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FUTURE LOGIC

© Avi Sion, 1990 (Rev. ed. 1996) All rights reserved.

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CHAPTER 48. THEORY SELECTION.

1. The Scientific Method.

2. Compromises.

3. Theory Changes.

4. Exclusive Relationships.

1. The Scientific Method.

The ‘scientific method’ consists in trying out every conceivable imaginary construct, and seeing which of them keep fitting all new facts, and which do not. Those which cease to fit, must be eliminated (or at least corrected). Those which continue to fit, are to that extent increasingly probable, until they in turn cease to fit. Whatever theory alone survives this eliminative process, is effectively proved, since all the shares of probability have been inherited by it.

In practise, the construction of alternative postulates, and the discovery of the full implications of each, are both gradual processes. We do not know these things immediately. Also, the given context is not static, but itself grows and changes as we go along. This feeds our imagination and insight, helping theory developments, and stimulating further research.

We may start with one or two partially developed theories, and slowly find additional alternatives and make further predictions, as events unfold and the need arises. The extent of our creative and rational powers affects the exhaustiveness of our treatment.

Several theories concerning some group of phenomena may, at any stage in the development of knowledge, simultaneously equally fulfill the criteria of relevance; namely, conceptual meaningfulness, internal consistency, ability to explain the phenomena in question, and compatibility with all other empirical givens so far.

In formal terms, this simply means that competing theories T1, T2, T3,… may, while being contrary to each other, each still logically imply the already experienced phenomena Q. That is, the hypotheticals ‘if T1, then Q’, ‘if T2, then Q’, etc., are formally compatible, even though ‘T1 or else T2 or else T3…’ is true.

The statement that our list of theories for Q is exhaustive, has the form ‘If T1 or T2 or T3… , then Q’, plus ‘one of T1, T2, T3… must be true’. Although it may be hard to prove that our list is exhaustive, we may contextually assume it to be so, if every effort has been expended in finding the alternative explanations.

Each theory contains a number of postulates: T1 = p11 + p12 + p13 +…, T2 = p21 + p22 + p23 +…, and so on. Some of these postulates might well be found in more than one theory; it may be, for instance, that p13 = p29 = p36. But each theory must have at least one distinctive postulate or a distinctive combination of postulates, which makes it differentiable from all the others.

Also, the phenomenon or group of phenomena labeled Q are already known empirically, and supposed to be equally embraced by the various theories put forward. But each theory may have other implications, if we can determine them through reason, open to empirical testing, though not yet tested.

Each theory has a set of predictions: T1 = q11 + q12 + q13 +…, T2 = q21 + q22 + q23 +…, and so on. Some of these must be in common, constituting the given phenomena Q which gave rise to our theorizing in the first place. That is, say, Q = q15 = q27 = q31.

The rest may likewise be all identical, one for one; or some overlaps may occur here and there, while some predictions found here are missing there; or, additionally, some conflicting predictions may occur, so that one or more theories affirm some prediction that certain other(s) deny.

In principle, it is conceivable that the various theories all make only the same predictions, in which case they are factually indistinguishable, and we cannot choose between them on an empirical basis, though we may still refer to utilitarian criteria.

Most often, however, we may eventually find distinctive further predictions for each theory, or at least some which are not common to all. A difference in postulates usually signifies a difference in predictions. Here, we must be careful to differentiate between:

a. a prediction implied by, say, T1, but neither implied nor excluded by T2, T3, etc. — if such a prediction passes the test of experience, T1 is confirmed, but T2, T3,… are neither confirmed nor rejected, though their probabilities are diminished by the increased probability of T1; whereas if such a prediction fails the test of experience, T1 is rejected, while T2, T3,… become more probable by virtue of being less numerous than before; and:

b. a prediction implied by, say, T1, and logically excluded by T2, T3, etc. — if such a prediction turns out empirically successful, T2, T3… are rejected, and (if only T1 is leftover) T1 is proved; whereas if such a prediction turns out empirically unsuccessful, T2, T3,… are confirmed by their anticipation of the negative event, while T1 is rejected.

Thus, theory selection depends on finding distinctive predictions, which can be used in adductive argument or apodosis. These should be empirically testable predictions, of course.

If one or more theories have an implication which the others lack, though are compatible with, or if one or more theories have an implication which the others are incompatible with — we have at least an eventual source of divergent probabilities, allowing us to prefer some theories over others, even if we cannot eliminate any of them; and in some cases, we may be able to eliminate some of them, and maybe ultimately all but one of them.

These methods are of course well known to scientists today. But all this concerns not only scientists at work, but the development of opinions by individuals in every domain. It is the ‘trial and error’ process through which we all learn and improve our knowledge.

Even if at a later stage we might manage to validate some of our beliefs more deductively and systematically, this is the method we usually use to initially feel our way to them and develop them. Knowing the ‘scientific method’ explicitly and clearly can help individuals to make their personal thinking on topics remote from abstract science more scientific.

2. Compromises.

We have described the ideal pattern of scientific evaluation of theories; but, in practise things are not always so neat, and we often have to make do with less than perfect intellectual situations.

a. For a start, the coexistence of conflicting theories may be viewed less generously as a source of doubt for all of them; they may each be corroborated by the delimited data they explain, but their mutual incompatibility is a significant inconsistency in itself.

We may remain for years with equally cogent, yet irreconcilable theories, which we are unable to decide between. Our minds are often forced to function with a baggage of unresolved contradictions.

In such case, we suspend judgment, and make use of each theory for pragmatic purposes, without considering any as ultimately true as a theoretical image of reality.

Even as we may give more credence to one theory as the more all-embracing and most-confirmed, or as the simplest and most-elegant, we may still withhold final judgment, and not regard that theory as our definite choice, because the evidence does not seem to carry enough conviction.

b. Sometimes the available theories only partially explain the given data. They may embrace some details in common, with comparable credibility, but one may be more useful than the others in some areas, while another is more thorough in other respects.

Although this suggests that the theories have distinct implications, they are each supportable on different grounds, perhaps with the same overall probabilities. We may not find a way to choose between them empirically, or to unify them somehow.

In such case, narrowing the field by elimination of alternatives is hardly our main concern; rather, we are still at a stage where we need a unifying principle, we effectively do not have a theory in the full sense of the term. An example of this is the particle-wave dichotomy, and the search for a unified field theory to resolve it.

Sometimes, we know our list of available theories is faulty, because their connections to the data are not entirely satisfactory and convincing. In that case, our ‘if-then-‘ statements are themselves probabilistic, rather than necessary. Our ideas then had better be called notions or speculations.

c. Sometimes, no theory at all can be found for the phenomena at hand, for years. There may be seemingly insurmountable antinomies. We are forced to wait for an inspiration, a new idea, a new insight, a new observation, which might lead us to a satisfactory solution.

Because it is in some domains very difficult to develop a meaningful and consistent conceptual framework, we may be forced to accept one which is conceptually or logically flawed, as a working hypothesis.

Sometimes, the problem may be shelved, because its impact lies elsewhere, creating doubts and questions in distant disciplines. For example, Heisenberg’s Uncertainty Principle seems to assault our common-sense conceptions of determinism for inanimate matter: this might later be resolved by Physics itself, or might remain an issue for Philosophy to deal with.

In practise, an imperfect tool of knowledge is often better than none at all. We prefer to have a theory formulated in terms of vague or seemingly contradictory concepts, with practical value, than to remain paralyzed by a dogmatic insistence on an elusive ideal.

d. Thus, sometimes, although a theory may apparently be strictly speaking felled by hard evidence, and we are unable to pinpoint its mistakes, we may nonetheless pragmatically hang on to it, if there is no other to replace it. We simply mentally attach a reservation to it, retain an awareness of its limitations, and move on cautiously to practical applications.

This is especially justifiable when the reason for its empirical rejection was an extreme situation, or ‘boundary case’, not encountered in the normal course of events. We then recognize the need to specify some limiting conditions to the theory, without being able to fulfill this need more precisely at the present stage.

3. Theory Changes.

Even when a theory is found empirically wrong, yet has alternatives, we may avoid outright rejection, and rather first seek to rectify it somehow, limiting it in scope or shifting some of its postulates slightly. This is feasible on the ground that there must have been some grain of truth in the original insight, and we may be able to tailor our assumptions to fit the new data.

Even if we cannot immediately conceive a correction, we may still choose to hang on to the original idea in the hope of its eventual redemption. We all carry a baggage of beliefs through life, which we know lead to contradictions or have been apparently disproved or rendered very improbable; we keep them in mind for further verification, anyway. This attitude taken to an extreme is of course contrary to logic, but within reasonable bounds it has some utility.

The pursuit of truth is not cold and vengeful, as it were, towards flawed theories, intent on rarefying the alternatives at all costs. Rather, it is a process of flexible adaptation to changing logical conditions. Our goal is, after all, to indeed arrive at truth, and not merely to give the impression that we did.

If we manage to modify a theory well enough to fit the new facts, then effectively we have developed a new theory. It may be a new version of the old, but still merits consideration as a theory in its own right.

We defined a theory as a number of distinctive postulates together implying a number of predictions. More loosely, the range of applicability of a theory might be varied, without radically affecting the substance of its proposals or its details.

Also, we may distinguish between essential postulates and postulates open to change. The former may be generic proposals, the latter specifics within them which we have not yet resolved — postulates within postulates, as it were. Likewise, we might distinguish between generic predictions, which are necessary consequences, and their specifics, which may be less firmly bound to the postulates.

With these thoughts in mind, we can talk of a theory ‘changing’, while remaining essentially the same theory. This may refer to changes in scope or changes in detail which do not affect the main thrust of a hypothesis. In other words, a theory may involve logical conditional propositions, as well as categoricals, leaving room for variations.

Denial of a postulate may mean: either denial of the broadness of the postulate, without excluding the possibility that a more moderate formulation is acceptable, or denial of a specific position, which can be replaced by another specific position with the same generic impact, or radical denial of a generic position, in the sense that all its possible embodiments are consequently denied.

Denial of a prediction may accordingly either merely cause us to regard the theory as having a more limited applicability than originally thought, or to make relatively small corrections in our assumptions, or force us to formulate a completely new theory.

Thus denial of a postulate or prediction does not necessarily mean rejection of the whole theory as such, it may be only partly discredited, requiring a less ambitious or a slightly altered formulation.

Accordingly, a new theory may totally replace an old one, or it may embrace it as a special case. For example, Einstein’s Relativity resulted in our particularization of Newtonian mechanics to commonplace physical levels; it was thenceforth seen as inapplicable to more extreme astronomical or sub-atomic situations, but retained much of its usefulness.

4. Exclusive Relationships.

We know from apodosis that affirmation of a postulate implies acceptance of all its necessary predictions (even those untestable empirically), and denial of a prediction obliges us to reject (or at least change) the postulates which necessitate it.

Denial of a postulate does not engender denial of its still untested predictions; it only diminishes their probability. However, empirically untestable predictions can still be discarded, if we can show them to be logically exclusive to some empirically rejected postulate(s). The argument is a valid apodosis:

Only if postulates p, then predictions q

(implying: if notp, then notq),

but not p,

hence, not q.

Doubt may remain, depending on how sure we are of the postulate’s denial, and especially on the strength of the exclusiveness. Also, what has been said does not prevent the possibility that a slightly different version of the predictions still hold.

Likewise, affirmation of a prediction does not in itself prove any of the postulates giving rise to it, but only confirms them. However, theoretical postulates can still be established, if we can show them to make some logically exclusive empirically tested prediction(s).

Only if postulates p, then predictions q

(implying: if notp, then notq),

but q,

hence, p.

This too is a valid apodotic argument. Again, such exclusiveness may often be hard to determine indubitably, but the principle remains valid.

It is not always easy or even possible to find such exclusive relationships. In such case, we are of course limited to the adductive approach. Note that, just as necessity is the extreme of probability, so apodosis is the limiting case of adduction: they differ in degree, not in essence.

Thus, it is not permissible to regard, as some philosophers seem to have intimated, science as incapable of certitude in disproof of empirical matters, or of certitude in proof of theoretical constructs. Admittedly, a good deal of theory selection is based on the processes of adduction and elimination; but this is only one arrow in the arsenal of the scientific method.

If we regard science as capable of establishing logical (or mathematical) connections for the purposes of mere confirmation or undermining of theories, then it is equally capable in principle of establishing exclusive connections which can be used for the above described demonstration purposes.

All the hypothetical forms are structurally identical, irrespective of the polarities of their theses. If any one of them is recognized as accessible to science, then they are all equally so. If we can rely on the ‘if p, then q’ of adduction, then we can just as well rely on the ‘if notp, then notq’ of exclusive apodoses.

There is no intent, here, to underrate the importance of competitive induction, only to point out that other, more certain, means are sometimes available to us, though not always. What is at issue here is the suggestion that we only have a choice of a-priori, axiomatic knowledge versus a posteriori, probabilistic knowledge.

There is an in-between alternative: knowledge which is at once theoretical, and certifiable, and empirical. It is arrived at through the logical discovery of exclusive relationships between postulates and predictions. This methodology has the stamp of approval of logical science, and is perfectly reliable.

Indeed, all our so-called mind-set concepts, even the axioms of logic, have such exclusive-empirical grounding, as well as self-evidence (i.e. self-contradiction of their contradictories). Every particular proposition, for example, appeals to this reasoning. More generally, any concept which appears as sole available interpretation or explanation of the experienced phenomena is justifiable on that basis.

2016-08-20T07:22:41+00:00