper'sLogical Definition of Verisimilitude. 3 Popper's Probabilistic Definition of Verisimilitude. 4 Conclusion. Introduction. Sir Karl Popper's epistemologicalposition 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 entertain and 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 supplanting a false theory by a true one. As a rule, the new theoryis also false but somehowless so than its antecedent.Popper's epistemologythus calls for a discriminatingapproach to 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 attempt to 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 another false theory. The probabilisticdefinition is 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's definitionA has strictly less verisimilitudethan B. x Preliminaries. Consider a languagehaving (as is usual) a finite number of primitivedescriptive constants. Any finite set of (closed) sentences of the language will 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 proposed the 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 relativecontent A, 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) (by 1.3 and I.2) = Cn(A)-AT (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'sLogical Definition of Verisimilitude. Popper never explicitly states but obviously presupposes: Definition2.1. A, and B, (or A, and B.) are comparablejust in case one of 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 respectivelycomparable with 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 a B, c and BF, AF or AT, B, and B, 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 1 This 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. 4 The proposition shows that the definition of the falsity content of A (as the class of false 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. On Popper'sDefinitions of 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 contra- diction to the choice of b. Thus B, t A,. Now assume BF 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 Proposition now 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. that the actual time is Then B is false. AT Suppose 9-48.1 Moreover, 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'sProbabilistic Definition of 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. The measure the truthcontent A is Definition3.I.3 ctT(A) of of I--p(A ). The measure the content A is Definition3.2.3 ctF(A) of falsity of I--p(A, AT). Popper's probabilisticexplication of 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 verisimilitude1vs,(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 introduce several 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.
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