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FUTURE LOGIC©
Avi Sion, 1990 (Rev. ed. 1996) All rights reserved.
CHAPTER 59. FACTORIAL FORMULA REVISION.
1.
Adding Fractions to Integers. 2.
Reconciliation of Integers. 3.
Indefinite Denial of Integers. 5.
Revision of Deficient Formulas. 1.
Adding Fractions to Integers.
We saw that, in natural (or temporal) modality, only 10 factors out of 15
are induced by factor selection from gross elements or compounds. Likewise, in
the wider context of mixed modality, only 21 out of 63 factors emerge out of
factor selection from gross formulas. Since all gross formulas were considered,
the question arises, what of the remaining 5 or 42 factors, how do they ever
occur in knowledge?
There is clearly a gap to fill; our theory of induction is not so far
exhaustive, and is in need of further refinement. The answer has to be that we
add fractions to induced integers, in analogy to gross formula amplification.
Let us examine this idea.
As a state of being, an integer is final (if only for the time or
circumstances concerned), it is a whole, complete as it is. This means that any
fraction not included in it, is factually incompatible with it, and may not be
joined to it.
But as a state of knowledge, an apparent integer may turn out to be
incomplete. The conjunction of two or more particular fractions may have been
thought to be complete, a correct image of reality, but then we find that the
truth was more complex. In this perspective, adding fractions on to others is
quite conceivable.
As we saw in the chapter on factor selection, a mere conjunction of
particular fractions always has more than one factor, except for the conjunction
of all the fractions in the system under consideration which has only one. If we
select the strongest factor available, we obtain, as our inductive conclusion,
an integer which looks identical to the original conjunction of fractions,
differing only in its being defined as having only one factor.
The movement from mere-conjunction-of-fractions to integer-by-definition
is an inductive one, since it involves elimination of weaker factors. When the
conjunction involves all the fractions in the system concerned it exceptionally
yields the corresponding integer deductively, because no other fractions are
available, so there can only be one factor, anyway.
The example given in natural factor selection was (IOp)(IpO)
®
(IOp)(IpO). Whereas the sum of the natural fractions f7+f8 has four factors, F10,
F13-F15, the
identical-looking integer F10 has by
definition only one factor, F10.
Similar examples could be provided for temporal modality, or the open system.
This signifies that we may amplify presumed integers by addition of
(particular) fractions, without our having to return to the gross formula level.
This simply involves removing the presumption that the original fractions
constituted an integer, whenever there is reason to believe that a further
fraction should be added on to them. The presumption of integrity is removed by
restoring the factors we previously selected out. This operation might be called
regression.
In that case, addition of any further fraction(s) to two or more
fractions is logically demonstrable, as a straightforward conjunction between
the disjunctions of factors involved. The result of such an operation, as we saw
in factorial analysis, is simply the common factors of the merged strings. If we
thereafter select the strongest of these common factors, we obtain a new
inductive integer as conclusion, which is descriptively identical to the
fractions conjoined.
For example, if (In)(IOp) + (IpO) = (In)(IOp)(IpO) is to be
proved, we say: (In)(IOp)
has only factor F6 as an integer, but
restored to the fractional form f5f7,
its factors are F6, F11, F13, F15.
The fraction (IOp), or f8, has the
factors F4, F7-8, F10, F12-15.
The only factors these have in common are F13,
F14. These equal the fractional formula f5f7f8, or inductively the integer F13, as was required to prove.
The following table displays the results of adding one fraction to a
given fractional formula, for the whole closed system of natural (or, similarly,
temporal) modality. When two fractions are to be added, proceed by adding one at
a time. The two columns on the left show the given formula, in integral and
fractional form. The headings of the next four columns show the particular
fraction to be added. And the cells where they intersect show the concluding
integers. Such conclusions are inductive, except in the cases where F15 results,
since no further additions are then possible. Note that the addition of a
fraction to a formula already containing it leaves the formula unchanged. Table 59.1
Adding Fractions in Closed Systems.
Note that the universal integers F1-F4
are translated into their corresponding particular fractions f5-f8,
respectively. A universal fraction may be viewed as a sum of two particular
fractions, identical in all but extension, and the latter may then be fused into
one particular extension, e.g. F1 = f5+f5 = f5.
But, let us now examine more closely the conditions of validity of this
process. The fractions (IOp) and (IpO) can be
directly observed, by experience of the same part of the extension under
different circumstances; but not so the fractions (In) and (On): they
depends on generalization. As well, all four may derive from deductive
arguments. While these avenues are individually easy to take for granted, it is
more difficult to rest assured that had we had all the data at our fingertips at
once, we would have interpreted it as we do when we receive it piecemeal. We are
very much assuming that the new additions do not affect the original positions.
This criticism only goes to show that such formula revision is an
inductive process, however neatly mechanical it looks. It admittedly involves
ambitious assumptions, but so do all inductive processes. We are free to return
to the gross formula level, and generalize from that. All logic does here is to
provide us with a cogent shortcut, which may just as well turn out to be
accurate.
The underlying justification is that, since the various fractions have so
far held their ground, we have no specific empirical or logical cause for
complaint; we remain protected by the conviction that, if we are wrong, an
inconsistency will sooner or later arise to awaken us to the fact. Returning to
the gross formula level would only make us lose already confirmed information;
we are free to try it and, if the results are found more reliable, to choose
that course, in any case.
Furthermore, the fact that this process of adding fractions to integers
is the only way we can conceivably arrive at the missing integers, namely F11-F15
for the closed systems as earlier mentioned, is a supplementary justification
for it. If this method of induction were not valid, the missing integers would
be unknowable, since neither factorial analysis, nor generalization, nor
revision of gross formulas are able to yield such conclusions, as was seen.
A table similar to the above can easily be drawn up for the open system.
The various combinations of the particular fractions f7-f12,
taken from 2 to 6 at a time, are amplified by one of these six fractions, to
yield an integer in the range F7-F63.
Since the results are implicit in our initial definitions of the integers by
reference to the fractions, and in the law of generalization, we may avoid
taking up more space here. 2.
Reconciliation of Integers.
Now let us consider how conflicts between induced integers might be
resolved, again in analogy to the doctrine of harmonization between gross
formulas.
a.
If two induced integers appear in the course of knowledge development,
and they are judged as having unequal weights, the resolution their conflict is
obviously to keep the more weighty one as it stands, and entirely reject the
lighter one.
Note well that such rejection does not mean simply labeling the
unfortunate integer as 'canceled out'. The denial of an integer signifies that
one or more of its implications is false. That is, some implicit element(s)
and/or overlap(s) must be false, to cause the downfall of the integer as a
whole. However, there is no need to seek for this precise cause of downfall: it
is fully defined by the leftover heavier integer.
Effectively, such harmonization between unequals is a special case of
factor selection, guided by ad hoc considerations of credibility, instead of the
regular appeal to the uniformity principle.
If any two (or more) integers make their appearance, then we are faced
with a deficient formula disjoining them, and whichever one is declared more
likely, on whatever basis, the other(s) is/are eliminated. For example, if F3
and F11 both emerge, then 'F3
or F11' is true; if now say F11
is judged more weighty, then by apodosis F3
is false.
b.
If the conflicting integers are of equal weight, we could accordingly,
simply select the stronger of the two as in any generalization. Thus, in the
example just given, lacking any other reason for preference, we would choose F3.
However, another solution to the problem seems more satisfactory. Instead
of reacting to such a situation in an extremist, either-or, manner, we could
seek to fuse the fractions implicit in the presumed integers, into a new integer
comprising all original data other than their implicit characterization as
exclusive.
The justification of such synthesis is that we thus avoid loss of
significant information, which has otherwise so far found confirmation (since it
has made its appearance here). Also, the repercussions on the wider context are
minimized, until and unless we are forced to be more decisive.
This topic is clearly an corollary of that of integer amplification by
fractions, and all that has been said in the previous section continues to be
relevant here. A presumed integer can always be made to regress into a mere
conjunction of fractions, so that it is associated with more factors, and thus
be made compatible with further fractions, with factors in common. Thus, though
real integers, having but one factor each, are mutually exclusive, their
fractional equivalents may be merged.
The following table shows how conflicts between presumed integers may be
resolved by merger, in the closed systems. The integers in the column on the
left are added to the integers heading the subsequent columns, and the results
are given in the cells of intersection. More precisely, of course, the fractions
(not shown here) corresponding to the original integers are merged, and the
result is then generalized into the new integer.
Note that the previously inaccessible closed-systems fractions F11-F15
are made possible by harmonization, as by amplification, of integers. Table
59.2 Harmonization
of Equal Closed-Systems Integers.
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