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The Logician © Avi Sion All rights reserved |
FUTURE LOGIC©
Avi Sion, 1990 (Rev. ed. 1996) All rights reserved.
CHAPTER 57.
FORMULA REVISION.
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.
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.
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. |
