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Verisimilitude Redefined Author(S): Pavel Tichý Source: the British Journal for the Philosophy of Science, Vol

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Verisimilitude Redefined Author(S): Pavel Tichý Source: the British Journal for the Philosophy of Science, Vol The British Society for the Philosophy of Science Verisimilitude Redefined Author(s): Pavel Tichý Source: The British Journal for the Philosophy of Science, Vol. 27, No. 1 (Mar., 1976), pp. 25- 42 Published by: Oxford University Press on behalf of The British Society for the Philosophy of Science Stable URL: http://www.jstor.org/stable/686376 Accessed: 28/10/2010 16:10 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. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://www.jstor.org/action/showPublisher?publisherCode=oup. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. Oxford University Press and The British Society for the Philosophy of Science are collaborating with JSTOR to digitize, preserve and extend access to The British Journal for the Philosophy of Science. http://www.jstor.org Brit. J. Phil. Sci. 27 (1976), 25-42 Printedin GreatBritain 25 Verisimilitude Redefined by PAVEL TICHf z Examples. 2 IntuitiveBackground. 3 The Definition. 4 The ExamplesRevisited. 5 A LookBack on Popper'sApproach. Are some falsehoods less false than others? Tarski's definition of truth tells us in rigorous terms what it takes for a statement to be false. Can we have an equally rigorous definition of what it takes for one false statement to be closerto the truththan another? The notion of verisimilitude is bound to figure centrally in any adequate theory of scientific progress. For in the empirical sciences change invariably takes the form of one false theory being discarded in favour of another false theory. If such a development is to be meaningfully qualifiable as a step forward, it must make sense to say that the new theory, although also false, is closer to the truth than its discarded predecessor. The problem of verisimilitude was first explicitly raised by Sir Karl Popper. Popper has repeatedly emphasised the fundamental importance of the notion and the urgent need for its rigorous explication.' However, Popper's own attempts at giving an explication have failed.2 The aim of the present paper is to propose a viable definition of the concept. I EXAMPLES The intuition underlying the concept of verisimilitude is a remarkably distinct one. In order to elicit this intuition in the reader, I shall cite three very simple examples. (Two notational conventions: a homogenous string of quantifiers, say '(Vx1)... (Vx,,)' will be abbreviated to '(Vx... x,,)'; a dot in a formula will represent a left-hand bracket whose right-hand mate is to be imagined as far to the right as is consistent with other pairs of brackets.) Received 5 August 1975 1 See Popper [1962], Popper [1963], and Popper [1972]. 2 Popper has proposed two definitions of verisimilitude: one in terms of logical consequence and one in terms of logical probability. The former has been found vacuous independ- ently by David Miller and myself (see Miller [1974] and Tich1r [1974]). The latter has been shown inadequate in Tich [1974]. 26 Pavel Tichi EXAMPLE A. Consider a language La whose only extralogical primitive symbols are three independent sentences h, r, and w. Assume that all the three sentences are, as a matter of fact, true. The following are then four false theories in La: (AI) ~ h& r& w (A2) ~h&r&~w (A3) h & r (A4) h & r & w Theory (A ) gets everything wrong; its is the worst consistent theory one can propose in La. (A2) is also rather bad, but definitely better than (AI): it gets at least something right. (A3) shares all (A2)'s merits, but only some of its deficiencies: in one point where (A2) is wrong, (A3) suspends judgment, thus allowing for the right answer. It is thus plausible to say that (A3) is marginally better than (Az). (A4) is almost right; it clearly surpasses all the other theories. In a word, (AI)-(A4) are (in this order) increasingly better approximations to the truth. EXAMPLE B. Let LB be a first-order language (with identity) whose alpha- bet aB of extralogical primitive symbols consists of two independent one- place predicates P and Q, and whose universe of discourse is a denumerably infinite collection LB.Assume that in actual fact P and Q have two-element extensions which partially but not entirely overlap (thus having exactly one element in common). We shall write '(3~x)A' for the standard formula saying that exactly n values of x satisfy A. Thus for instance, '(3,x)Px' is = short for '(3xy) . ~ x = y & Px & Py & (Vz) . Pz D . z = xV z y'. The following are four false theories in LB: ~ (BI) (3,x)Px & (Vx) Qx (Bz) (32x)Px & (Vx) ~ Qx (B3) (3,x)Px & (31x)Qx & (Vx) D ~ Qx (B4) (32x)Px & (32x)Qx & (Vx) . Px = QQx. Theory (BI) is very bad. It ascribes P a one-element extension and Q an empty one. (B2) corrects the former error of (BI) but shares the latter. (B3) partakes in all the assets of (B2) and mitigates its errors: while (B2) made sets the number of elements at i. Thus Qempty, (B3) Q's (BI)-(B3) are increasingly better approximations to the truth. As for (B4), it corrects something which (B3) gets wrong, viz. the size of Q; on the other hand, it gets wrong something which (B3) gets right, viz. the number of Q's which are outside P. (B3) and (B4) thus seem equidistant from the truth. EXAMPLE C. Finally, consider a language Le whose alphabet cc consists of one two-place predicate R and two individual constants o and n, and whose universe to is again a denumerably infinite set. Assume that, as a VerisimilitudeRedefined 27 matter of fact, the extension of R effects an irreflexive linear ordering of te and that o and n are, respectively, the first and the fifth term of that ordering. Consider the formula (y) (Vxyz). [-x = y . Rxy V Ryx] & [Rxy & Ryz = Rxz] & [RxyD ~ Ryx]& Rox. In order to save space, let us write 'R*(x,y)' for 'Rxy & (Vz) . Rxz & Rzy' , The following are four false theories in Lc: (CI) y &o=n (C2) y & R*(o,n) (C3) y & (3x) . R*(o,x) & R*(x,n) (C4) y & (3xy) . R*(o,x) & R*(x,y) & R* (y,n). Again, (CI)-(C4) are increasingly better approximations to the truth. Each of the theories is right in affirming that R is a linear ordering and that o comes first in that ordering. Now the remaining, false, information they give is that n is, respectively, the first, second, third, and fourth term of the ordering. But to say, successively, of the fifth member of an ordering that it is the first, second, third, and fourth member, is clearly zeroing in on the truth. Consider, moreover, the following three theories: (C5) (Ci) V (C2) V (C3) V (C4) (C6) (C2)V (C3)V (C4) (C7) (C3)V (C4). Theory (C5) places n between the first and fourth member of the ordering (including the extremes), (C6) places it between the second and the fourth and (C7) between the third and the fourth. (C5)-(C6) are thus markedly less specific than (CI)-(C4), and the differences among them as to proximity to the truth are, accordingly, somewhat less striking. My intuition tells me nevertheless that (C6) is closer to the truth than (C5) and that (C7) is closer to the truth than (C6). My intuition is here in full agreement with Popper's. Popper has considered1 the statements 'It is now between 9.45 and 9.48' and 'It is now between 9.40 and 9-48' where 'between' is supposed to exclude the two bounds. Supposing that the actual time is 9-48, Popper says, 'we can ... assert that the first [statement] has a greater truthlikeness than the second'. It is readily seen that the way Popper's two statements are related to the truth is completely analogous to the way (C7) and (C6) or (C7) and (C5) are related to the truth in our example (n corresponding, of course, to Popper's 'now'). How do (C5)-(C7) compare with (CI)-(C4)? Both (C5) and (C6) seem x See Popper [1972], p. 56. 28 Pavel Tichj definitely better than (CI) and (C2). Both allow for the very bad estimates of n's position made by (CI) and (Cz), but all the other estimates they also allow for are significantly better. (C6) locates n roughly where (C3) does; but being, as it is, substantially vaguer about it than (C3), it seems marginally worse than (C3). But (C7) is definitely better than (C3); it allows for (C3)'s quite close guess, and, besides, for a better one. (C4), on the other hand, is patently better than (C7). Intuitively, therefore, the ordrer of increasing excellence is this: (CI), (C2), (C5), (C6), (C3), (C7), (C4).
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