III. VERISIMILITUDE AND THE PROBLEM OF DESCRIBING SCIENTIFIC PROGRESS
by Noretta Koertge
Introduction:
Popper first discussed his concept of Verisimilitude in 1963 (Conjectures and Refutations), a flurry of critical papers appeared in 1974, and ever since there have attempts to develop a viable analysis of concepts such as verisimilitude, degrees of truth, approximate truth, etc. A recent book by David Miller (Critical Rationalism: A Restatement and Defence, Open Court, 1994) provides references to most of the major contributions in this area but is not complete. Since there is now a very extensive literature much of which is very technical, what I will do in this lecture is to summarize some of the most salient features of Popper's own approach and try to present an overview of what I think are the most significant morals of this enterprise.
I begin with a rather naive presentation of Popper's original problem situation, trying hard not to introduce too much wisdom from hindsight or "Monday morning quarterbacking", to use the American sports metaphor. I then present his qualitative theory of verisimilitude and the difficulties it encountered. Then, reversing the actual order of publishing, I turn briefly to Popper's quantitative definition and its difficulties. After a look at Tichy's distance approach we then step back and think more critically about the informal concept Popper started from and look more carefully at exactly what the function of a notion of verisimilitude/approximate-truth/whatever would be in our philosophical account of science.
1. Popper's Original Informal Presentation
In the section of the essay in Conjectures and Refutations called "Truth, Rationality, and the Growth of Knowledge" in which he introduces the term "verisimilitude", Popper begins by emphasizing how important it is that scientific conjectures be interesting ones that have a high degree of explanatory power or solve a difficult problem. (He quotes a rhyme about the triteness of "Twice two equals four".) Yet it is historically the case that bold conjectures eventually turn out to be false. So how is it that we believe that we learn so much through scientific inquiry? Popper's answer is that "even after t2 has been refuted in its turn, we can still say that it is better than t1, for although both have been shown to be false, the fact t2 has withstood tests which t1 did not pass may be a good indication that the falsity-content of t1 exceeds that of t2 while its truth-content does not." (p. 235)
In order to speak of scientific progress we need to be able to compare false theories, but Popper does not want to rank them merely on instrumental or pragmatic grounds. Therefore he wants to develop a concept of similarity to the truth. Popper reveals additional motivations for the concept in the essays published later in Objective Knowledge. Here he reiterates his nursery rhyme and again stresses how important the goal of high content is for science. Drawing a little diagram with the set of all true statements portrayed as a circle in the middle of a field of false statements, Popper says: "The task of science is, metaphorically speaking, to cover by hits as much as possible of the target (T) of the true statements...and as little as possible of the false area." (p. 54)
Objective Knowledge introduces a second motivation, one that has a "whiff" of inductivism about it. He first criticizes the following inductivist argument: "It can hardly be an accident that the theory predicts these utterly improbable predictions unless it is true. From this it is argued that there is as great a probability for its truth as there is an improbability for these successes to be due to an accumulation of accidents." (p. 101) What we can say, however, according to Popper is this:
"The good agreement with the improbable observed result is neither an accident nor due to the truth of the theory, but simply due to its truthlikeness. (p. 102)
"There is something like verisimilitude, and an accidentally very improbable agreement between a theory and a fact can be interpreted as an indicator that the theory has a (comparatively) high verisimilitude. Generally speaking, a better agreement in improbable points may be interpreted as an indication of greater verisimilitude." (p. 103)
Although Popper warns against turning this approach into yet another theory of induction, it clearly would perform some of the functions of such a theory. It would enable us not only to describe scientific progress in retrospect, but also to say something positive about the prospects of our current best theories.
2. The Qualitative Definition
Reiterating his admiration for Tarski's account of truth, Popper sets out to define verisimilitude in Tarskian terms, using the following notation: Associated with the statement a is its consequence class A. The subset of true statements is AT (its axiomatization is designated as aT ). The subset of false statements is AF. (This set is not closed under deduction.)
Popper laid down two conditions for saying that the Verisimilitude of statement a was greater than that of statement b:
either (i) a has more truth congent but not more falsity content, or (ii) a has less falisty content and at least as much truth content (Conjectures and Refutations, p. 233 and Objective Knowledge, p. 52).
Although these are presented as providing a definition of verisimilitude, we can immediately see that it is more plausible to think of them as being only sufficient conditions, not necessary, because we can imagine a third situation that cannot be expressed using the language of subsets but which falls under the naive concept:
(iii) a has more truth content and more falsity content than b, but its excess truth content "outweighs" its excess falsity content.
One notes the parallel to Popper's attempt in the Logic of Scientific Dsicovery to rank statements by the number of their potential falsifiers. If we consider sentences of the form, All S are P, then the broader the subject class and the narrower the predicate class then the more content of the sentence. It is easy to compare two sentences using a subset relationship if (i) one has a broader S class but not a narrower P class or (ii) one has a narrower P class and at least as broad an S class. However, a qualitative approach will not rank cases where one sentence has a broader S class while the other has a narrower P class.
Given this, from the outset we should not be surprised if some one were to present a case where intuitively we feel that a has a higher verisimilitude than b, but yet a and b do not satisfy either condition (i) or condition (ii). We would simply assume such a counterexample was a case satisfying condition (iii). What was surprising at the time (though perhaps not in retrospect) is the discovery (by Miller, Tichy, Gruenbaum and others) that there are no pairs of theories which satisfy either condition (i) or (ii)!
Rather than go through the proofs, let me instead informally point outsome crucial logical facts on which they depend. The definiton directs us to look for proper subset relationships between the truth- and falsity contents of a and b but the comparisons don't work as we might intuitively expect for the following simple reasons: (i) If a is even a "little bit" false, then it contains at least one false consequence - call it f. But now we note that no many how true consequences a has (call them t1, t2, t3,....) we can construct a string of additional false consequences, t1 & f, t2 & f, etc. So the bigger the class of true consequences, the bigger the class of false consequences.
(ii) A similar explosion of consequences occurs using logical disjunction. As long as a has one true consequence (which it always will) - call it t - then no matter how "massively" false a intuitively is, for each false consequence (call them f1, f2, f3,....) we can construct a true consequence, t v f1, t v f2, etc. So the bigger the class of false consequences, the bigger the class of true consequences.
Perhaps these two simple considerations will make it less surprising that one can prove that there are no pairs of false theories which satisfy either clause of Popper's qualitative definition. Hence, Popper's explication in terms of the subset relationship is useless in the case of false theories. One might conclude, however, that all cases of pairs of false theories are really instances of the third type of case and that perhaps if we turned to a quantitative comparison of truth- and falsity contents things would work out better. Let us see!
3. Quantitative Definition
Relying on logical probability as a measure of content, Popper proposed the following definitions: the measure of the truth content of a is simply 1 - prob(aT). He defined falsity content in terms of the extent to which a went "beyond" its own truth-content: 1 - prob(a, aT).
Verisimilitude then was defined as the difference between truth-content and falsity-content: Vs(a) = prob(a,aT) - prob(aT)
Here again the definition turned out to be unsatisfactory for ranking false statements because it depended too strongly on content simpliciter and not on the "extent" of the falsity. In cases of universal generalizations, to assess the verisimilitude of any false claim, all we need to do is assess its overall content - the details of its test record become irrelevant! Also, given that the definition contains a conditional probability similar to Carnap's c-functions and rested on the notion of logical probability, it was vulnerable to all of the technical difficulties encountered by the Carnapian partial entailment program - e.g., the numerical values assigned are sensitivity to the number of predicates in the language.
4. The Distance Approach
Tichy and others proposed an alternative to Popper's reliance on the overlap between total sets of true and false consequences (in the qualitative approach) and the overlap in state descriptions (in his quantitative account). Why not instead set up a measure of the overlap in atomic propositions? The theory which gets most atomic propositions correct will be a shorter distance from the truth and hence be of higher verisimilitude. This actually seems to be closer to what we intuitively have in mind when we talk about more or less veridical theories.
So, to use a simple toy example popular in the literature, if the true state description says the weather is hot, dry and windy, and a says it's cold, dry and windy, while b says cold, wet and windy, intuitively a is closer to the truth because it gets 2 out of 3 basic weather facts correct while b only gets one atomic fact right. (We can now see why working with state descriptions themselves didn't match our intuitions. Both a and b are state descriptions and so of course neither overlaps with the true state description.)
We will not be concerned here with the details of how distance should be defined or how the definiton could be generalized to richer languages because Miller introduced a criticism that he and many others found to be devastating: the attempt to define verisimilitude in terms of number of overlapping atomic descriptions is so sensitive to the choice of primitive terms that through translation we can not only change the numerical values of the verisimilitudes of a and b but also their order! In fact Miller introduced a new weather language for the above example (for example, the weather is said to be be "Arizonan" just in case it is either hot and dry or cold and calm) in which all of the above statements can be translated but in which b gets more of the new atomic weather facts correct than does a.
The parallels with the grue-bleen riddle of induction are obvious and so likewise are the responses to the Minnesotan-Arizonan challenge to the distance approach. Miller has also extended his criticism to cases involving numerical parameters and here an attempted defense saying that the nightmarish language is somehow pragmatically inferior is somewhat less convincing. Let me present a very simple numerical example which seems a little less contrived than the Arizonan case.
Suppose I am teaching a unit on Greek science to elementary school kids and want them to remember that there are 5 Platonic solids (P) and that in ancient times there were thought to be 7 celestial bodies (C) that were not stars. Since the kids are also doing arithmetic I set the questions this way: What is the sum of P and C? What is their difference? Compare now the responses of two children. Suppose Mary correctly believes P = 5, but thinks C = 6. She will get both test questions wrong. Martha, incorrectly believes P = 4 and C = 8, yet she will get one question right and thereby score higher than Mary! Is the exam unfair? Well, maybe. On the other hand, the phenomenon of compensating errors or compounding errors is a fairly common one in science and we can not always arrange direct tests of the most basic parameters of our scientific theories, so maybe the Miller contrivances are not quite as artificial as they first appear. But let me now move on to a new concern.
5. The Whole Truth vs. Nothing But the Truth
Various commentators have pointed out that Popper's informal notion, even though it has a nice intuitive feel - what could be simpler than saying that verisimilitude should increase with truth content and decrease with falsity content - nevertheless is a composite of two quite different figures of merit that we may award a theory, namely its scope and its reliability (some speak of the vertical and horizontal dimensions of truthlikeness).
I want to explore the methodological ramifications of this aspect of the informal notion with a fable. Suppose Mother Nature is called in to arbitrate a dispute among citizens of World-3. The argument is not about which is true but about relative merit. It seems that Newtonian theory is disgruntled at having to sit at the low table for false claims where the company is very mixed while true singular statements and tautologies get to sit at high table. Mother Nature decides to seat statements according to their TQ (truth likeness quotient) and she designs an objective test for these purposes that will allow every theory to show its strengths, yet probe its weaknesses. Since theories can not guess, Mother Nature quickly decides that in addition to answering questions, theories must also be given an option of answering "don't know" or leaving the answer blank. Thus the raw score for every theory A will consist of three subscores:
(i) A/t: the truth score of A due to the questions it got right
(ii) A/f: the falsity score of A due to the questions it got wrong
(iii)A/c: the cop-out or compatability score of A due to the questions it left blank
Let us assume for now that Mother Nature has no difficulty in formulating a series of yes-no questions and grading each individual test item. (One wonders however how to pose questions to theories with false presuppositions or whose ontologies are askew.) Nevertheless she is baffled when it comes to combining the subscores into a composite. The plus and minus signs in an equation such as the following seem plausible but what values should be assigned to the weighting constants?
TQ(A) = kA/t - k'A/f - k''A/c
What first occurs to her is to let the coefficients of the true and false answers be equal and to set the coefficient of k'' at one-half. If theories could guess, or if we were to base our actions on the theories and guess in case where the theory remains silent, we should be right half of the time. These considerations would suggest the formula:
A/t - A/f - 1/2A/c
But Mother Nature isn't totally convinced by this reasoning, especially since we're doing the comparison for strictly world-3 purposes, not technology or action, and so she calls in some methodological theories for advice. Here are some of the answers she receives:
(I) True Blue Inductivism says k'' should definitely be set at zero. Scientific theories should not be penalized for saying nothing about things where they have no correct answers, so we should appraise theories solely in terms of their truth and falsity scores. Furthermore, our weightings should reflect out belief that caution is a virtue and discourage gratuitous boldness. Therefore k' should be bigger than k. Since guessing would lead us to a correct answer one-half of the time but a theory which purports to have the answer but doesn't will prevent us from guessing, we should perhaps make k', the quotient on the falsity sub-score, equal to at least 2. So a reasonable function would be:
A/t -2A/f (or we might wish to normalize it by using A/t + A/f as the denominator).
(2) Pure Popperianism, by contrast, wants to discourage agnosticism. If a theory says something false, then by the normal corrective procedures of scientific method we can hope to replace those claims with true ones. But nothing is gained by working with cautious theories that don't stick their necks out. Therefore, k'' should be at least as big as k' - maybe even bigger. So the proposed function here might be:
A/t - A/f - A/c.
The moral of the story is that the equation we choose either reflects or commits us to a particular methodology. Once said, this point seems very obvious. In everyday life, we distinguish between lies of commission and lies of omission or between telling nothing but the truth and telling the whole truth and there is no natural, context-independent way of saying which is more important. Suppose two children are told to pick all the good cherries and nothing but good cherries from a tree. Who has done a better job of following instructions, Clara Caution who comes back with a quart of good cherries with only one rotten one or Bertha Bold who comes back with three quarts of cherries of which three cups are rotten? It all depends on what we are going to do with the cherries - how many do we need for a pie (will one quart be enough?), how long will it take to sort out the rotten ones, and how much dammage will the stray rotten one cause? What are the consequences of leaving good cherries on the tree? (If we are conducting an experiment to find out whether birds will eat rotten cherries by counting pits, even a few missed good cherries might be important.) As in the case of Type I vs. Type II statistical errors, the pragmatic context will determine which is more serious.
Put in falsificationist language, we see there are two distinct ways a theory can be sub-optimal. (i) It may rule out as impossible states of affairs which actually can occur. (ii) Or it may fail to rule out states of affairs which actually cannot occur (not because the theory explicitly allows such phenomena but because it fails to say anything about them). Even if we do not adopt Popper's utopian language of aiming to capture the whole truth about the whole universe and instead relativize our truth target to the problem at hand, there are still two desiderata, one having to do with correctness and one having to do with coverage. And when there are two things we wish to maximize, there is no unique way to characterize the distance from twin goals.
6. Interim Conclusions and Revisiting the Original Problem
The story of Popper and verisimilitude is strange in many ways. From the very beginning it seemed strange to me that Popper held simulatenously that the verisimilitude of a false theory could be defined in terms of its empirical content and that his account would be an alternative to the instrumentalist account of scientific progress. Isn't counting successful predictions (and balancing them out against unsuccessful predictions) exactly the sort of activity that we might expect to find an instrumentalist (as opposed to a realist) engaged in? As Watkins said somewhere [reference?], when we think of truth-likeness don't we think of somehow simulating the ontology of the world, the way things really are? Shouldn't we require for high verisimilitude not just a modicum of predictive success but also a pretty good simulation of the causal structure of the world? But all of this is very vague. Do we even want to say (assuming we knew how to say it clearly) that although the ontology of the phlogiston theory is extremely far from the truth, the causal structure described by the theory is not too bad an approximation to what we now think is the true causal structure? For example, phlogiston theory was correct in positing that the mechanisms underlying the processes of combustion and calcination are causally similar while the process of reduction occurs through the inverse mechanism. Is this an important level of truth-approximation above and beyond that which an instrumentalist would appeal to? Perhaps, but it needs to be worked out in more detail. Here the literature on the General Correspondence Principle might prove helpful.
It is also strange, at least in retrospect, that Popper invoked the concept of the whole truth in his presentation. In many places he is very modest about whether we even have the brains to understand the laws of the universe let alone the luck to stumble upon them. Since his whole methodology is a via negativa and given his repeated appeals to Darwinian metaphors, why did he introduce this utopian omega point? Granted that his original problem was to show how one could speak of positive progress (not just error elimination) within a falsificationist framework, but could he not have instead taken a tack in which he talked about the increasing viability of solutions to increasingly deep problems? As I indicated in my last lecture, when we try to solve scientific problems as I described them, we are driven automatically to solutions that have high content. We don't need to posit the goal of high content ab initio.
Let us return to Popper's original problem situation and see what we now want to say about it. Popper introduced the concept of verisimilitude in an attempt to answer the question: if all scientific theories are eventually discovered to be strictly speaking false, in what sense is there cognitive progress in science? (And can it be described in realist terms?) His original answer was: yes, there is progress because later false theories are closer to the truth than the earlier theories they replace. (We can not prove this is the case but we can cite reasons for our verisimilitude rankings of theories, based on their respective test records.) Given the various criticisms of Popper's concept of verisimilitude - not only are there technical difficulties with the various explications which have been proferred, there are also objections to his intuitive concept - we should come up with another way of articulating wherein the progress lies.
I propose we should say something like the following: first of all, there is not a single dimension to progress such that we can line up all scientific theories in a row. Secondly, I think we need not appeal to a God's eye version of scientific progress. We can instead simply answer the question, what reasons do we have to think there has been scientific progress? To answer this question, we need to make our evaluative comparisons within a specific historical, scientific context. So, for example, we might say something like the following. Why do we today think Galileo's account of falling bodies was better than Aristotle's and would people living at Galileo's time have agreed? Well, it seems non controversial to say that Galileo's account of free fall was better than Aristotle's. Aristotle correctly said that bodies speed up as they get closer to the earth but he didn't say by how much they speeded up - Galileo could and his predicted numbers matched the experimental results available at his time pretty well. Galileo also showed that Aristotle was wrong to say that heavy bodies fall faster under all conditions.
But what about the fact that Aristotle offered an explanation of the acceleration (in terms of the projectile seeking its natural place and the nature of the elements of which it was composed) while Galileo offered no explanation whatsoever and seemed even to rule the question inadmissible? Here I think we might want to say something like the following: Even though Galileo pointed out the poverty of Aristotle's own explanation, still from a Popperian point of view we should give Aristotle credit for raising the question. In retrospect, we can see why it may have been good methodological strategy for Galileo to limit himself for a time to How-Questions instead of Why-Questions, but after Newton we can also see that the question of what caused the acceleration was an extremely important one. So we may wish to say that it was only after Newton that it became completely non-controversial to say that the move to Galileo was clearly progressive, bearing in mind that Newton followed Galileo. Or we might even wish to say that it is only with Newton that we have a theory that supercedes Aristotle's on all the counts listed.
Are we admitting with Feyerabend that there may be other respects in which Aristotle is better (say in his general teleological approach) and that these aspects of Aristotelianism might stage a come-back? I would say that we cannot prove Feyerabend is wrong - in fact some cosmologists are toying with a Strong Anthropic Principle which includes a different kind of teleology - but on the other hand there is no reason to be sceptical of our own judgment about the superiority of Galileo's formula for free fall and Newton's theory of graviatational attraction. All we need grant Feyerabend is that there might be some other question raised by Aristotle which Galileo/Newton do not answer but which some future theory will answer. At this point our evaluation of Aristotle might take another turn, but that is OK.
What we can pretty straightforwardly do is to evaluate theories on the basis of the problems which were current at the time they were proposed or which ahve turned out to be important later. Sometimes one theory is clearly superior on all counts. At other times perhaps the older theory will ahve raised a question which gets obviated or simply dropped by later research programs. Here I am prepared to admit that to compare the theories we now need to make a judgment about which problems are more important. But given our account of the cognitive evaluation of problems, we can even do this without appealing to social or ideological factors.
So, to take one of Kuhn's examples where there is supposedly a loss in problem-solving power in some domains as well as a gain in switching paradigms, it is true that the phlogiston theory tried to give an account of color changes in chemical reactions and the shininess and ductility of metals, thus placing on the agenda explanatory problems that were then abandoned by Lavoisier and the Newtonian approach to chemistry. Does this mean that it's a matter of subjective preference which is better, Stahl or Lavoisier? I think not. First of all, at the time one could note that although phlogiston theory raised these questions the actual explanations offered were immediately seen to be inadequate. Although one may note analogies in ductility/fluidity between phlogiston-rich metals and phlogiston-rich organic compounds, charcoal, which was sometimes thought to be almost pure phlogiston, was not in the least ductile. And even Stahl himself noted that the vagaries of color changes were enough to drive one crazy. So even at the time, the evaluation as to which was more progressive in its problem-solving capacity was very clear, I think. It was not just a case of people adopting a prejudice against secondary qualities or being enamored with Newton. And of course, now that we have theories about ductility and color we can see that phlogiston was in no way a stepping stone to a good explanation.
The approach I am recommending is unabashedly Whiggish - we are using later scientific results to evaluate earlier developments, but this is surely quite appropriate given that the question concerns a comparison of earlier and later theories. But, might not a sceptic say that this is an unfair comparison! Of course, we think today's theory is better than yesterday's almost by defintion. (If we really thought yesterday's was better, we would revert to it!) However, we also provide reasons for our ranking and, as the above examples indicate, in most cases we can argue that today's theory either solves all of the problems that the old one does or gives very good reasons for thinking them insoluble, at least in the terms posited by the old theory. There is no reason to think Aristotle would not grant the superiority of Galileo's quantitative description of free fall and even grant the existence of Newtonian forces. (If he were to demeur at all it might be on theissue of the metaphysical attractiveness of a more unified explanatory account of human, animal and natural phenomena.) So I don't think we are stacking the deck in favor of the contemporary.
But what about David Miller's point that we cannot unambiguously compare the numerical accuracy of theoretical predicitons? (Recall the example with composite questions about the numbers of celestial bodies and Platonic solids.) Here I will simply give an answer parallel to the way Miller himself deals with the grue-bleen riddle. If there is an extant problem to which propositions concerning the sum and/or difference of P and C is a solution, then I will take that into account when saying which theory is better. Again we are not trying to make claims from a God's eye perspective in which historical context does not count. Rather we are talking about relative success from our own perspective.
The differences between the task of classifying theories as true or false and assigning appraisals of relative merit to false theories are thus quite significant. When we judge a theory to be false, say on the basis of an empirical refutation, our reasons for making that judgment are of course dependent on our own historical context, what experiments we know about, etc. But the content of the meta-claim we are making is independent of historical context. When we make a claim about the respective merits of two theories, however, we are involved in an additional, stronger sort of historical dependence. First of all, to say one false theory is cognitively or scientifically "better" than another is to make an intrinsically ambiguous claim and only the historical context can tell us which sense(s) of "better" is appropriate.
At the risk of confusing matters, let me introduce a crude analogy. When we say one box is of a "bigger size" than another, we are usually thinking of volume comparisons, but when one is shipping packages, there are often girth requirements (where "girth" can be also be defined in different ways, generally as the sum of the height, depth, width but sometimes a diagonal distance is used). For most comparisons of box size, it doesn't matter which explication is used. Usually, boxes of big volumes also have big girths. Which explication of "size" is appropriate depends on which practical situation we are in. (One can even imagine situations where one wnated some sort of composite function of volume and girth.) Once we have used context to choose the precise manner in which we want to compare boxes, then it is of course a context-independent question to determine relative box size.
Just as there is a fairly limited range of acceptable explications for "bigger size", so there are limited aspects of a theory that could conceivably figure in when we say it supercedes its predecessor - how many empirical questions it answers, how many explanatory questions, how precisely it answers them, how accurately, how cognitively important the problems are, etc. But there is no unique emphasis and no unique way of composing these meritorious features. In some episodes of the history of science the later theory will be better in every way that we can think of, but that need not be the case for us to speak sensibly of scientific progress.
7. Conclusions
So where have we arrived in our meandering discussion of verisimilitude? How verisimilar was Popper's original account? How have we corrected it? Well, first I will remark that I have been more influenced by the informal critiques of his informal conception than by the various logicians' nightmares that have been generated to criticize his formal explications. (But that may be more of a commentary on me than on the debate.) I think that relative to a given problem it is very often the case that one can trace a historical trajectory that corresponds to increasing verisimilitude as Popper originally conceived of it. These episodes are also well-described by the General Correspondence Principle. Van der Waal's Equation describes the relationship between the pressure and volume of a gas more accurately than does Boyle's Law in every case, plus it gives a generalized account of which special cases Boyle provides a pretty workable prediction and in which cases (e.g.,those involving high densities and high pressures) his "law" is very far from being empirically adequate. (Here I am "blowing off" Miller's "Arizonan" puzzles as well as his mathematical transformations.)
However, if we wish to speak of progress in the history of science as realists do, it is not enough just to point to increasing predictive accuracy in certain carefully selected domains. We also want to ask whether the later theories give us a more truthful picture of the causal structure of the world or its ontology. After all, it could happen that what we gain on the straight-a-ways (cf. predicitive accuracy in certain domains) we lose on the round-abouts (descriptive accuracy of basic processes). [It is easy to imagine actual examples of this occuring in artificial intelligence research where some investigators are only trying to get computers to simulate input-output relationships of human cognition, freely using computational processes that are beyond human capability, whereas others require that the underlying processes in the computer bear a strong analogy to human thinking. So a number-crunching chess machine might easily defeat one which used the sorts of heuristics that human chess players use. Deep Thought, as I understand it, combines both approaches.] Thus to defend realism using the history of science, we need to be able to describe the degree to which a later theory gives a better account of causal structure as well as being more predictively accurate. Again, sometimes episodes used to illustrate the General Correspondence Principle seem to have this feature, but one needs to describe the correspondence relation in a manner that highlights the retention and correction of causal structure in addition to the retention and correction of observations. (And I don't really know how to do this!)
Once we introduce more than one desideratum or evaluate theories' answers to more than one problem, we have to impose a weighting on the relative importance of the multiple desiderata. And I can think of no unique way to do this, even from a God's eye point of view. However, one of the fascinating aspects of the history of science is how frequently a later theory will "out-plus" its predecessors on every (or vitually every) desideratum. It is almost as if every new Olympic winner of the Decathalon were to break their predecessors' records in all ten events! When this happens, in order to say that the most recent winner surpasses earlier ones we need not quarrel about which decathalon event is the best indicator of athletic ability. But when this does not happen, then we must provide an evaluation of which problems it is most important for a theory to make progress on. And here our account of cognitive importance in lecture 2 will give us a checklist of considerations to bear in mind.
None of the above remarks should provide any grist for the relativist's mill. Although we evaluate theories relative to a problem, once the problem is specified then the assessment of the intellectual merits of various solutions is an objective one. And although the weighting of problems is not uniquely determined even relative to context, there are some constraints on how the weights are assigned (for example, the fact that a theory gives good predictions in a certain domain can never be used to down-grade it - even if the predictions are politically undesirable at a given time); often the issue of relative weights does not even arise because one theory is superior on all counts. However, I think we must also admit that introducing talk about verisimilitude or better problem- solutions does not in itself allow us to say anything prospective about future evaluations of theories. It does not furnish us with a surrogate for induction.
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