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FUTURE LOGIC

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

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CHAPTER 57. FORMULA REVISION.

1. Context Changes.

2. Kinds of Revision.

3. Particularization.

1. Context Changes.

As knowledge evolves, our position shifts from one set of givens to another, and the inductive or deductive conclusion concerning any subject to predicate relation must be adapted to the new situation. All knowledge is contextual and tentative, anyway, in principle. Changes in context are to be taken in stride, as normal and to be expected. The current formula is revised, reformulating our state of knowledge in the light of new input, and then induction and deduction proceed as usual.

There are two kinds of context change. Starting with some formula, we discover new data, concerning the same subject to predicate relation. The new input may either be compatible with the preceding context, and be implicit in it and so without effect on it, or add to it, making it more specific. Or the new input may be incompatible with previous positions, in which case some conflict resolution is required.

We may discover such factual or logical errors in our beliefs by deductive or inductive means, from whatever sources.

Some new line of thought or generalization or observation may have taken place, which shows our preceding belief to be too limited or too vague or over-extended. Or the novelty involved may be relative: we may have come across this additional data before the data under consideration, but simply did not instantly make the conceptual connection; here, the novelty lies in our only now becoming aware of its impact.

The old and new information may have the same or different form: each may be positive, negative or bipolar; it may be particular, singular or general; it may have any modality; it may be elementary or compound; it may be a fraction, an integer, or even already in factorial form.

Whatever the case, formula revision is needed. We must step back and reconsider our situation in the light of the new data, formulating a new gross statement of our position to fit it, and drawing a new inductive conclusion from that.

Nevertheless, we want to retreat from previous positions as conservatively as possible. We do not want to radically revise our ideas or beliefs every time we face new material, though in some cases we may have to do just that. We do not want to overreact and lose valuable information, unless we have to. So we must learn to evaluate the seriousness of our predicament, and develop techniques for handling the various kinds of problems.

Formula revision, like factor selection, is largely an art, rather than an exact science. In some cases, the result is clear-cut; but in many situations, we are faced with a variety of paths which may seem equally credible, and the choice among them is intuitive and esthetic to a great degree. The task of logical theory is to facilitate decision making in such cases, by clarifying the options and their significances. It provides the artist with the tools, without rigidly prescribing their use.

2. Kinds of Revision.

We may distinguish two kinds of formula revision: amplification and harmonization.

Amplification occurs when the additional information is consistent with the original givens, and so can be simply conjoined to them. Note the connotation of growth. (Perhaps the name ‘apposition’ would have been more appropriate, but I settled on the latter because of its musical analogies.)

Amplification is of two kinds. It may narrow down the potential scope of a proposition; we call this process ‘specification’. Or it may broaden the actual scope of a proposition; we may call this ‘elaboration’. For example, given first that some S are P — if we thereafter find that some other S are not P, the initial proposition is further specified, whereas if we find that all other S are P, it is broadened. The logical possibility of the particular proposition to become general, is stifled in specification, but confirmed in elaboration.

Harmonization occurs when merging the two formulas would yield an inconsistent conjunction, so that some decision or compromise between them must be sought. We often call this process ‘reconciliation’.

Amplification may occur between propositions of similar or different polarity, provided they are not contrary or contradictory. Harmonization, in contrast, always concerns propositions of somehow opposite polarity, which are wholly or partly in conflict.

The premises and conclusions of these operations may be of similar strength, or weaker, or stronger, depending on our point of view.

Amplification of a formula is straightforward enough, formally speaking. Still, having assumed the original formula complete, in the sense of summarizing available knowledge, we may have made a generalization, and then deductions from this, which must now be reconsidered: they are now open to doubt, though not deserving of outright rejection. For the new, amplified formula will very likely suggest other inferences. Such review of the wider context is very often difficult; sometimes it is impossible to retrace our past course, and we must hope that inconsistencies will eventually arise, allowing us to streamline our knowledge base.

With regard to harmonization, or conflict resolution, one or both of the clashing, or adverse, theses must be changed to remove the problem and harmonize our knowledge. If one or the other is dominant, because of the greater credibility of its foundations, the other will be downgraded alone, or even totally eliminated if required; the latter may then be said to have conceded or yielded to the former. If they are of equal weight, for lack of a reason to prefer the one over the other, the common ground between them is sought: they in principle have to both be downgraded (though in certain cases it is permissible and sufficient to downgrade only one). Whatever the conflict, questions arise as to how deep a correction is called for, and in what direction it should be effected. Obviously, the retreat in quantity and/or modality should be the minimal permissible.

Here again, the consequences on the wider context of knowledge must be considered, to the extent possible, and these may in turn boomerang on the propositions under consideration, through successive formula revisions.

If a premise was itself obtained by deduction, and has been denied or downgraded for the purposes of conflict resolution, those prior sources are now known to certainly contain some error, and some or all of them must in turn be revised. Also, if either or both of the two original theses were generalized, before our becoming aware of their conflict, we can expect the inductive conclusion from their harmonization to disagree with one or both of these anterior inductions. If any deductions were made from a premise or its generalization, they are now put in some doubt, even if not automatically to be rejected.

Formula revision always means the conjunction of an old and new thesis. They may both be gross formulas (elementary or compound), or both be fractional formulas (isolated fractions or seeming to make up an integer). Or we may be dealing with the interactions between these various kinds of formula. Even deficient formulas not expressible as gross formulas may be involved. We have to look into all the possibilities.

All these issues will become clearer as we proceed with applications.

While the pursuit of consistency is recognized as in the logical domain by tradition, it has been dealt with in relatively vague terms. Effectively, we were given the tables of opposition as tools, but no step by step tactical instruction. We were told that in the event of inconsistency we should review our assumptions, but we were not provided with more specific guidance. The reason for this is that the classical model, where categorical propositions are all actual, is too limited and simplistic. The modal system provides us with a larger field of activity, complex enough to suggest the kind of difficulties which occur in practise.

3. Particularization.

Formula revision involves two initial theses, to be somehow fused in the conclusion. Formula revision occurs because of time lags between the emergence of items of knowledge, which may be consistent or inconsistent. But at the moment of revision, the time ingredient becomes irrelevant, and the theses are logically at the same level. One may not be regarded as more of a premise than the other.

Since formula revision involves two theses as premises, our understanding of each operation depends on which premise we compare to the conclusion. Looking at the one, we will notice this or that change has been effected on it by the process; looking at the other, the process has a different character. Both must be looked at, rather than subjectively focusing on either as ‘the premise’, to avoid misinterpreting the process.

Also, we may be tempted to compare the possible generalizations from the premises to the anticipated generalization from the conclusion. Or the one as-is to the generalization of the other. Inquiry of this sort is not without value, but should be done consciously, without confusion as to what precisely are the starting points and end result of the formula revision per se.

We should view formula revision as only including the work of amplification or harmonization as such. The generalizations which might have been made from the premises, or the generalization which normally follows the conclusion, are in principle optional and independent operations. Although, as we shall see, these may play a central role in the direction the formula revision takes.

Now, we would characterize as ‘particularization’ any process whose result is weaker than (or at best equal in strength to) the givens. This refers to decreases quantity and/or modality, essentially. Such contraction can be expressed as an increase in the number of weak factors, or as disappearance of stronger factors.

While formula revision does indeed usually involve particularization of the elements involved, there are certain special cases where it in fact yields a stronger conclusion. Sometimes there is a particularizing effect in one respect and a generalizing effect in another. The term ‘formula revision’ therefore has a more neutral connotation than the term ‘particularization’, and they may not always be equated, though they are often loosely-speaking confused.

Amplification of gross formulas is purely deductive revision, and only the subsequent generalization from its conclusions may be called inductive. But amplification of fractional formulas is itself inductive, quite apart from any subsequent generalization.

Harmonization, on the other hand, is only deductive in its application of the laws of opposition; with regard to its evaluations of credibilities, and its choices between alternative conflict resolutions, it is inductive, as much so as subsequent generalizations from its results.

We saw that generalization starts from a consistent body of knowledge, which, viewed simultaneously, has been summarized and factorized; thereafter, the strongest factor among those available is selected, so that the conclusion is generally superior to the premise.

Formula revision does not exactly refer to a mirror image of this process. It has a different structure and goal, the marriage of two premises. Particularization is not its essential goal, and not always its result. Furthermore, as we shall see, formula revision often solves problems by factor selection under the law of generalization.

Particularization is not a distinct process, but refers to certain specific applications of processes already defined. Consequently, it has no clear-cut ‘law’ or ‘rules’ analogous to those for generalization. We cannot simply convert the latter to predict the former. For instance, we cannot say that, since the latter prescribes that we favor quantity over modality, the former will affect modality before quantity. As will be seen, in some cases the result is one way, in other cases, the other way.

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2016-08-20T07:27:22+00:00