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FUTURE LOGIC

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

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CHAPTER 47. THEORY FORMATION.

1. Theorizing.

2. Structure of Theories.

3. Criteria.

4. Control.

1. Theorizing.

Every theory involves an act of imagination. We go beyond the given data, and try to mentally construct a new image of reality capable of embracing the empirical facts. The more nimble our imagination, the greater our chances of reaching truth. Think how many people were stumped by the constancy of the velocity of light discovered by the Michelson-Morley experiment, until an Einstein was able to conceive a solution!

Without creativity our understanding would be very limited. We need it both to construct hypotheses, and to uncover their implications. Neither of these achievements is automatic. Conceiving alternatives and prevision both involve work of imagination.

In practise, no theory is devoid of hidden assumptions, besides its stated postulates. We may try to be as explicit as possible, but often later discover new dependencies. Thus, with Newton’s assumption of Euclidean geometry, which was much later discarded in the General Relativity theory.

Thus, our theorizing is always to some extent limited by our ability to make mental projections, and the depth and breadth of our conceptual insight.

These faculties of course depend very much on the mind being fed by new empirical input. Creativity depends on the ideas provided us by new experience, and revision of fundamentals depends on the stimulus of discovered difficulties.

Each individual has his own limits. People often remain attached to preconceptions, and are unable or refuse to consider alternatives. This can be a weakness or vice, but it is also a normal part of the way the mind works.

We have to hold something steady while considering the impact of new perspectives. We cannot re-invent the wheel all the time, without justification. We review our presuppositions, only when the need arises, when some empirical problem presents itself.

This does not exclude ‘art for art’s sake’. The pursuit of theoretical improvements is always permissible. But it is anyway serial. We are mentally unable to change all our knowledge at once, but are forced to proceed in an orderly, structured manner, gradually focusing on this or that proposed change while the rest is taken for granted.

Logical and mathematical skills also count for much in the development of theories. Many a wild speculation is built on unsound reasoning. These skills include, among many others: clarifying inter-relationships, finding analogies and implications, distinctions and contradictions, ordering information.

A good grasp of the methodology of adduction is very important. It opens minds to the ever-present possibility of alternative explanations and further testing. Adduction is essentially a process of trial and error.

The tentative, and often transient, nature of theories, as well as their ability to make impressive predictions, has been exemplified in some stunning scientific revolutions in the past few centuries. Even seemingly unshakable theories have been known to fall, and some of the discoveries occasioned by the new perspectives would have seemed unthinkable previously.

There is much to learn by observing the ‘life’ of theories, their historical courses, the ways they have augmented or displaced, complemented or contested, each other, their dynamics.

2. Structure of Theories.

Any one general proposition can of course be viewed as in itself a theory, and the processes of generalization and particularization are samples of adduction. The relation of a general proposition to particular observations, is logically one of antecedent and consequent, though the chronological order may be the reverse.

However, we normally use the term ‘theory’ in larger, more complex, situations. We think of a rational system for understanding some subject-matter. The sciences of course consist of theories, which attempt to explain the empirical phenomena facing them. But we also build small personal theories about events in our lives of concern only to ourselves.

Let us examine the structure of theories. A theory (say, T) consists of a number of conceptual and/or mathematical propositions. Among these propositions, some cannot be derived from the others: they may be called primary; the others, being of a derivative nature may be called secondary. The derivation, of course, is supposed to be logically or mathematically flawless.

Among the primary propositions, some are distinctive to that theory: they are called its postulates (label these p1, p2, p3, etc.). Postulates should be as limited in number, as simple in conception and broad-based, as we can make them. Though postulates may be particular (as for instance in a theory concerning historical events), the postulates of sciences are normally general propositions. These are usually obtained by generalization from directly observable particulars, but not always (consider, for instance, the idea of curved space).

If a primary proposition is not distinctive to that theory, but found in all other theories of the subject under investigation, then it is not essentially part of that specific theory, but stands outside it to some extent. Such external primaries may be transcendent axioms, or they may be borrowed from some adjacent or wider field of investigation, taken for granted so long as that other theory holds.

The secondary propositions are called the theory’s predictions (label these q1, q2, q3, etc.), even if not distinctive to that theory.

Some predictions are testable, open to empirical observation, perhaps through experiment; some predictions are intrinsically difficult to test. To the extent that a theory offers untestable predictions, it tends to be viewed as speculative. Among the testable predictions, some are normally already tested: they provided the raw data around which the theory was built; others may be novel items, which anticipate yet unobserved phenomena, providing us with opportunity for further testing.

Predictions are derived from the postulates by a process of production, mediated by the relatively external primaries. We regard the external primaries as categorical, as far as our theory is concerned, so that they may remain tacit, though they underlie the connections between our postulates and predictions.

Thus, postulates are hypothetically linked to predictions, in the way of antecedent to consequent. The antecedent need not include the external primaries, since the latter are considered as affirmed anyway, and were used to establish the connection. For example, Newton’s laws of motion were the postulates distinguishing his mechanics, while his epistemological, ontological, algebraic and geometrical assumptions lay outside the scope of his theory as such.

Theories often draw on findings in other domains outside their direct concern, and may have powerful repercussions in other domains. Thus, Newton had to develop calculus for his mechanics; this mathematical tool might well have been researched independently, as indeed it was by Liebnitz, but it was also stimulated or given added meaning when its value to physics became apparent.

A theory, then, may be described as follows, formally:

T = If p1 and p2 and , then q1 and q2 and

Note that this overall relation may in some cases be supplemented by narrower ones. It may be that all the postulates are required to make all the predictions; or it may be that some of the postulates are alone sufficient to make some of the predictions.

3. Criteria.

Theories serve both to explain (unify, systematize, interpret) known data, and to foresee the yet unknown, and thus guide us in further research, and in action. The criteria for upholding a theory are many and complex; they fall under three headings:

a. Criteria of relevance. A theory may be upheld as possibly true, so long as it is meaningful, internally consistent, applicable to (i.e. indeed implying) the phenomena under investigation, and consistent with all other observation to date.

This possibility of truth signifies no more than that the theory is conceivable, and has some initial degree of probability. This may be called relevance.

b. Criteria of competitiveness. But the work of induction is not complete until the theory has been compared to others, which may be equally thinkable and defensible in the given context. Induction depends on critically pitting theories against each other.

Two or more theories may each fulfill the conditions of relevance, and yet be incompatible with each other. They might converge in some respects, having some postulates and/or predictions in common, but found divergent in other respects.

It might be possible to reconcile them, finding postulates which succeed in encompassing the ones in conflict, while retaining the same uniform predictions. Or we may have to find exclusive predictions for each, which can be tested empirically to help us make a choice between postulates.

This is where adduction comes into play. It is the process used to evaluate, compare, and select theories through their predictions. It is the main tool for the induction of theories, commonly known as ‘the scientific method’.

c. Utilitarian criteria. Although utility is a relatively ‘subjective’ standard for evaluating theories, being man-centered, it plays a considerable role. For us, knowledge is not a purely theoretical enterprise, but a practical necessity for survival. We use it to support and improve our lives.

We judge a theory to some extent by how accessible it is to our minds, by virtue of its simplicity, or the elegance of its ordering of information. All other things being equal, we would choose the theory which approaches this ideal most closely, on the general grounds that the world is somehow simple and beautiful. The onus of proof is on the more complex, the more ‘far-fetched’, theories: avoidable complications need additional justification.

However, simplicity should not be confused with superficiality. People often opt for overly simplistic viewpoints, which only take the most obvious data into consideration, and ignore deeper issues. A theory should preferably be simple, but not at the expense of accuracy; it must cover more known phenomena and answer more questions, than any other, to be credible. The easy solution often has a limited data base, and reveals a naive outlook.

Apart from such rationalistic and esthetic bias, we also look at the implementation value of a theory. Even if a theory or group of theories is/are known to contain some contradictions, we may hang on to them, in the absence of a viable substitute. We assume that the problem will eventually be resolved; meanwhile, we need a tool for prediction, decision-making and action, however flawed. Thus, for example, with the particle-wave dichotomy in physics.

We will look at some of the dynamics of theory selection in more formal terms, in the next chapter.

4. Control.

It must be stressed that the primary problem in theorizing is producing a theory in the first place. It is all very well to know in general how a theory is structured, but that does not guarantee we are able to even think of an interpretation of the facts. All too often, we lack a hypothesis capable of embracing all the available data.

Very often, theories regarded as being ‘in conflict’, are in fact not strictly so. One may address itself to part of the data, while the other manages to deal with another segment of the data; but neither of them faces all the data. Their apparent conflict is due to their implicit ambition to fit all the facts and problems, but in reality we have no all-embracing theories before us.

However, quite often, we do easily think up a number of alternative theories. In that case, we are wise to resort to structured theorizing and testing, to more clearly pose the problems and more speedily arrive at their solutions.

This is known to scientists as ‘controlled experiment’, which consists in changing (by small alterations or thorough replacement) one of the variables involved, while ‘keeping all other things equal’. The method is applicable equally to forming theories and to testing them (by simple observation or experiment).

Structuring consists in ordering one’s ideas in a hierarchy, so as to systematically try them out, and narrow down the alternatives.

a. List the independent issues. A subject-matter may raise several questions, which do not seemingly affect each other; these various domains of concern must first be identified. For example, in geometry, whether or not space is continuous, and whether or not parallels meet, seem to be two separate issues.

b. For each issue, list the alternative postulates, which might provide an answer. Combine the various postulates of each issue, with the various postulates of all other issues involved, to yield a number of theories (equal to the product of the numbers of postulates in the various issues). Some of these combinations may be logically inconsistent, and eliminatable immediately; in other words, there may be some partial or conditional dependencies between the issues.

c. Within each issue, distinguish between alternative postulates which are radically different, and between postulates which may be viewed as minor alterations of one common assumption. In the former case, we may expect to eventually find some radically different predictions from the alternative postulates. In the latter case, varying the main postulate may merely cause small variations in the predictions, and the work involved is more one of fine tuning our theory.

d. The best way to test ideas is to organize them in terms of successive specific theses and antitheses, as follows:

Starting with the seemingly broadest, most independent issue, focus on one postulate p1, and find for it a prediction q1, which is denied by the denial of that postulate, thus:

If p1, then q1, but if not p1, then not q1.

Next, suppose that p1 wins that contest, and concentrate on the next issue; within that issue, consider one postulate p2, and again look for some exclusive prediction q2 for it:

If p2, then q2, but if not p2, then not q2.

Proceeding in this manner, we can gradually foresee the course of all possible events, and eventually of course test our results experientially. This is an ideal pattern, in that it is not always easy to find such distinctive implications; but it often works.

The trick, throughout the process of theorizing and testing is to structure one’s thoughts, so as to advance efficiently to the solutions of problems. A purposeful, constructive, orderly approach, is obviously preferable to a hesitant, vague, muddled one. It often helps to use paper and pencil, or computer, and draw flow-charts; it generates new ideas. Sometimes, of course, it is wise not to insist, and to let the mind find its way intuitively.

I would like to here praise the inventors and developers of the modern personal computer, and all software. Imagination and verbal memory greatly improve the mind’s ability to formulate and test thoughts. The invention of the written word, and pen and paper to draw and write with, provided us with an enormous expansion in these capabilities.

The word-processing and other computer applications increase our mental powers still further, by an enormous amount. A patient person can keep improving ideas on a screen, again and again, to degrees which were previously beyond reach. This has and will make possible tremendous advances in human thinking.

2016-08-05T10:03:48+00:00