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FUTURE LOGIC©
Avi Sion, 1990 (Rev. ed. 1996) All rights reserved.
CHAPTER 58.
GROSS FORMULA REVISION.
In gross formula amplification, we merely add the new data to the old,
usually to obtain a more complex gross formula, of greater informational
specificity. The tables of opposition tell us what elements are compatible; so
that, as long as all the elements involved may coexist, they can be conjoined
together consistently. The revised formula should then be analyzed factorially,
and its strongest factor selected in accordance with the law of generalization.
The new result may or may not differ from the original. Let us consider some
examples.
The easiest case is when the additional data is already implicit in the
original, as in A+I = A
or InOp+I = InOp.
Here there has effectively been no change of formula, though one can say that
the original formula is further confirmed. Any generalization made from the
original remains the same.
Next, we have the case of a formula of single polarity amplified by data
of the same polarity not implicit in it, as in A+In = AIn. Here, there is
a change of formula, but this has still no effect on eventual generalization,
except to render it more certain by diminishing the number of factors dropped by
it.
Thirdly, we have the case of a formula being so affected by additional,
though consistent, data, that its inductive results may change. For instance,
take ApIn+Op = ApInOp:
the original conclusion from ApIn
would be (An); after addition of Op,
the gross formula becomes ApInOp, and
its conclusion is (In)(IOp),
quite different. Or again, take ApInOp+O = ApInO: here, instead
of concluding (In)(IOp),
we are now forced to conclude (In)(IpO).
There is no need to list all possible cases of gross formula
amplification, for they are already effectively spelled out in the chapter on
factor selection. Such situations may occur either prior to generalization, when
we are still dealing with gross formulas, or at a more advanced stage of the
proceedings, provided we are able to retrace our past course.
In the event of a conflict between two formulas, if we are able to
remember the consistent gross formulas which gave rise to these generalizations,
or these deductive consequences of generalization, which are now in conflict, we
can resolve the conflict by merging the root propositions into a new, amplified
formula, which henceforth will serve as our inductive premise. But such radical
revision is not always possible.
All that has been said so far is applicable equally to naturals, to
temporals, and to mixed modality gross formulas.
Harmonization, or reconciliation, concerns situations where the original
and additional data are in conflict. In a looser sense, we might view even
amplification as involving a kind of 'conflict': that between the assumption
that the original thesis told the whole story, and the discovery through the new
input that it did not. But conflict here means specifically oppositional
incompatibility.
Some of the elements are contradictory or contrary, and the issue is how
to resolve such discrepancy once it is noticed. Something has to 'give', on one
side or the other or on both sides, for the controversy to be defused. An
adaptation of sorts is required.
Harmonization follows the familiar dialectical
pattern: thesis, antithesis, and synthesis.
Conflicts arise because, somewhere along the line, there has been an
over-generalization. We exaggerated the impact of certain observations,
insights, or narrow inferences, and, sooner or later, this was bound to lead to
inconsistency, whether with these same terms, or in consequent propositions
involving other terms. However, when the inconsistency arises, it may not be
immediately clear which side to blame for the error.
The way conflicts are to be resolved depends, firstly, on the relative
weights of the original and new data.
The degree of credibility of a thesis is a function of how clearly it is
formulated, how tightly it is argued, how much connection it has to the rest of
knowledge, how well established its conceptual and logical sources are, how
empirical it is, how often it is confirmed, how dependent we are on it for
practical purposes. Credibility is thus a variable appearance, a phenomenon by
which the many forces affecting our trust in a thesis are summed up at any given
point in the development of knowledge.
The credibility of a thesis is also a function of the existence of
alternative ideas or theories, and their relative credibility. A thesis may be
credible on its own, yet, when viewed in perspective, in comparison to others,
it may seem less likely than some other. When two or more theses are in
conflict, an evaluation of their relative weights must be made, and an order of
priority assigned to them.
Note that very often, we assign greater weight to an older thesis, one to
which we are more accustomed; such conservatism is somewhat justified on the
basis that the older thesis is time tested. But it can happen that a new thesis
may be shown, by thorough examination, to be in all respects of equal weight or
better; from a scientific point of view, age is ultimately accidental, and not
an over-riding cause for inertia.
The simplest case of conflict resolution is when either the original or
the new data, whichever, has greater weight. In such case, the dominant thesis
remains untouched by the conflict, and its submissive antithesis alone must bear
the burden of adjustment.
Each thesis may be an element or compound; and it may be positive,
negative, or of mixed polarity. Obviously, however, only elements of opposite
polarity may be inconsistent with each other; though of course, if elements of
opposite polarity are low enough on the scale of quantity and modality, they may
not create a conflict.
The general rule of harmonization of theses of unequal weight is, any
element(s) explicit or implicit in the submissive thesis, which are
contradictory or contrary to some element(s) in the dominant thesis, must be
diminished in quantity and modality until compatible, or even, if needful,
totally denied.
Some examples. If the dominant is An,
then all negative theses from Op on
up must be rejected. If the dominant is AIn,
then only Op may coexist, since Ep
would contradict In, and O would contradict A, and
perforce any statement still stronger than those would be contrary. If the
dominant is ApI, then only O
and/or Ep are acceptable, since On
or more would conflict with Ap, and E
or more would clash with I. If the
dominant is ApInO, then A, Ep, On,
and all their subalternants are impossible.
Thus, when one thesis is given more weight, the outcome is easy enough to
predict, using the rules of opposition. However, harmonization does not end
there. We must now consider the significance of our rejection of certain
elements. If, say, the submissive thesis was An,
and we were forced to particularize it to ApI,
then we must find out if any of our other beliefs include or imply An
or A, and correct these too in turn;
also, if any deductive inferences were drawn from An or A, they are now
rendered less sure (though not automatically deniable).
Furthermore, such weighted harmonization usually, though not always,
leaves a remainder. If, for instance, An
is pitted against En, with the latter
dominant, the result is simply En,
since even Ip would maintain
conflict. But if A is pitted against E,
again with the latter dominant, only A
is rejected, while its subaltern Ap
remains in force, since there is no reason to surrender it, and is to be added
to E, forming the compound ApE.
In this way, we do not push the harmonization beyond the necessary
minimum, but subtract only those levels of quantity or modality which cause the
interference. Such conjunction of remainder (of the submissive) to the dominant
follows the pattern already described under the heading of amplification.
Harmonization may change our original inductive conclusion, or leave it
unaffected. For instance, if we start with thesis ApIOp,
and find E from other sources, and
deciding the former stronger, resolve the conflict to ApIEpO,
the inductive conclusion is changed from (AEp)
to (IOp)(IpO). However, if AOp was
the original thesis, and E arose in
submissive conflict with it, the resolution AEp does not affect our initial induction of (AEp).
Again, all that has been said so far is applicable as well to naturals,
to temporals, and to mixed modality gross formulas.
By far more complex, and interesting, is harmonization between gross
formulas of equal weight which are in contention. We know of no reason to favor
one thesis over the other, so that their credibilities are in equilibrium, and
we seek their common grounds, their points of agreement. We are supposedly
unable to retrace the course which led up to them, or pin-point where we may
have erred, and so must deal with them as we find them, narrowing their scopes
to acceptable levels, down to where they can coexist.
More often than not, there may be more than one way to resolve the
conflict. This is where factor selection comes into play, providing us with a
convincing basis for arbitration. Without this beautiful instrument, such
formula revision would have been pure guesswork.
Conflicts between gross formulas of equal weight are resolved step by
step, as follows:
a.
Fuse the original two gross formulas, whatever they be, into one inconsistent
compound of mixed polarity; we can do this, because all elements have equal
weight. Then, separate the positive and negative elements or compounds from each
other, to form two gross formulas each of which has uniform polarity.
b.
Now, try out the hypothesis of harmonization of quantity. This means,
firstly, diminish the quantity of all the elements in both formulas to
particular; we shall call the resulting compound the 'quota'. Secondly, find the
'remainder' of this operation, if any; that is, what explicit or implicit
elements in the two formulas are compatible with the quota. Thirdly, fuse quota
and remainder into one compound, and identify its strongest factor.
c.
Next, similarly try out the hypothesis of harmonization of modality.
Thus, firstly, diminish the modality of all the elements in both formulas to
potential; the resulting compound being the 'quota'. Secondly, find the
'remainder' of this operation, if any; that is, what explicit or implicit
elements in the two formulas are compatible with the quota. Thirdly, fuse quota
and remainder into one compound, and identify its strongest factor.
d.
Lastly, compare the strongest factors of these two alternative
conclusions. Whichever has the stronger strongest factor is the revised gross
formula. The selected revised formula is, alone, the conclusion of the
harmonization process, the resolution of the original conflict, the goal we
pursued. Its strongest factor is accordingly the optional inductive conclusion,
if we choose to generalize at this point.
Thus, the decision as to whether to revise the formulas by reference to
quantity or to modality is made for us by factor selection. It is not
subjective, but systematic, controlled by what would be the inductive result in
either case. There is but one universal law for both inductive processes.
Our results demonstrate that, had we sought guidance through some
separate 'law of particularization', we would have made many mistakes, as will
be seen. Some conflicts are resolved on the side of contraction of quantity,
others are resolved through modality contraction, others still yield the same
result on either side; there is no general rule in that respect.
Particularization has no special basis, but is entirely determined by the same
law as generalization: namely, selection of the strongest factor.
The following table shows the results of conflict resolution obtained, by
the method described above, for gross formulas in the closed system of natural
modality. As usual, similar results could be worked out for temporal modality.
The positive and negative elements of the conflicting theses are grouped
separately, and listed under the heading 'given conflict'. The resolutions by
quantity and modality, and their respective strongest factor (SF)
are listed in adjacent columns. In each case, the quota is displayed first, and
the remainder if any is added on to it with a '+'
sign. The corresponding strongest factor, note well, concerns the complete
compound of quota and remainder (this compressed version is not shown in the
table, being easily constructed). We know the selected factor in each case from
our prior inquiry on this topic (the reader is referred to the chapter on factor
selection).
Finally, the inductive conclusion is given: notice that this is the
selected factor with the lowest ordinal number of the two. The data has been
classified by similarity of inductive conclusions, to show the inherent
continuities. The range is F3-F10.
The revised gross formula is the one whose strongest factor is selected as the
inductive conclusion. Observe the uniformities found in each grouping.
Note the inclusion in this table, for the sake of completeness, of cases
where the positive and negative sides are not in conflict (these are marked
'okay'). Excluding these, there are in all 50 valid moods. Among them, 18 have
been earmarked with an asterisk (*):
these may be viewed as the main moods, in that they present the strongest
conflicts for each distinct revised formula. Examples are given after the table. Table
58.1 Conflict
Resolutions for Equal Gross Naturals. (Click
to see table.)
Some examples are in order. ·
ApIn
versus Ep. Contracting their
quantities, we obtain InOp; we can
still keep the Ap explicit in ApIn, since it is not in conflict with Ep; hence, we get ApInOp
(strongest factor F6), under this
hypothesis. Alternatively, contract their modalities, to obtain ApEp;
this allows for I; so, ApIEp (strongest factor F3)
is yielded by this hypothesis. Comparing the two, the latter is found stronger,
and so the revised gross formula is ApIEp,
and F3 is its eventual
generalization. Effectively, the harmonization particularized In
to I, necessity to actuality, in this case. ·
An versus En. Quantity
harmonization is InOn; there is no
remainder since even Ap is excluded
by On and Ep by In. Modality
harmonization is ApEp (AE
being impossible); with remainder IO
(rather than A or E, which would be
asymmetrical). InOn has F5
as its strongest factor, whereas ApIEpO
has F10, so the former wins. In this case, the harmonization
particularized An to In
and En to On, that is,
quantity instead of modality. ·
An
versus Op. This is a case where both
directions lead to the same conclusion. Whether our quota and remainder are InOp
and A, or ApOp and AIn,
the result is still AInOp (SF = F6). Here, particularization has downgraded both the quantity (An
to In) and the modality (An
to A).
An especially interesting case is that of A
versus E. We saw that, within the
closed system of actual propositions, this conflict yields conclusion IO,
or (I)(O). However, here, in
the wider context of natural modality, we see that the conclusion by any means
is, more fully, ApIEpO, or (IOp)(IpO). Thus, we agree concerning IO, but we note that ApEp
were previously hidden in the traditional system. In any case, this shows that
our broadened theory of induction is consistent and continuous with the
classical.
A similar approach may be used for conflict resolution among gross
formulas in the open system of mixed modality. Since the results stem directly
from tables of opposition and factor selection, they will not be listed in this
treatise, to avoid excessive volume. The processes involved are now firmly
established and clearly exemplified, and that is sufficient for our present
purposes. |
