On Popper's Notion of Verisimilitude

Keiichiro KAMINO Osaka City University

1. Popper's Notion of Verisimilitude 2. Criticism on Popper's Theory of Verisimilitude 3. Suggested Ways of Defining the Notion 4. Popper's Rejoinders 5. Concluding Remarks

The purpose of this essay is in the first place to make a survey of the problem situation in which the notion of verisimilitude is now found, and then try to clarify what Popper intends to do with the notion. Therefore, I shall sketch Popper's general epistemological position in so far as it is relevant to the present discussion, and also mention Popper's original proposal for the definition of the notion. Secondly, in order to make the situation clear, I shall introduce some of the criticisms raised against Popper's original proposal and also show what I think is the general trend in the recent attempts to define the notion. Finally, I shall examine Popper's reaction against those criticisms and new proposals which were produced by Tichy , Miller, and others, to try to see Popper's real intention concerning the notion of verisimiltude, and to find out a way which hopefully will lead to a better problem situation agreeable from Popper's point of view, which is based on critical rationalism.

1. Popper's Notion of Verisimilitude

1.1 Popper's epistemological point of view

In order.for us to discuss on Popper's notion of verisimilitudein some detail, it would be necessary to characterise his general philosophical position so that all arguments will not drift astray. On the one hand Popper is surely sceptical with regard to the possibility of our attaining scientific truths. For according to Popper, scientifictheories are not the digest of observations; they are inventions, or conjectures boldly put forward for trial to be eliminated if they clash with observations. And he contends even that we can never have sufficiently good arguments in the empirical sciensesfor claiming that we have actually reached the truth. All theories are hypothetical, and all may be overthrown. Indeed, any attempt logically to justify a scientific statement as ture, would lead to an infinite regress. We can never have sufficient reasons for holding any scientific

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statement to be true, even if the statement be in fact true. Popper's rejection of the quest for justification has some affinity to the sceptical claim, which is very pessimistic with respect to the possibility of knowledge. In fact, he calls his own position dynamic scepticism, although he identifies this with 'critical inquiry'. On the other hand, however, he admits the possibility of the growth of know- ledge. He is always optimistic with the possibility of knowledge. Indeed, he is still optimistic, even when his theory of verisimilitude is exposedto severe attacks. With his theory of knowledge,however, one thing should be kept in our mind. The traditional view on knowledge claims that scientia is knowledgeof universal truths which are true of neccesity, and that knowledge can only be obtained by demon stration (though not necessarily logical demonstration). In the long history of Western thought since Plato and Aristotle, knowledge has been contrasted with belief or opinion, whose attribute is not 'true' but 'probable'. Knowledgeought to be true and to have certainty; opinion is more or less probable and subjective. In view of this, I contend that Popper stands on a very unique position. Certainly he claims that all scientifictheories are hypothetical, and strips off their garments of certainty. This naturally sounds as if he takes a kind of probabilism. It is clear, however,that he stands firmly on the traditional view that scientiais demonstrative, when he tries to demarcate scientia against pseudo-sciences. Because in trying to do so, he relies upon the power of logic, i.e., modustollens. I shall call Popper's position 'optimisticscepticism. This nice expressionI borrowed from P. Tichy.

1.2 Why Popper needs the notion of verisimilitude Popper's optimistic scepticism, however, does not claim that the truth itself is just an illusion. Some such description would indeed be a misleading one to express the nature of Popper's theory of knowledge. Popper maintains that even if there is no general criterion of truth for scientific theories, the notion of truth is logically legitimate. This, as he himself says, is very opposite to the positivistic position as was propounded by Wittgenstein that a concept is vacuous if there is no criterion for its application. The positivistic conceptionis as is known refuted by the modern development of logic, and especially by Tarski's theory of truth, which contains the theorem: for sufficientlyrich languages there can be no criterion of truth. (Popper, O.K., p. 321.) What then are we doing, when we say that we are seeking for truth We say, for instance, that we have come nearer to the truth, or that some theory Tl is superseded by some new theory T2, because T2 is more like the truth than T1. These are somehow intuitive assertions, of course. Can we, however, give good reasons for these propositions at least in some cases. If the notion of truth is dubious, so will be the notion of a better approach to the truth, or of a nearness to the truth or of a greater verisimilitude. Fortunately, however, the legitimacy of the notion of

-2- No. 1 On Popper's Notion of Verisimilitude 3 truth has been established by Tarski. Can we then hope that we may say the notion of verisimilitudeis also legitimate? I hope the answer is in the affirmative. In advocating the notion of verisimilitiude,what does Popper take to be its advant ages or utilities? First let us see how it explicates our notion of scientificinvestiga tion. The task of scienceis, in a sense, to cover by hits as much as possibleof the true statements, by the method of proposing theories or conjectures which seems to us promising, and as little as possible of the false conjectures. It is therefore very important that we conjecture true theories; but truth is not the only important properties of our conjectural theories. We are not particularly interested in pro- posing trivialities or tautologies; as "all tables are table" will not be a scientific truth. On the contrary, we know for instance that Newtonian physics is false. We know that few of scientifictheories we discover are true, or perhaps none is. Even so, we think that it is a much better 'approximation' to the truth, because it contains a number of interesting and informative true consequencses; the truth content of Newtonian physics is very great indeed. The fact is that we are not simply looking for truth, but are looking for interest- ing and enlightening truth. The introduction of the concept of versimilitude, there- fore, brings us a considerable advantage over the simple formulation that the aim of science is truth. The notion of verisimilitude will free scientific theories from the search of trivial truths, and at the same time, will explain why we are interested in some fales theories such as Newtonian physics or Einstein's relativity theories which have a great truth content. We admit false theories or statements if they are not ' too false' (if they have not too large falsity content) and contain a great truth content. Popper also adds that the search for verisimilitude is a clearer and a more realistic aim than the search for truth, and that we can have strong and reasonably good arguments for claiming that we may have made progress towards the truth. Moreover, he claims, we can explain the method of science, and much of the history of science, as the rational procedure for getting nearer to truth. (cf. ibid., pp. 57-8)

1.3 Popper's definition of verisimilitude Popper's theory of verisimilitude was first presented in Popper (1962) and (1963). It has since been expanded in Popper (1972). Popper's logical notion of verisimili- tude is a combination of two notions, both of which were originally introduced by Tarski; one is the notion of truth and the other is the notion of the logical con- tent of a statement, i.e., its consequence class. Now the content or the strength of a deductive theory can be determined by the size of its consequence class. The greater the consequence class of a theory is, the stronger it is. In other words, the comparability of theory will be reduced to set-theoretical inclusion relation. The size of the consequence class, however,

-3- 4 K. KAMINO Vol 6 does not establish the superiority of theory, or so Popper thought in the sixties. In assessing scientific theories, we are interested in the increase of true con- sequences, of course. Thus Popper first tried to define both the truth content and the falsity content of theory. Thus Popper said: the set A of all true statements following from any given statement a is called the truth content of a. It is a deductive system., in symbols: A=Cn(A)=Cn(a), and vice versa,

(to every statement say a there corresponds a finitely axiomatizable system say A) therefore,

A=A•¿T=Cn(a•Ét).

To define the falsity content of A, he drew upon the idea of the relative content A, given B, or A, B.

A,B=Cn(A,B)=Cn(A•¾B)-Cn(B)=Cn(a&b)-Cn(b)

The falsity content, then, AF=A,AT=Cn(A+AT)-Cn(AT)=Cn(A)-Cn(AT).

Thus, for the notion of verisimilitude, he said: a theory T, has less versimilitude than a theory TQ if and only if (a) their truth contents and falsity conetents (or their measure) are comparable, and either (b) the truth content but not the falsity content of Tl is smaller than that of T2, or (c) the truth content of Tl is not greater than that of T2, but its falsity content is greater. This proposal is only intuitive and qualitative. Above all, it is rather restricted, for it is effective noly if one truth content (or falsity content) properly includes another in the usual set-theoretical sense. Poppper was of cource aware of this shortcomings, and therefore his quantitative theory was introduced. He produced two definitions of verisimilitude. And, using the notions of logical probability, Popper offered the following definitions: The measure content ct(A) of the truth content of A is 1-p(A), and the measure content ct(A) of the falsity content of A is 1-p(A, A).. And then his two, quantitative definitions of verisimilitude are:

the verisimilitudelA: ƒÒsl (A) of A is ctT(A)-ctF(A) the verisimilitude2: ƒÒs2(A) of A is ctT(A)-ctF(A)/(2-ctT(A)-ctF(A)). In his (1966), Popper could show that if B exceeds A in content then it exceeds also in truth content. So, the theory with the greater content will also be the one -4- No.1 On Popper's Notion of Verisimilitude 6 with the greater versimilitude unless its falsity content is also greater. But unfortunately it turned out that if B exceeds A in content, then either the former exceeds the latter in falsity content or they are both true (of. Miller, 1974). Amongst false theories both truth content and falisty content increase strictly monotonously with content.

1.4 On the misunderstandings of the notion of verisimilitude Popper's defence of the legitimacy of the notion of verisimilitude has sometimes been grossly misunderstood. Therefore, before I introduce the objections raised by several philosophers of science against Popper's notion of verisimilitude, I should like to quote what Popper says about these misunderstandings (cf. Popper, O.K., p. 58). He says: "In order to avoid these misunderstandingsit is advisable to keep in mind my view that not only are all theories conjectural, but also all appraisals of theories, including comparisons of theories from the point of view of their verisimilitude. It is strange that this point, which is all important for my theory of science, has been misunderstood." I hope that all of us will keep this in mind: whether or not Popper's definitons of verisimilitude are successful, he takes the appraisal of theories in terms of verisimilitude to be conjectural. This implies, in the first place, that all appraisals of theories are appraisals of the status of their critical discussion: we have to criticise not only scientific theories themselves, but also the criticisms and the appraisals advocated for those theories under investigation. From this it follows that the clarity is an intellectual value in philosophical discussion. Without clarity we can not make any critical discussion. It should, however, be noted that the exactness for the sake of exactness will not be much appreciated, especially not when that kind of exactness declines to trivialities or superficialities devoid of philosophical significance. In fact, Popper has never been very much interested in giving definitions only to be (unneccessarily) exact. Secondly, Popper's point of view, it should be, also implies the rejection of the supposition of the ultimate true theory and of the fixed, or closed language to express it. Such supposition does not conform with Popper's idea of the cyclic growth of objective knowledge. We shall be back to this point later.

2. Objections to Poppers' proposal

2.1 Objections appeared in 1974 (Tichy, Miller, Harris) In B.J.P.S., vol. 25, no. 2, there appeared several objections against Popper's definitions of verisimilitude, and they arrived at more or less the same result. For instance, Miller?(1974)reached to the conclusion that no two distinct axiomatisable

-5- 6 K. KAMINO Vol. 6 theories may be compared for verisimilitude unless they are both true. In other words, no false theories are comparable by verisimilitude, either qualitatively or quantitatively, in the way Popper suggested. But if two theories are both true, we shall hardly need the sophistication of verisimilitude to tell us which of them we would in principle prefer. For one of them will be logically stronger than the other, so clearly an improvement. Harris (1974) proved that one can not increase the truth content of a false theory by conjoining a logically independent sentence without also increasing the falsity content and vice versa, whether or not the conjoined sentence is true. (Harris (1974),p. 163). In his (1974), Tichygave an simple example to show that the inadequay of Popper's definitionsof verisimilitude, vs1 as well as vs2. His counterexample to Popper's definition is based on a rudimentary weather language L which contains only three primitive sentences and no predicates. On this simple example, he demonstated that the value of vs1,and vs2at false sentencesof L dependssolely upon the logical probabilities of the sentences. And he argued that this fact alone makes vs1 and vs2 unfit to explicate the intuitive notion of proximity to the truth, since we surely want it to be possiblefor one false theory to be closerto the truth than another false theory despite the two theories having the same logical probability. For if Popper's proposalswere right, then in order to decide which one of two false theories is closer to truth, no factual knowledgewould be required over and above the knowledgethat the two theories are indeed false (Tichy, 1974, p. 158) As he showed, some conjectures which are by far nearer to the truth than another can have the same degree of verisimilitude with the latter. Moreover, Popper's vs1 and vs2 not only fail to discriminate between theories which are vastly unlike in proximity to truth, in many cases they accord strictly greater verisimilitude to a theory which is patently far from the truth than another theory (ibid. p. 159). When claiming the comparability of the contents of Newton's theory (N) and Einstein's (E), Popper has at least two things in mind: "(a) to every question to which Newton'stheory has an answer, Einstein has an answer which is at least as precise; this makes (the measure of) the content, in a slightly wider sense than Tarski's, of N less than or equal to that of E; (b) there are questionsto which Einstein's theory E can give a (non-tautological)answer while Newton's theory N does not; this makes the content of N definitelysmaller than that of E". (Popper, O.K. p 52.) Thus in this way we can compare intuitvely the contents of .these two theories, and Einstein's theory E has the greater content. Miller takes up this suggestion seriously, and tries to examine if it works. He first distinguishesthe precisionof a theory from the accuracy of a theory. He says that 'accuracy' might indeed be a factor worth consideringin judgement of truthlikeness, but hardly in judgement of content. "Accuracy applies to obser

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vations singly, and presupposes the existence of a true value; precision infects the observational procedure as much as the individual observation, and takes it into account such aspects as refinement of instrumentation and scatter of results (Miller 1975a, p. 208) If the preciseness of E supersedes that of E, and we can show that it is the case, the physical theories such as E and N may be compared for content by ignoring numerical differences in the predictions they make about certain quantities. Can we show that E answers at least as many questions as N does because every consequence of N is itself a consequence of B or is contradicted by some consequence of E? Miller's answer is in the negative. In order for E to be comparable with N in this way, it has to be complete, which of course is not. We can always find a consequence of N which neither follows from E nor contradicts it. After having established this result, Miller solicits us to return to his earlier proposal that we concern ourselves only with the quantitative consequences of the theories we are trying to compare. But the conclusion he reached is that even the numerical consequences of conflicting theories cannot be compared, and that all the obvious ways of comparing them in fact break down (Miller, 1975 p. 166). This conclusion boils down to the imcomparability of their truth contents (and of falsity contents), and therefore of their verisimilitudes.

I shall introduce his argument in a sketchy way. Suppose we have two con

flicting numerical theories. Then Miller can show that there will always be quan

tities evaluated by one but not by the other. In order to illustrate this point, he

produced a simple example. He says: let ƒ¦ and ƒ³ be two physical constants. Consider the theories A:ƒ¦=8 and B:ƒ³=7. Then the constant (8-ƒ¦).ƒ³ is given

the value 0 by A, but no value at all by B. Moreover, we cannot evade this

conclusion by requiring all our constants (or, more generally quantities) to be

primitive; that is, not calculable within the theory in terms of the other quantities. In this example, theory A gives values for some quantities for which theory B

gives no value at all From this it may be asserted that although A and B give the appearance of answering the same question, they cannot be regarded so; the

appearance is not a reality. Miller further argues that even if we could suppose that A and B as answering the same quantitative questions, a simple example will be given to show that the constants that A predicts truly cannot all be constants

B predicts truly, unless B always predicts truly. And thus, he draws from this the following conclusion: no false quantitative theory can, under the obvious variant of

Popper's original recommendation, be closer to the truth than another (Miller , 1975, pp. 166-7). Miller's result may be further elaborated. In any case it will bring in the following consequences.

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(1) Suppose that we have two false competing theories A, B, which predict more than one numerical constants. Now if A be the more accurate theory than B with respect to some constants ƒ³,ƒ¦, then we can always define new constants with respect to which the ordering of the accuracy will be reversed. (2) This, i.e. (1), will be further expanded to the cases where two theories predict the values of more than one variable parameters. In view of these results, Miller is naturally very sceptical with the possibility of evaluating the verisimilitude of theory in the way Popper originally suggested: perhaps there does not exist any solution to the logical problem of verisimilitude.

3. Suggeted ways of defining the notion When in his contribution to the 1960 international congress for Logic, Meth odology, and Philosophy of Science at Stanford, Popper proposed the idea of verisimilitude for theories, it did not draw much serious attentions of the philosophers outside the Popperian school. Although Popper himself further developed this idea and produced not only the qualitative but also quantitative definitions of verisimilitude (Popper. 1966), in the sixties the main topics were the Carnap-Popper controversy over induction or the Kuhn-Popperian controversy over the rational growth of knowledge. And people thought that the notion of versimilitidue would be suspect: if you can not know the truth, how can you ever measure the distance of theories from the truth? In the seventies,the situation has changed. Several philosophers who belong to the anti-Popperian school began to discuss on the notion of verisimiltude. Philosophers such as Hilpinen, Tuomela, Niiniluoto, who belong to the Carnap Hintikka school, showed their interests in the notion. If one see "Synthese, vol. 38, no. 2 which is the special issue for the discussion of this notion, you will find, among the names of contributers, such names as R. Tuomela, I. Niiniluoto, P. Mott, W. Krajewski as well as P. Tichy and D. Miller. The stimilus which caused this situation, was not only the news that Tichy (1974),Harris (1974),Miller (1974, 1974a, 1975) had refuted Popper's original proposal for the comparative definition of verisimilitude, but also the realisation that the explication of this notion is important for all supporters of the 'critical' scientific realism. In spite of such negative results as were produced by Miller, Tichy, and Harris, positive solutions have been attempted and discussed in a number of recent papers. Tichy, (1974) contains both the criticism of Popper's definition of verisimilitude and the suggestion for a new solution. And he further developed his idea in his (1976) and his (1978). According to Niiniluoto (1978), R. Hilpinen (1976) presented an approach to truthlikeness which uses as a primitive notion a comparative relation of distance between possible worlds; Niiniluoto himself observed that one

-8- No. 1 On Popper's Notion of Verisimilitude 9 could replace possible worlds by maximally strong description of possible world, i.e. Hintikka's constitutents, and then specify a quantitative distance measure between constitutents. This idea was worked out in his (1974), Now Tichy's suggestion in his (1974), Niiniluoto says, turned out to be almost identical to Niiniluto's 'linguistic' modification of Hilpinen's possible world approach. Tichy, however, further developed and in his (1976) presented a general treatment of truthlikeness in full first order language. (His (1974) covered only the propositional case.) Tichy's theory was criticised by Miller (1974, 1975, 1976, 1978) and Popper (1976). Now we have Niiniluoto (1978) and Tuomela (1978). This is only a rough sketch of the problem situation of verisimilitude. One thing, however, can be clearly seen from this lie of the land. All these writers except Popper seem to be in agreement on the dirction in searching for the definition of verismilitude. Whether this point can be a progress made within the logical theory of truthlikeness, I do not know. I rather think the way of the search is still far. In any case, however, they seem to be. in agreement on the point that the problem of verisimilitude, or of distance from the truth is in some way reducible to the problem of specifying a satisfactory measure of the distance between constitutents for the language under consideration, (Miller, 1978): every theory A can be represented as a set of constitutents, and the truth T is itself a single constitutent. Alternative directions in seeking for the reasonable notion of verisimilitude are rather scarce, although not none. One of such examples is Mott's proposal, which covers only the propositional case yet. Mott's ideas are based on Martin Hyland's suggestion which Miller rejected. Miller (1978) thinks that the distance of A from B is reasonably supposed to be some function of the distances between the various constitutents of the language. But Hilpinen, Niniluoto, Tichy, Miller have not come to agreement yet, as to what function is most suitable. So far as the present situation goes, the approach based on Hintikka's notion of constituents, though far from perfect, appear to be most promising. No body would deny this. I wonder, however, if this approach is really promising from Popper's point of view.

4. Would Those Proposed Definitions Solve Popper's Problem? - Popper's Rejoinders -

4.1 In order to see what Popper's own notion of verisimilitude should be , it will be helpful to have a glance at his reactions to the criticism raised against his theory of verisimilitude and to the recent moves taken by several philosophers outside his school. In the first place, as far as the notions of content and verisimilitude are concerned, Popper's rejoinders (1976, 1978)are mainly to Tichy (1974, 1976) and to -9- 10 K. KAMINO Vol. 6

D. Miller (1974, 1974a, 1975). In any case, so far I know no reply of his own to the atempts put forward by such persons as Hilpinen, Niiniluoto, and Tuomela. As I mentioned in the last section, however, they and Tich are in agreement on the general strategy for their attacks to the notion of verisimilitude, though their tactics are different. They rely on the Hintikka's concepts of distributive normal form and constituent. Therefore, I shall try to pinpoint, as far as possible, what would be Popper's opinion about their attempts. First let us see Popper's rejoinder to Tichy (1974). In replying to Tichy, Popper produces some counter-intuitive examples which arise within the framework of Tichy's theory of verisimilitude. Tichy contrives in his (1974), a simple language L which consists of three primitive sentences h, r, w (it is hot, it is raining, it is windy), and tries to define the distance of an arbitrary sentence a from the truth. Now suppose h&r&wis the truth, i.e. the strongest true statement in the logical space of L. Accordingto Tichy's definition, dT(a),the distance of a from the truth, will be determined by n(n) the number of the negation signs of the disjunctive normal form, and by n(c) the number of the conjunctive constituents of the disjunctions, i.e. dT(a)=n(n)/n(c) Of coruse dT(t)=dT(h.r.w)=0. True statements e.g. h can be distinguished from the statement t which express the truth; they are those statements in whose disjunctive normal form the statement t appears as one of the constitutents. For instance, a true statement h will be expressed as h=h.r.w.•Éh.r.w.•Œ•Éh.r•Œ.w.•Éh.r•Œ.w•Œ: in this case the n(n)=4, and the n(c)=4, therefore. dT(T)=n(n)/n(c)=1

In the same way, the distance of a tautology, say h •É h•Œ is: dT(taut.)=1.5, for in this case, n(n)=12, and n(c)=8.

So, in his (1974), Tichy seems to think that a true theory a is the nearer to the truth the stronger it is. And so far Tichy's strategy seems to be in accord with

Popper's view. Popper, however, proceeds to show that Tichy's approach is not consistent. His arguments are as follows. (1) Suppose that a=h .r.w.•Éh.•Œr•Œ.w•Œ., and then make a series of statements b,c,d, each of which is logically derivable from its predecessor (i.e. the successor is weaker than its predecessor).

b=a•Éh.r.w•Œ

c=b•Éh.r•Œ.w

d=c•Éh•Œ.r.w

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We obtain for these four statements the Tichy-distances from the truth. dT(a)=1.5, dr(b)=1.3, dT(c)=1.25, dr(d)=1.2,

This, as Popper points out, shows that the distances from the truth of these true

statements decline with their declining logical strength, although they ought to

increase with declining logial strength. Popper, moreover, produces number of

other counter-intuitive examples, to show that Tichy's is definition is inadequate.

For example, Popper proves that some false theories have all equal dT, even

though they show decreasing strength, or that the tautology has the same distance

from the truth as the relatively good and strong theory h.r.w.•Éh•Œ.r•Œ.w•Œ. Now so

much for the counterintuitve examples. What does Popper actually say about

Tichy's strategy in general ?

4.2 Popper makes two claims against the proposal to define nearness to the truth

as nearness to the set of all true statements (Popper, 1976, p. 155). (1) "This set T

is too big. We may admit into our universe of discourse only such statements as

we conjecture to be relevant (italic is mine); relevant, that is, to the problem situa

tion in hand. This contraction of the universe of admissible statements may be

interpreted as the introduction of a language which is confined to the problem in

hand (such as Tichy's closed language h-r-w•c). However, I prefer not to think

in terms of a closed artificial language but in terms of a language into which we can

freely introduce new problems and new statements whenever we conjecture that

this is relevant to our problems." (2) In the place of T (the complete deductive system of all true statements), Popper wishes to introduce weaker comparisontheories which are (true) answers to the problems under consideration. Popper's idea is that "by confining ourselves to relevant conecjtures we can solve the problem of how to avoid strengthening of a theory by inserting just any strong irrelevant conjuncts". Popper maintains that any such procedure would have just to be justified as relevant to a solution of one of the probems in hand. I think that this contention of Popper's is worth noticing. For I take it to imply that the definition of verisimilitude would not be adequate without taking into account the problem situation or the substantial content of the theory in question. Some definitions of verisimilitude surely enable us to compare true theories, without taking into account the problem situation of those theories . But so far we have no adequate definition of verisimilitude by which we can compare the degrees of truthlikeness of false theories. In any case , Popper's suggestion that some relevancy restrictions would be necessary for the statements which we can admit into our open universe of discourse, as belonging to or contradicting with theories under consideration, does, I think , imply that kinds of epistemological consideration on the problem situation and on the structure of

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those theories will be necessary for us to obtain an adequate notion of verisimilitude. This will become clearer when we see Popper's rejoinders to Miller (1975). 4.3. Popper's reply to Miller (1975) consists of four points (Popper, 1976, pp. 156-158).

(1) Popper admits that scientific theories invented for the solution of problems are entitled to take as fundamental those parameters that are central to those

problems. But this should not be interpreted as implying that in taking a

problem Pl as fundamental, we have to reject another problem, say P3, and with it

possibly that set of new parameters which produce Miller's results in his (1975). This interpretation is not correct; it, moreover, violates one of Popper's principles.

In reply to Miller, Popper produces his usual schema: Pl•cTT1•cEE1•cP2

•c TT2•cEE2•cP3•cetc. And Popper contends that Miller's results present no

difficulty: Miller's result interfere neither with the idea of an approach to truth nor

with Popper's methodology. At the same time, however, Popper notes that "Mil

ler's results may possibly force us to an historical relativisation of the notion of

verisimilitude, in the sense that two historically isolated and different chains of

problems cum solutions may become comparable with respect to verisimilitude only after the two chains have merged; that is, after we have found theories that

solve the problems of both chains better than all their predecessors." The relativisation of theories to the problems they are designed to solve is a point which Popper stressed long ago, or so Popper says (Popper, 1976, p. 157). Popper's statement quoted here certainly explain why the comparison of the verisimilitude of theories in terms of Hintikka's distributive normal form and constituent looks promising. It is always possible to suppose a language which contains all primitive sentences and primitive predicates necessary to cover all relevant facts, objects, and attributes which those objects may or may not satisfy. And such a language will help us to compare theories which have been historically isolated yet; for ex hypothesisthese theories can be translated into the language, whose logical space includes the totality of the possible worlds. But Popper will reject to take this path, which leads us away from his purpose. (2) Popper admits qualitative or topological conjectures as the approach to truth, besides the approach to the true numerical value, which only in cases of statements of constant is the approach to truth. Qualitative or topological conjectures, Popper contends, are not subject to Miller's result. In addition to this, Popper further claims: "moreover, there are conjecturesabout structure, which are most important". (italic is mine.) (ibid., p. 157). (3) As to Miller's argument concerning the combination of two numerical constants, with respect to which the accuracy of two false theories can be reversed, Popper tentatively replies that in testing the value of a constant, scientists in stinctively avoid testing the values of two constants together: if there is more

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than one constant involved, scientists would treat only one as problematic and the other as 'background knowledge'. (4) Against Miller's pessimistic argument, Popper defends the theory of verisi militude with the analogy of certain computerised guided missiles whose target may be described by more than one parameter. From items (1) and (2) it will be clearly seen that in order to compare theories Popper invites considerations on the problem situation and the substantial content of the theories under investigation.

5. Conclduing Remarks

(1) So far I have tried to make a sketch of the situation in which the theory of verisimilitude is discussed. Where can we guide it by feedback ? What directions can we give it at least tentatively ? In his "Conjecture and Refutations", Popper declared that his notion of verisimilitude is neither epistemological nor epistemic, but semantic, just as 'truth', 'logical consequence' are. And he suggested also that we should distinguish two questions, namely the question 'what do you intend to say if you say that the theory T2 has a higher degree of verisimilitude than the theory Tl ?,' and the question 'how do you know that the theory T2 has a higher degree of verisimilitude than the theory Tl ?' If Popper's notion of verisimilitude may be said 'semantical', then in a sense, the former is surely a logical question. As has been shown already, however, it can not be merely logical; when we consider the verisimilitude of a theory, we shall have to examine not only its problem-situation but also its structure. Mott's move in his (1978), is an example to show the necessity of the consideration about these aspects. His proposalfor the definitionof verisimilitudeis based on Hyland's suggestion which was once rejected by Miller (1974a). One of Mott's devices to avoid such undesirable results as Miller pointed out in examing Hyland's idea is to exclude as uncomparablethose theories that contain no characteristic theorem. Inconsistent theories, for instance, have no characteristic or important theorems, and the set of logical truth, too, has no characteristic formulas. On two grounds Mott rejects the 'truth only' approach: we base comparisons of the verisimilitudes of theories solely upon the truth in theories. First, the inconsistent theory contains all the true sentencesof a language,and if we confine our attention to the truth in theories it would appear as good as any theory at all. But this is absurd. We have to exlude the inconsistent theory, or to find the way to do so. Mott's theory developed on this line leads to the conclusionthat the set of logical truths has no characteristic formulas. Secondly, Mott seriously takes up Popper's sugges tion that we may admit into our universe of statements only such statements as we conjectureto be relevant, and Mott stresses the interest-dependenceof verisimilitude. We would not argue that the verisimilitude of a theory is essentiallyrelative to our

-13- 14 K. KAMINO Vol. 6 interests and problem situation. But can we not formulate an interest-dependent account of verisimilitude: X has less verisimilitude than Y relative to interest I ? It may be that this kind of definition is the only workable and realistic one for us to measure the degrees of verisimilitudes of theories, for we have no absolute truth or no sufficient reason for claiming that we have,. Thus Mott pursues the interest-dependent account of verisimiitude; and he referring to Tichy's proposal regards the language-dependent account as a disguised form of the interest-dependent account. In view of all this, I rather think that, unless we have a perfect language in which the truth can be described adequately, it should be natural that we take the interest- or problem-dependent approach. We need epistemological considera tions to discuss the verisimilitude of theories. I even contend that this is what Popper to some extent explicitly suggests, though this suggestion may not conform with his original intention to achieve the logical definition of verisimilitude. There may be a dilemma in his aim.

(2) One of Popper's replies to Tichy makes it clear that Popper prefers not to think the theory of verisimilitude in terms of a closed artificial language , but in terms of an open language into which we can freely introduce new problems and new statemwnts, whenver we conjecture that such an introduction would be relevant to our problem situation. As is noted above, he thinks that our scientific activities proceed in a kind of cyclic progress, whose process is, in oversimplification, pro blem•¨tentative theory•¨error elimination by critical discussion (Popper's scheme is 'P1•¨TT1•¨BE1•¨P2•¨etc ?); and he also thinks that the notion of verisimi litude should allow us to understand this process as rational one . It should help us to take rational steps twoard a better scientific theory. All this means that Popper should not admit us to claim that we have actually reached the ultimate truth. For the claim of the ultimate truth is very closely connected with essentialism. Should we have such a theory and the language to describe it, then we could easily measure the distances of other theories from the true theory, providedthat those other theories could be translated into the language. But if we really have obtained the ultimate scientifictheory which should cover all the scientific truths, the notion of verisimilitude would lose its methodological importance. We no longer need to know the verismilitude of each theory, though the caluculation of the degrees of verisimilitude might be a nice pastime. And Popper's old scheme of problem situation would become just useless. The supposition of the ultimate theory is not consistent with Popper's notion of verisimilitude, relative to the growth of the problem situation. The relativisation of theory comparison to scientifically relevant problems, whether it is relative to the content of theories under investigation, or to the growth of problem situation of theories, seems to involve epistemological or

-14- No. 1 On Popper's Notion of Verisimilitude 15 methodological considerations. Behind Popper's intention to formulate the logical notion of verisimilitude, therefore there always lurks another intention to apply it to methodological problems. It is obvious that the definition of the notion may be logical, but the significantion of the notion is methodological. It is true that Popper asserts that the notion of verisimilitude is not an epistemologicalor an epistemic idea, but is a 'semantic' idea, like Tarski's termino logies such as 'truth' or 'logical consequence', but we have by now reached to the conclusionthat it presupposes some methodologicalconsideration. This, I gather, would be one of the reasons why so far Popper has made no comment on the approach based on Hintikka's distributive normal form and constituent; he says nothing explicit about it, even in his (1978). Obviously, Popper's methodological principles flatly contradict with such an approach. Well, the approach is not of course without any defect. And it may be that it will turn out, in the very near future, to be ineffectiveor contradictory. That possibility should not be denied. It is, however, clear that at the moment it looks most promosing; and many scholars are seeking for the solution in that direction. Why is Popper silent about the approach? The reason can not be otherwise than he takes it to be un- worthy of commenting on, because it does not accord with his methodological principles. In fact, when criticising Tichy, who is one of the proponents of the approach, Popper only pointed out anomalies in Tichy's theory to show that his theory is inadequate, but said almost nothing about Tichy's fundamental idea. Popper evinced neither agreement nor disagreement, even in his revised edition of "Objective Knowledge" which appeared in 1979: Popper could have enough time to read through a copy of the 'verisimilitude' issue of Synthese, vol. 28., no. 2, June, 1978. With the epistemic framework of the language which can express all the truths we have to express, it would be quite natural for us to suppose that the distance of false theory or partially true theory from the ture theory should be calculable within this epistemic framework. Such is, indeed, the idea with which Tichy, Tuomela, and Niiniluoto are in agreement. It is a very natural idea. The problem left to them, they would think, would be only technical; the problem may be one to see whosemethod would be the most recommendable,or to elaborate the results already obtained. Popper, however, will be antagonistic to posit such an epistemic framework. To his eyes, it will probably look just a variant of essentialism. Even if their approachcould come to achievethe logicallyindefectible definition of verisimilitude, Popper would probably refuse to incorporate it with his theory, for it jeopardises his 'old' scheme. Essentialism will destroy Popper's arguments for the objectivty of scientifictheory. As he himelf says, not only scientifictheories themsevles but also the appraisals of the verisimilitude of those theories are conjectural, and subject to criticisms and therefore to revisions. If so, I contend that the notion

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of versimilitudecan not be purely logical, and that it presupposes the methodologi cal import which rejects any essentialistic approach. (3) If, however, the language of science is neither fixed nor closed, but is open to accept new problems and new statements, and if scientificfacts are impregnated with theory, then we have to confront with the problem of incomparability of theory which was forcefully put forward by Feyerabend. In other words, if we reject the notion of the ultimate protocol sentences which describe the world, and regard every basic statement of scientific theory as being subject to further investigations, then, without having an epistemic framework common to mutually independent theories T1 and T2, we cannot compare those two theories. The epistemic framework required should give a language which enable us to express all the true facts under consideration,and therefore to specify the range of objects and the range of conditionswhich those objects may or may not relevantly satisfy. The expression 'epistemic framework' is introduced by P. Tich y. I shall follow him, also in calling the range of objects 'the universe of discourse' and the range of conditions 'the intensionalbase'. Only with a language of such an epistemic framework, we can compare the theory T1 and the theory T2 in terms of verisimilitude. In case T1's epistemic framework differs from that of T1, we shall have to construct a third epistemic framework which covers both. If, however, there is no possibility of our con structing such epistemic framework but one which is nothing but the simple addi tion of those two epistemic frameworks, what is the merit of the amplified epistemic framework? It seems that any theory appraisal achieved on such a device would be just idle; in that case theories have no critical standard to estimate the degrees of verisimilitude. Popper does not seem to be interested in this problem; he says nothing on this score. But if Popper admits that the growth of knowledge may accomany a peradigm change, he has to work out a theory which makes it possible for us to estimate the verisimilitude of theories whose epistemic frameworks are mutually independent and incommensurable. Of course, the acceptance of the paradigm change in theory may be incom patible with the ides of the growth of objective knowledge. Radically different theories may not be compared after all. But if so, I mean if there be no solution of the incommensurability of theories, then we shall have to restrict the use of ' nearer than'. And we shall have to say, with Quine, that the term is defined only for number, not for theories. In order to avoid these results, we shall have to produce a method to com mensurate theories; perhaps we have either to reduce all global radical changes into a set of small scale local theory changes, or to construct a theory that covers both the predecessor and the successor theory and yet is comparable with both of them.

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Whicheverway we may take, I think it would be helpful to keep in mind that an epistemicframework consists of the universe of discourse and an intensional base; changes in the epistemic framework, therefore, will be classified for the further analysisinto three possiblecases. (a) No change in the range of objects; but changes occur in the senses of propositions,even if these propositions are expressed in the same terms as before. For instance when people rejected Ptolemy's astoronomy and adopted Copernican Astronomy, the sentence "The sun will rise at 6 a.m. on December8, 1979,in Kyoto." changed its meaning. We have similar cases in the other instances of theory change such as transitions from Aristotelian physics to Cartesian mechanical view of nature, and from Newtonian physics to Einstein's relativity theories. In this latter case, although both theories use more or less the same vocabulary, at least such concepts as 'space', 'time', and 'mass' have changed their meaning. (b) Changes in the range of objects, i.e., either the elimination of some theoretical entities or the introduction of new theoretical entity. E.g., the rejection of phlogiston theory, and the rise of quantum theory. (c) Both the intensional base and the iniverse of discourse changed. The problem of theory change, however, will be discussed in another occasion.

References

Ayer, A.J., 'Truth, Verification and Verisimilitude', in P.A. Schilpp (ed.), The Philosophy of Karl Popper (The Library of Living,Philosophers, vol. XIV, Book ‡U), Open Court, La Salle, 1974, pp. 684-692. Harris, J.H., 'Popper's Definition of "Verisimilitidue", The British Journal for the Philosophy of Science 25 (1974), 160-166. Hintikka, K.J., Logic, Language-Games and Information, Oxford University Press, Oxford, 1973. Miller, D., 'Popper's Qualitative Theory of Verisimilitude', The British Journal for the Philio8ophy of Science 25 (1974), 166-177, [1974]. Miller, D., 'On the Comparison of False Theories by Their Bases', The British Journal for the Philiosophy of Science 25 (1974), 178-188. [1974a]. Miller, D., 'The Accuracy of Predictions', Synthese 30 (1975), 159-191, 207-219. Miller, D., 'Verisimilitude Redeflated. The British Journal for the Philosophy of Science 27 (1976), 363-380. Miller, D., 'Bunge's Theory of Partial Truth is No Such Thing', Philosophical Studies 31 (1977), 147-150. Miller, D., On Distance from the Truth as a True Distance', in K.J. Hintikka, I. Niiniluoto, and E. Saarinen (eds.), Essays on Manthematical and Philosophical Logic, D. Reidel, Dordrecht, 1978. [1978a]. Miller, D., 'The Distance Between Constituents', Synthese 38 (1978), 197-212. [1978b]. Mott, P., 'Verisimilitude by Means of Short Theorems', Synthese 38 (1978), 247-273. Niiniluoto I., 'Notes on Popper as Follower of Whewell and Peirce', Ajatus 37 (1977), 272- 327. [1977a], Niiniluoto, I., 'On the Truthlikeness of Generalizations', in R.E. Butts and K.J. Hintikka (eds.), Basso Problems in Methodologyand Lingusitics, Part Three of the Proceedings of the Fifth International Congress of Logic,Methodologyand Philosophy of Science, London, Ontario, 1975. D. Reidel, Dordrecht, 1977, pp. 121-147. [1977b]. -17- 18 K. KAMINO Vol . 6

Niiniluoto, I., 'Truthlikeness in First-Order Languages', in K.J. Hintikka, I. Niiniluoto, a nd E. Saarinen (eds.), Essays on Mathematical and Philosophical Logic, D. Reidel, Dordrecht, 1978. [1978a]. Niiniluoto, I., 'Truthlikeness: comments on Recent Discussion', Synthese 38 (1978), 281- 329. [1978] Niiniluoto, I., 'Verisimilitude, Theory-Change, and Scientific Progress', in I. Niiniluoto and R. Tuomela (eds.), The Logic and Epistemology of Scientific Change, Proceedings of a Philosophical Colloquium, Helsinki, December, 12-14, 1977. [1979] Popper, K.R., 'Some Comments on Truth and the Growth of Knowledge', in E. Nagel, P. Suppes, and A. Tarski (eds.), Logic, Methodology, and Philosophy of Science: Proceedings of the 1960 International Congress, Stanford University Press, Stanford, 1962, pp. 285-292. Popper, K.R., Conjectures and Refutations: The Growth of Scientific Knowledge, Routledge and Kegan Pau 1, London, 1963. Popper, K.R., 'A Theorem on Truth-Content', in P. Feyerabend and G. Maxwell (eds.), Mind, Matter, and Method, University of Minnesota Press, Minneapolis, 1966. Popper, K.R., Objective Knowledge, Oxford University Press, Oxford, 1972. Popper, K.R., 'A Note on Verisimilitidue,' The Brritish Journal for the Philosophy of Science 27 (1976), 147-159. Popper, K.R.: Appendix 2. Supplementary Remarks (1978): in Revised Edition of Objective Knowledge. 1979. Tichy, P., 'On Popper's Definition of Verisimilitude', The British Journal for the Philosophy of Science 25 (1974), 155-160. Tichy, P., 'Verisimilitude Redefined', The British Journal for the Philosophy of Science 27 (1976), 25-42. Tichy, P., 'Verisimilitude Revisited', Synthese 38 (1978), 175-196. Tornebohm, H., 'On Piecemeal Knowledge-Formation', in R.J. Bodgan (ed.), Local Induction, D. Reidel, Dordrecht, 1976, pp. 297-318. Tuomela, R., 'Verisimilitude and Theory Distance', Synthese 38 (1978), 213-246.

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