PART B: MY FRAMEWORK
VII. Types of Problems and the Dimensions of Appraisal
I will now provide an analytic framework for the evaluation of problems. (Detailed theories of evaluation as proposed by various philosophers will come later.) From our discussions above, we have already introduced three broad dimensions along which problems can be evaluated. The first two were already noted by Bacon when he distinguished between experimentum luxus and experimentum fructus. "Experiments of light" add to our intellectual understanding of the universe. When referring to this aspect of problems, I will speak of their K-importance, i.e., how big an impact do they have on our Knowledge system?
Bacon's "experiments of fruit" will give us more power to achieve our present goals. Here I will speak of the A-value of a problem, i.e., how big an impact on our ability to Act is its solution likely to have? In the example of games we saw that people also judge problems at least partly on the basis of how intriguing/pleasurable the process of trying to solve them is. I will speak of the P-value of a problem when referring to its interest as a pure puzzle.
Although ordinary usage does not always draw the distinction consistently, I believe when people speak of a problem as important they are referring primarily to its K-value or A-value. To say a problem is interesting, however, is generally to say something about how difficult or tricky it is to solve - its P-value.
Of course, interesting problems can also be important. And an interesting problem, the answer to which has little impact on either our knowledge or our ability to act, may turn out to be of indirect importance if the method which we invent to solve it turns out to be applicable to other important problems.
Most of what I say below will be about importance (K-value and A-value). It is much harder to say what makes a problem interesting. As a start I would say that it must look easy at first, but resist obvious methods of solution, yet yield partially encouraging results just often enough to keep us hooked. Even without a good analysis of the P-appeal of puzzling problems, we can still ask what the comparative role is of P-, K-and A-values in both science and education.
One last preliminary remark. By distinguishing Knowledge evaluations and evaluations for Action, I do not wish to take sides in the debate over whether intellectual values, such as truth, should be reduced to or grounded in the value of being able to carry out life projects (or vice versa)! As we begin our analysis, these appear to be distinctive evaluations. I leave questions about how they are related until later.
We have described three broad dimensions along which problems are evaluated, namely according to their impact on knowledge, their impact on our decisions about action, and their intrinsic interest as puzzles. But when people choose a problem for inquiry, they also have to make a judgment about how likely it is that this particular problem can be solved, given the intellectual and material resources available.
As an example, consider the problem of scientifically predicting (as opposed to prophesying!) the behavior of the stock market. Clearly a theory which permitted us to do this would be of great practical importance, and probably also of great pure theoretical interest since it would have to combine elements from economics, politics, psychology, sociology, etc. But although this problem ranks high in both K-value and A-value, it would be a bad problem to choose for inquiry because based on what we know today, it seems very unlikely that it could be solved. The problem ranks low on the solvability scale (or S-scale, for short).
Of course, we can't really know whether a problem is solvable or not until we try! Yet we routinely make judgments about such matters. Non-tenured professors are advised to undertake low-or-medium-risk research, saving "big" questions until they are more secure. And funding agencies make appraisals of whether a problem is "ripe" for solution! In the discussion which follows, we will look at some of the factors which enter into our judgments about solvability.
These various appraisals fit smoothly into a decision-theoretic approach. K-, A- and P- evaluations combine to give a utility value for the option of choosing to work on a given problem. The S-potential acts like a probability. Multiplied together they give the expected utility of choosing to work on the problem in question.
Let us now turn to the classification of problems. Real-life problems may sometimes fit under more than one heading (e.g., a problem might involve both inconsistencies and explanatory gaps). Nevertheless, it is useful to distinguish the different problematic aspects of the complex starting points for our inquiry. My classification is based on logical features of the problems. The major distinction is between those arising from inconsistency vs. those arising from incompleteness. Under each type I will describe various factors which contribute to its importance. Formal/logical factors will be described first; then I will turn to pragmatic/contextual factors. The order of discussion is not meant to prejudge the issue of which factors are weightier. By distinguishing logic and pragmatics, I do not mean to rule out the possibility of reducing logical factors to pragmatic ones, or vice versa.
All of my evaluative rules have caeteris paribus clauses, e.g., "Other things being equal, the more content..., the more important..."
1. Problems Arising from Inconsistencies
Suppose our knowledge system K contains propositions O and W (intuitively, oil and water). Suppose O entails p and W entails ~p. If we notice these inconsistent consequences we have a problem. (Here I follow Dewey, not Popper.)
Although I will say that every noticed inconsistency is a problem deserving of inquiry, some may be of very minor importance. In classical logic, every inconsistency is equally disastrous - from p and ~p every well-formed formula of the language logically follows. However, various current philosophers have made more subtle appraisals of inconsistencies. Feyerabend, for example, has pointed out that in practice scientists insulate the inconsistencies in a system and are judicious in the consequences they draw. And relevance logicians have provided rules of inference which block the explosion of consequences from an inconsistency.
Since most of us were brought up believing that tolerating a contradiction was a mortal cognitive sin, I want to illustrate their abundance by confession some of my own inconsistent beliefs:
(i) A photo slide of my grandmother is stamped September, 1959, but I remember distinctly that she died before my 21st birthday (1956). (I was to receive $100 if I reached 21 without becoming a smoker.) Yet I can't believe that I would have left film in my camera for over two years. I really can't explain it.
(ii) Value-added-tax is a form of sales tax. Sales taxes are regressive. The VAT is popular in Northern Europe. But I can't imagine countries like Britain would put in a regressive tax. I must ask an economist friend to sort this out for me.
(iii) It seems pretty clear to me that AIDS is spread through sexual activities in gay bathhouses. I believe the government should intervene when a public place of business constitutes a serious health hazard. (It was right to close swimming pools during the polio epidemics.) I also believe that gay bathhouses are a major cultural institution for a beleaguered minority. To close the baths would not only be a symbol of oppression; it would also very likely lead to increased homophobia and deprivation of civil liberties in other spheres. Thus it would be quite wrong for the government to intervene. I don't know how to resolve this dilemma - maybe they'll find a cure soon.
I must admit that even in the process of writing down these examples of inconsistencies which I claim have not sparked inquiry, my mind wandered off trying to come up with a solution! Still, we don't have time to follow up on every glitch in our knowledge system. So which factors are relevant to our evaluation of inconsistencies?
a. Epistemic Factors
In the examples above, I was pretty firmly convinced of both O and W. This made the situation more problematic than if I were only somewhat inclined to believe both O and W. (And if I were strongly inclined to O and only weakly inclined to W, it would be a less interesting problem because it seems obvious which revision to try.) Our first two rules for evaluating inconsistencies are as follows: (I have omitted the other-things-being-equal clause.)
Rule l: The more epistemic support there is for both O and W, the more K- important the problem posed by the inconsistency.
Rule 2: The more balanced the epistemic support for O and W, the more P- interesting the problem posed by the inconsistency.
b. Content Factors
If K contains O and W and O p while W ~p, then either p and O are false of ~p and W are false (and possibly all four are false). One immediate solution to the inconsistency problem is to drop all of the conflicting propositions, e.g., p, ~p, O, and W (plus any other of their logical consequences), from K.
If I delete conflicted propositions about the exact year of my grandmother's death from my K, relatively little information is lost, because few of my other beliefs are connected to the inconsistent pair.
However, if my problem concerned conflicting reports on the age of a human fossil, dropping the inconsistent pair and each of their consequences might leave me with a seriously truncated historical anthropology.
Rule 3: The more content which would be lost from K by simply dropping both p and ~p and each of their consequences (e.g., ~O and ~W), the more K- important the problem posed by the inconsistency.
An especially pressing problem arises when the conflict between p and ~p casts doubt on one or more widely used epistemological methods. Let us return to the case of my grandmother. One horn of the dilemma rests on my memory. Although the memory is a rather vivid one, reinforced by the disappointment at not collecting the $100 - still it would not be too big a shock to discover yet another case where my memory failed. I would be more surprised to find an error in the Kodak, company's dating procedures, but still few of the other propositions in my K would be jeopardized. However, if I were doing a massive historical investigation which relied heavily on the dates in slide archives, many more data would become suspect and the problem would become more serious for me.
Rule 4: Until the inconsistency between p and ~p is resolved, the methodologies under which each was introduced into K are suspect. The more reliable the methodologies were believed to be and the more widespread their use, the more K- important the problem posed.
c. Pragmatic Factors
We have seen how the discovery that our K contains both p and ~p sends tremors through substantial parts of our cognitive system. Until we discover the source of error, both p and ~p are unreliable as are the logical consequences of each. (This works in two directions: (1) If O p and p turns out to be false, then O must also be false and K must be revised to include ~O. (2) If p q and p turns out to be false, then we can no longer use p as a reason for believing q, so unless we have independent evidence for q, prudence dictates it also be deleted. And we pointed out that the methodologies which generated each member of the inconsistent pair are undermined and therefore other products of the methodologies are cast in doubt to a certain degree.
The cognitive impact of the contradiction depends on how many other beliefs are affected. The pragmatic impact of the contradiction depends on the degree to which my well-being depends on the truth of the propositions which are now under question. I offered the uncertainty over the date of my grandmother's death as an item of relatively small cognitive importance. However, if I'm on trial for murder and my alibi is that I was attending her funeral at the time, there are powerful pragmatic reasons for trying to resolve the inconsistency.
Let us suppose that given my present practical situation, if p is true, I can expect to alter my present state of well-being by Up utiles. (Utiles may include everything from crude appetite satisfaction to money to moral virtue.) While if ~p is true, my state will change by U~p utiles. I now propose the following evaluative rules:
Rule 5: Other things being equal, the larger the absolute value of Up - U~p, the more A- important the problem.
Rule 6: Other things being equal, if at least one of the U's is negative, the problem is more A- important. (Here I am assuring neg-utilitarianism.)
The urgency with which we inquire into an inconsistency also depends on our power to act in the light of what we discover. Some people would rather not know whether the lump was cancerous or not if nothing can be done about it. Others do want to know, but generally I suspect it's because the news will affect their projects and decisions in the time remaining.
Rule 7: The more p vs. ~p is relevant to our decision-making, the more A- important it is to resolve the inconsistency.
To summarize: Our knowledge system contains many inconsistencies. I claim that we judge how serious an inconsistency is according to how much of our system is logically connected to the inconsistent pair, how well-tested each member of the pair is, how frequently the experimental methodology which generated components of the inconsistency is used, how much difference the resolution of the inconsistency would make to our future happiness and decision-making, and how interesting the solution process appeared to be. Although I proposed some evaluative rules for comparing problems along these various dimensions, at best they would produce only partial orderings.
In order to make a decision to start work on a problem, we also need to appraise its solubility.
d. Availability of Additional Relevant Information
The inconsistency concerning my grandmother's death should be easy to resolve - I could call the local newspaper or look in the family Bible. Note that once that problem is solved, another immediately arises - either I must ask exactly how was my memory mistaken (Did I actually collect the $100 and then forget about it, or?) or ask why was the photo slide dated so late? Whether this resulting problem is easy to resolve or not is a quite different matter.
Rule 8: The more readily available information is which will either confirm or refute one of the inconsistent pair, the more solvable the problem.
e. Availability of Plausible Alternative Hypotheses
In our example, O and W were found to be inconsistent because they implied p and ~p respectively. If we can easily imagine an alternative to W, call it W', which plays a similar role in K but which does not imply ~p, the inconsistency could be easily eliminated. This might even provide an immediate solution to the problem, but generally we would also need to design a crucial test between W and W'.
I am pretty sure that the list of factors relating to problem choice given above is incomplete, but they already illustrate how complicated the evaluation of problems can be. It is also worth noting that the factors may not be independent. If my ambition is to win the Nobel Prize, I may give a higher A-value to problems which appear difficult to solve (low S-rating). Inconsistencies which involve big chunks of our knowledge system (hence have a high K-value) will also tend to have a low S-rating.
Despite these complexities, I still think this framework illuminates the process by which we actually choose problems.
2. Problems Arising from Gaps in Our Knowledge
In the above section we looked at problems which arise because of inconsistencies within our fallible knowledge system K. We now turn to another kind of imperfection in K- -- its incompleteness. Logicians define an axiom system as complete when for every well-formed formula in the language, either it or its negation is derivable from the axioms.
It was the Laplacian dream that we would someday have a complete K. If the universe were a Newtonian clock and if we knew the initial conditions of all particles (This second condition was already a Utopian assumption.), then Laplace thought it would be possible in principle to derive all future states of affairs.
The quest for completeness in such a strong sense has since been abandoned for a variety of reasons. First, as long as our most fundamental physics is indeterministic, the propositions we derive from them will be only statistical - in principle our best knowledge of the future can only give us the probability that an event will occur. Secondly, mathematical logicians have proved that arithmetic is incomplete - Godel's first theorem says that in any formal system adequate for number theory, there exists an undecidable formula (i.e., neither it nor its negation can be proved) ("Godel's theorem" in Encycl. of Phil. 3, p. 348). Thirdly, there are a variety of philosophical arguments showing that there are severe limitations on the ability of a predicting machine, such as a computer or a brain, to predict its own future state. (See Popper, .)
There is another kind of argument against hoping for a complete K. As science progresses, not only do the truth values which we assign to theories change as we move from flat earth to round earth, from geocentric to heliocentric, from animistic to mechanistic, etc., there are also innovations in our basic concepts. If Aristotle were asked to affirm or deny propositions concerning entropy, quantum numbers, synopses, x-rays, genetic drift or black holes, he would be literally speechless - his K did not even contain these concepts, let alone theories describing how such entities behave. We certainly cannot have knowledge today of things which our present language can't describe.
So complete knowledge is a completely Utopian goal and hence not a good source for problems. Nevertheless certain gaps in K do motivate inquiry. I now ask which incompletenesses in K are seen as puzzling or deserving of inquiry.
a. Deductive Gaps (whether-p)
Often K contains theories which imply the existence of entities which have as yet not been detected. A simple case is that of the genealogist constructing a family tree. Barring parthenogenesis, there must have been a great-grandfather on my mother's father's side, even though we don't know yet who he was.
Scientific inquiry is often directed towards gap-filing. After Mendeleev proposed his famous Periodic Law, he noted that there were three elements missing from the Periodic Table and predicted many of their properties. On the basis of these theory-generated chemical profiles, experimental chemists set out to look for the new elements in appropriate ones and quickly discovered them. (See Henry M. Leicester, The Historical Background of Chemistry.)
Other gaps occur when a theory in K predicts as yet unobserved properties of already known entities. In the genealogy case, I may be able to infer that my great-grandfather Karl must have been born before 1848 even though no birth records nor direct evidence exist. Mendeleev was able to predict the correct formula of beryllium chloride before it was experimentally determined using difficult vapor density measurements (Mendeleev, "Faraday Lecture," II, 502-3).
I will say that K contains a deductive gap (?p) when T K and T p, but p has not been observed nor confirmed independently of T. I suggest that the importance of investigating whether-p depends on epistemic, content, and pragmatic factors.
Rule 8: If whether-p is a deductive gap in K generated by T, then other things being equal, the less sure we are of T and the larger the content of T, the more important investigating whether-p becomes. Thus problems of testing K- bold hypotheses will be important. (If we discover p, then ~T will be refuted.)
Rule 9: If p itself has a relatively large content, investigating whether-p becomes more K-important.
Rule 10: The bigger the difference which directly confirming p would make to my decision-making and expected well-being, the more A- important whether-p becomes (Cf. rules 5, 6, 7).
Let us now apply these rules to the genealogy example and Mendeleev case. By rule 8, searching for the missing elements is more important than searching for the missing great-grandfather -we are unlikely to discredit the theory that every person has a father. Mendeleev's theory, however, though very attractive to scientists was as yet relatively untried.
By Rule 9, finding missing elements is more exciting - we thereby discover more new things about the world.
By Rule 10, the evaluation could go either way. Does an inherited fortune or the order of royal succession hinge on the identity of the ancestor? Will the new element be named after the nationality of the discoverer and hence contribute to a cold war propaganda victory? Or will it have useful properties for technology?
Assessing the solubility of whether -p can be a relatively straight-forward business if p describes an easily observable state of affairs. Sometimes, however, the determination of the truth value of p requires the invention of new apparatus or new auxiliary theories. For example, from stellar parallax observations and the Copernican theory, one can derive a minimum value for the distance of the stars from the earth (p). But with the methods of determining stellar distances available at Galileo's time, it was impossible to determine whether p was true.
b. Explanatory Gaps (why p?) (or Abductive gaps?)
In the deductive case considered above, some theory in K logically implied some observable consequence p. Checking directly on the truth of p is a good way of critically assessing K. If p turns out to be false, we have a prima facie refutation of T. However, if p is detected, we have generated positive support for T. Testing the deductive implications of our theories is (or should be) a routine cognitive activity not only for scientists but also for detectives, investors in the stock market, parents and gardeners.
Another important motive for completing/expanding K is the search for explanations. Instead of asking whether p is true, we have here a case where some phenomenon p is well-known, but we don't know the causes/reasons/preconditions/triggers for it.
If p violates our expectations then the problem of explaining why-p is a special case of the problem of resolving inconsistencies which was discussed above. An example might be: why do planets exhibit retrograde motion? (I thought all celestial bodies moved in smooth arcs across the sky!)
Not all surprising phenomena are based on strict inconsistencies, however. I would be surprised if Reagan carried all 50 states, but I do not believe it impossible. This suggests an immediate evaluation rule:
Rule: The lower the probability of p, given K, the more important the problem why-p. (Note the tension with Salmon's S-R model.)
Yet many of our explanatory problems do not obviously involve unexpected or improbable events. Some of the same early astronomers who were puzzled by the planets also asked why the sun rose every morning. Why do mother rabbits always have baby rabbits and not baby chicks? Why can't (most) plants live in the dark? The child or early scientist who asks such questions is not surprised by the phenomenon - they know it happens regularly.
There are enormous numbers of non-surprising, yet unexplained events and regularities in our lives: Why did the dog destroy my left sandal (and not the right)? Why did I miss two (instead of one or three) spelling words on the test? Why does dandelion dust make your nose yellow? Yet we soon pass the two-year-old's stage of posing a seemingly inexhaustible stream of why-questions (which, as Dewey points out, are probably not real requests for explanation but non-focussed prods - "Tell me more interesting things - anything."). And we no longer seek to fill every explanatory gap in our K. For two good reasons: first, to do so would lead to an infinite regress-one can always ask for an explanation of the explanans ("Why is energy always conserved?")
Secondly, there are in practice way too many gaps to think about. And, of course, as we conduct our investigation, further unexplained things happen to us ("Why did I think of that explanation first?") - a sort of Tristram Shandy effect.
So we must now ask, which explanatory gaps are considered to be more important and/or more interesting? Or as Rueben Edel (1982) puts it, "Is it only the village idiot, or is rather the scientific genius, who is likely to ask why grass is green?...When is a fact just a `brute fact?' [And when does it call for explanation?] (p. 89).
As we discuss the evaluation of why-questions, it is useful to distinguish carefully between appraisals of solubility (how easy would it be to come up with an explanation?) and appraisals of importance (how theoretically or practically significant would the sought-for explanation be?).
Let us begin with evaluations based on metaphysics, the criterion for problem choice discussed by Agassi. As an example, recall Descartes' proposed cosmology in which all motions were caused by corpuscular contacts - there was no action at a distance. Given such a world view, the fact that magnets can attract iron filings through glass or paper is very puzzling. If one could propose a mechanism which would explain magnetic attraction in terms of contact forces, this would be a great coup for Cartesianism.
As a result, many models involving effluvia, corkscrew particles, etc., were proposed. All were unsuccessful, the degree of solubility of this explanation problem was low - there were no good hints or useful analogies to pursue. But the interest in the problem was high because the phenomenon did not fit in smoothly with the prevailing metaphysics. A failure to explain magnetism would not logically refute Cartesianism, but a success would be a great victory. (See Watkins, "Confirmable and Influential Metaphysics.")
It was also important for Cartesians to provide detailed theoretical explanations for other phenomena, such as vortex models for the motions of the planets.
Rule: The problem of why-p is more important when the explanation of p is expected either to instantiate or undermine a scientific world view.
In principle, there is a clear logical distinction between a metaphysics for science and a scientific theory - the latter can be refuted by singular statements, the former cannot because it contains an existential quantifier. However, in real-life cases, the demarcation is often difficult to apply. It sometimes depends on whether the doctrine in question is taken to include a methodological rule protecting certain core statements from falsification. It may depend on what auxiliary theories are taken to be non-controversial. And often a broad doctrine contains a variety of elements, some plausibly construed as falsifiable, others not. There have been various attempts to give general analyses of the bundles I have just called "doctrines." Lakatos speaks of research programs, Laudan of research traditions, Shapere of domains and fields. In the examples which follow I will frequently just use the ordinary science language term theory, without worrying about whether the doctrine is highly falsifiable or not. The theories I have in mind are generally more specific than the views labelled above as metaphysics.
Let us look at the relationship between theories and the evaluations of why-questions. For Darwinians, the problem of explaining the origins of the eye was important. Creationists considered such complex, beautifully adaptive organs to be evidence of providential design. It was difficult to see how the imperfect proto-stages of the eye required by evolutionary theory could have enhanced survival value.
According to Aristotelians nothing moved without a mover, all locomotions (as well as other types of change) were caused. But what caused an arrow to move once it was separated from the bow string? What impelled a stone as soon as it left the sling? On the face of it, these common phenomena contradicted the theory. (It depends on what one assumes about the visibility of movers.) Explaining them was an important problem for Aristotelians. Some of their solutions were easily testable (e.g., the hypothesis about air currents); others less so (e.g., the impetus theory).
Rule: the closer p is to being a refutation of T, (where T is in K) the more important it is to explain p.
Rule: If p is a near-refutation of T, then the more content T has, the more important the problem why-p.
By a near-refutation, I have in mind cases where for each plausible boundary condition or auxiliary hypothesis we can think of (call these A, A', A"...) T&A ~p, T&A' ~p, etc.
In all near-refutation cases, the degree of solubility of why-p is small (we have already exhausted all the obvious solutions to the problem).
Let us now turn to situations where the S-rating of the problem is relatively high because untried heuristic strategies are available. I will begin with a slightly artificial example. It is a well-known fact that the moon appears larger when it is close to the horizon. Why should this be? Various sorts of explanation to this puzzle either have been proposed (or could be - here's where the example becomes artificial).
Perhaps we should look to astronomy for the answer. Maybe the moon is actually closer to the earth at dawn and sunset. Or maybe the moon actually expands and contracts during every twenty-four hour period. Or maybe this is a problem in optics. Perhaps the light coming through the dense atmosphere at the horizon is refracted in a manner which enlarges the image.
On the other hand, maybe we should look for the answer in the psychology of perception. The apparent size of an object is affected by what we compare it with. Perhaps trying to estimate the diameter of the moon when it is close to familiar objects near the horizon somehow leads to the illusions.
If we assume that each of these approaches has some plausibility (as I said before, the example is a bit contrived), then one would hardly know where to begin looking for an explanation. We don't even know which discipline the phenomenon falls under!
More realistic examples concern the attempts to explain schizophrenia: should one look for genetic factors, chemical imbalances, childhood experiences, or ?? Or why did SAT scores plummet? Bad schools, deteriorating family, poor nutrition, faulty standardization of test, too much TV, radio-active fallout from bomb testing, or ??
Other explanatory questions seem much more routine. "Why did the racehorse Swale die?" One can at least begin an autopsy. "Why does Mercury's perihelion precess at that rate?" Newtonians knew what to do - postulate as yet unobserved massive bodies of just the right characteristics to produce the perturbation and then look for them. (After all this is the way Neptune, Uranus, and Pluto were discovered.)
Rule 13: Other things being equal (such as importance), we S-prefer explanatory problems for which heuristic strategies are available - we judge them to have a higher degree of solvability.
If the phenomenon is supposed to fall under a set research program, it appears more likely to be soluble at this time and hence worth working on. Scientific grant committees ask not only whether a problem is important, but also whether it is "ripe" for solution. Working within a research program can have not only heuristic, but also epistemological advantages. A good research program not only provides some more-or-less definite hints for generating conjectural solutions. If the RP has been successful in the past, we believe the conjectures so generated have a prior probability greater than chance of being true. Conversely, if the RP fails to lead to an explanation, we consider it to be (mildly) discredited. (For a discussion of the confirmation and disconfirmation of RP's, see ,) so the original explanatory problem takes on added implications if it falls within the domain of an RP.
Scientific research programs may offer quite specific heuristic advice. Perhaps more common is the case where some familiar system in K, call it S, serves as an analogue or model, for some less well-understood domain D.
For example, Darwin developed his evolutionary theory by conjecturing that there was a process of natural selection which operated somewhat like the artificial selection procedures of breeders. (There were also important disanalogies - the breeders selected according to a design they already had in mind while nature did not.)
Today, one often draws analogies between the behavior or reactions of rats and those of people. If overcrowding makes rats aggressive or saccharine gives them cancer, then we may be led to ask about the effects of such factors on people.
Or in artificial intelligence studies, one may draw analogies from the problem-solving, or memory-processing procedures in computers to those in the brain.
Analogical inferences are very slippery customers. On the one hand, we profess not to be the least surprised when they don't work; yet they are somehow irresistible. Freud summed it up thusly: "Analogies prove nothing, that is quite true, but they can make one feel more at home." I would add that the existence of analogies also increases the perceived solvability of a problem.
Rule 14: The availability of a strong analogy to p increases the S-rating of why-p.
Sometimes analogies appear to be purely formal - one notes structural parallels between familiar system S and D, the object of our inquiry, and uses the similarity as a psychological aid in generating hypotheses about D. Thus, it is sometimes claimed that Harvey was led to his hypothesis about the circulation of the blood by drawing an analogy between the body (microcosm) and the universe (macroosm). Presumably, the analogy went as follows:
The sun is the source of life and energy in the Universe, the peripheral items, circulate planets, around it in closed paths.
The heart is the center of heat and life in the body.
So maybe the blood (peripheral element) likewise circulates around the heart in closed baths.
It's hard to believe this analogy played any significant inferential role in Harvey's discovery, although it may well have served as a psychological trigger. Why should the blood move like the planets? Presumably the causal mechanisms involved are quite different. Yet somehow the human mind delights in such empty and at times farfetched parallels. It was common in the 17th century scientific exposition to invoke yet a third analogue - the King as sun and heart of the state with courtiers "circulating" around the kingdom on his business!
In other cases, we use analogies to talk about a shared underlying mechanism which we may not yet be able to articulate. When chemists proposed a "liquid drop" model for fissioning nuclei, they were tacitly assuming that the forces operating at the nuclear level were relevantly similar to surface tension and other forces operating in ordinary drops of water.
The success or failure of a serious analogy like the liquid drop case does have some impact on our K, because it will affect our willingness to use the analogy again.
Rule: Analogies based on hypotheses about shared causal mechanisms form a part of K and as such can be confirmed or refuted. The availabilities of a causal analog to p increases the K-value of a problem. Let us now turn to a discussion of pragmatic factors.
There are many philosophical models for scientific explanations ranging from Hempel's deductive-nomological covering law model through Salmon's statistical relevance model to Popper's Rationality Principle explanations of human actions. But although differing widely about the details, I think all philosophers would agree that a good scientific explanation of phenomenon p will supply information which is at least pertinent to the following questions even though it may not answer them completely:
1) What factors make p more likely to occur? In the ideal case the explanation may give sufficient conditions for p.
2) Which factors are necessary for p or whose absence makes p less likely to occur?
Explanations are not only intellectually satisfying, but may be of immediate practical value. If p is a desirable sort of phenomenon, (e.g., having pure drinking water) knowledge of the process of attaining it is valuable. If p is undesirable (e.g., pneumonia) we will want to know how to remove necessary conditions for its occurrence. And if we want to praise or blame someone for p, then we have to ascertain their causal role in the event.
Rule: The more utility (or dis-utility) we assign to events described by p, the more A-important it is to have an explanation of p.
Having an explanation of p doesn't always give us a means of controlling it. I doubt that even a wonderfully complete and deep analysis of the origins of hurricanes would help us prevent them. On the other hand, most people believe that a good etiological theory of AIDS would help us stamp out the disease.
Rule: Other things being equal, the more likely that an explanation of p will help us control p (i.e., either prevent or instantiate it) the more A-important the problem of why-p.
c. Filling in Unaffiliated Blanks
We have noted that K is always incomplete in two senses: First, it is not the case that for every well-formed formula in the language either it or its negation is derivable (Cf., Hintikka). Secondly, we have no reason to believe that our present language is conceptually rich (or correct!) enough to answer all the questions which can be posed about the world.
How then do we choose which gaps to worry about? I have already described what I called deductive and explanatory gaps and described the factors which make them important. In these cases there is either a logical or a causal connection between the lacuna and the rest of our knowledge system. These connections increased the K-value of such problems. But there can also be blanks in K which are (almost) independent of the rest of K. How many sheets of letterhead are in my desk drawer? At the moment I can think of no other proposition in K to which the answer to this question would be relevant. Of course, if the correct number turned out to be 3X109, many of my beliefs (starting with the ones about the size of my desk, office, and building), would have to be revised. But I have expectations that the number will not be jarring.
Rule: When none of the expected possible answers to the problem disrupt the rest of K, the K-value of the problem is a minimum. Of course, the answer may have important action consequences - perhaps I need to type a letter of recommendation for a student over the weekend and sent it off Federal Express. Note that if a problem has little or no K-value or A-value, the mere fact that it looks soluble (high S-factor) does not make it a good choice. (S-factors are multiplied by, not added to, value factors).
4. Problems Arising from Systematic Flaws
Most inquiry, I believe, is triggered by either inconsistencies or gaps in our available knowledge. Above I have discussed some of the factors which help us decide which among these problems are most important.
There is another source of problems - those arising from what are perceived as structural flaws within tightly organized theoretical systems. As far as I can tell, problems concerning systemic flaws rarely arise in ordinary belief systems. They can be important, however, in quasi-axiomatic theories such as those found in mathematics, theology, and theoretical science.
Let me immediately proceed to some examples: There were many problems with the Ptolemaic system by the time of Copernicus. Some fall into categories already discussed above. For example, tables of the positions of heavenly bodies calculated using the system were inconsistent with current observations. In order to get a decent planetary theory, Ptolemaians had introduced the equant, a mathematical device for adding an oscillating linear component to the circular motion of the deferent. But this was inconsistent with the original program of "saving the phenomena" with the uniform circular motion which was believed to be the only one appropriate for celestial objects. There were also explanatory gaps. How could the trajectories of comets be accounted for using Ptolemaic devices?
We now come to two additional objections which fall into our new category of systemic flaws. The first stems from the necessity to proliferate epicycles. When Ptolemy introduced the one epicycle into the mathematical description of planetary motion, it had an obvious qualitative function. One could see upon simple inspection that simply adding an epicycle introduced the possibility of retrograde motion. Since all the planets exhibited retrograde motion, the epicycle seemed to be a brilliant and economical explanatory device.
However, in order to get the quantitative details of the orbit right, more and more epicycles were added. But there was no rhyme or reason to these additional epicycles. How many would be required? Would there be a different number for each planet? Astronomers felt that this profusion of epicycles made the system clumsy and ad hoc.
There was another problem with the system, hardly what one would normally call a flaw, yet puzzling, nevertheless. Venus is called the Morning Star and/or Evening Star because she is always viewed close to the part of the horizon which is hiding the sun. All the other so-called inferior planets (those which Ptolemy considered to be lying "beneath the sun") move similarly. In order to account for these observations, Ptolemy "tied" the center of these planets' deferents to the sun.
The fact that the orbits of all the inferior planets had this nice simple feature might seem to be an attractive feature of the system. Yet in a way the uniformity was gratuitous. Why should the deferents of the inferior planets be coordinated with the sun while those of the superior planets were not? It couldn't just be an accident, yet the theory provided no reason for it. Scientists seem to find unmotivated uniformities in a theory almost as distressing as ugly or inelegant features! Both are taken as indicators that our present theory needs revision.
Mendeleev's theory of the Periodic Table provides another example of problems arising from systemic flaws. Not only was his Periodic Law inconsistent with three sets of atomic weight data (the problem of the three reversed pairs), there was something odd about the whole conception of order by atomic weight being of physical importance. Paneth describes the problem as follows:
...the limited significance of the atomic weights has in a sense been implicit in any table of the periodic system because here - in contradistinction to its representation by curves - the elements were always arranged at equal distances from each other and not according to the actual values of the atomic weights. The first scientist who drew the consequences from this fact was the Swedish physicist Rydberg who, in a paper of l897, stated with admirable clarity: 'In investigations on the periodic system not the atomic weights, but the ordinal numbers of the elements, have to be used as independent variables.'
It is hard to tell from historical studies how much importance scientists assign to systemic flaws because, as our examples above indicate, there generally are enough other problems with the theory in question to provoke inquiry. What is clear, however, is that once a new, better theory is in hand, partisans of the later system laud its ability to illuminate or remove the flaws or incongruities inherent within the previous account. For example, in classical physics, inertial mass and gravitational mass are conceptually distinct although they always turn out to be numerically the same. Relativity removes this accidental degeneracy by fusing the concepts.
I will not attempt to give any further analysis of either the nature or weight of flaws but simply market their existence with a rather vague rule.
Rule: If T has a systemic flaw, then there is some motivation to replace T in K. The larger the flaw and the more content in T, the more important the problem.