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

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1. Prediction.

2. The Uniformity Principle.

3. The Law of Generalization.

1. Prediction.

We indicated in the previous chapter that induction depends on factorial analysis of our knowledge context. Once this is done, we are usually faced with a number of factors to choose from, which represent the various outcomes our knowledge may move towards.

But reality can only exist in terms of integers; it is only the deficiencies of knowledge which make possible the indefinite situation of integers in disjunction. On this basis, we know for sure that one, and only one, of the factors of a formula can be factually correct. The other alternatives, if any, are a sign of doubt; they do not represent a fact of reality.

There is no recognition of an ‘Uncertainty Principle’ in this logic. Uncertainty is a phenomenon of consciousness, with no equivalent in the Object. It is perhaps conceivable that certain motions of matter occur indeterministically, without order or cause, as modern Physics suggests. But, according to Logic, whatever has occurred, once it has occurred, is firmly fixed, be it discernible or not.

The inductive process of factor selection consists in anticipating reality, trying to predict, from the available knowledge of contextually allowed factors, which of the factors is most likely to emerge as the right one. In some cases, while such a definite result is inaccessible, we try to at least approach it, by diminishing the number of factors. In other cases, the given formula has only one factor, anyway, so there is no problem, and the result is deductive.

The question arises, how do we know which factor is most likely? Formally speaking, they are all equally possible; this is the verdict of deductive logic. But induction has less strict standards of judgment.

2. The Uniformity Principle.

The principle involved in factor selection may be glimpsed in the paradigm of generalization from actual particulars. We will call it the uniformity principle, understanding by this term a broad, loose reference to repetitiveness of appearances, coherence, continuity, symmetry, simplicity.

Consider for example generalization from I. The general alternative (A) is more likely then the contingent one (I)(O), because the former involves no unjustified presumption of variety in polarity like the latter. We are not so much inventing information, as refraining from baseless innovation and maintaining continuity.

Thus, the qualitative inertia of the first factor is more significant than the quantitative change (from some to all) it introduces. In contrast, the second factor introduces just as much quantitative change (through the O), so that it is no better in that respect; and additionally, to its detriment, a novel fragmentation of the extension, absent in the original data and the preferred factor.

We obviously select the factor most resembling the given data, as its most likely outcome. Unless or until we have reason to believe otherwise, we assume the given information to be reproduced as far as it will go. We can thus express the principle that, in factor selection, the most uniform factor is to be accorded priority.

Ontologically, this signifies the assumption of maximum uniformity in the world, in preference to an expectation of diversity. Events are believed representative, rather than unique. The world seems to tend in the direction of economy.

On a pragmatic level, the reason for it is that a generality is easier to test than a particular statement, since deductive logic, through which the consequences of assumptions are inferred, requires general statements. Thus, the preference for uniformity also has an epistemological basis. In the long run, it assures us of consistency.

The uniformity principle, then, is a philosophical insight and posture, which sets an order of priority among the factors of a formula.

But, it is important to stress that this principle is merely a utilitarian guideline to factor selection, it does not in this format have the binding force or precision found in the laws of deductive logic. Inductive logic merely tries to foresee the different situations which may arise in the pursuit of knowledge, and to suggest seemingly reasonable decisions one might make.

Choices other than those proposed remain conceivable, and might be intuitively preferred in specific cases. There is an artistic side to induction, to be sure. Our general recommendations, however, have the advantage of having been thought out in an ivory tower, and of forming a systematic whole.

3. The Law of Generalization.

Fortunately, we can neatly summarize the results, obtained by application of the uniformity principle, in a single, precise law for generalization. This has greater practical value.

The reader will recall that when the integers were defined, they were organized, in order of the number of their fractions. Those with the least fractions came first, then those with two fractions, then those with three, and so on. Within each such group, comparable integers of opposite polarity were paired off, with the more positive one preceding the more negative. Also, they were ordered according to their level of modality in the continuum concerned.

Thus, in the closed systems of natural or temporal modality, the 15 integers F1F15, and in the open system of mixed modality, the 63 integers F1F63, are ordered in such a way that their numbers reflect their degree of ‘strength’. The lower the ordinal number, the stronger the factor.

A stronger factor is less fragmented (i.e. has less fractions, out of a possible 4 in the closed systems, and 6 in the open). It is closer to universal (in the closed systems, F1F4 are universal; in the open system, F1F6). It has higher modality; for instances, (An) is higher than (AEp), (In)(On) is higher than (IOp)(IpO).

Thus, in any factorial formula, the factors in the series are already numerically ordered according to their relative strengths. This was not done with factor selection in mind, but because of the clarity it generated in the doctrine of factorial analysis. As detailed work will presently reveal, it turns out that:

In any factor selection, the strongest factor is the one to prefer.

This is the law of generalization.

In a few exceptional cases, the first two factors must be selected, in disjunction, for reasons that we shall see. But, on the whole, this law holds firm, and successfully sums up all our findings.

This law is a summary of results. In point of fact, it only emerged at the end of painstaking analysis of a large number of specific inductive arguments, attempting to make sense of them, case by case, through the intuited uniformity principle. However, once arrived at, it seems obvious. But the true justification of it all, is the consistency and cogency of the totality of the theory, with all its details, of course.

Note well, incidentally, that henceforth, to avoid neologisms, the term ‘generalization’ is used in a general sense not limited to quantity. It is applied to either increase in quantity, from some to all; this is extensional generalization. And/or to increase in modality from possibility to actuality to necessity; this being modality generalization, (natural and/or temporal, as the case may be). Likewise, the term ‘particularization’ may be used for any such type of decrease.

But most precisely, generalization may now be defined as inductive selection of the strongest factor(s) of a formula, by suppression of weaker factor(s). Particularization will be dealt with under the heading of formula revision.

Generalization can, therefore, be applied to deficient states of knowledge not expressible in gross formulas. We saw in the chapter on factorial analysis that, while all disjunctions of integers represent deficient states of knowledge, some of them do not correspond to any gross formula. In other words, gross formulas with two or more factors are not all the possible states of relative ignorance, other combinations of factors are conceivable.

The law of generalization makes selection of the strongest factor legitimate in such already factorial formulas, too.