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The Logician © Avi Sion All rights reserved
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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.
These results may be validated by factorial analysis; or more simply by
appeal to the previous table, adding on one fraction at a time. An example of
validation: the presumed integers F5
and F9, say, make their appearance. Regressing them to their fractional
equivalents, we get f5f6 and f6f7.
Factorization of these yields F5, F11-12, F15
and F9, F11, F14-15. The common factors in these two series are F11, F15, and these are
the factorial formula of f5f6f7. By
selection of the strongest factor, we conclude F11, as tabulated.
We may here raise objections similar to those raised with regard to
amplification of integers, and they may be similarly countered.
A table similar to the above can easily be drawn up for the open system.
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. 3.
Indefinite Denial of Integers.
It is conceivable that we may discover an induced integer to be wrong,
without knowing precisely what is wrong with it. This situation would arise if
we had drawn deductive inferences from the integer, and perhaps again from its
implications in turn, and found some such consequence factually in error. We
then know that some implied element(s) or overlap(s) in the integer must be
false, but we may have no way to pin-point the culprit. This may be referred to
as indefinite denial of an integer.
a.
If we remember the factorial formula from which we selected this integer,
because it was the strongest available, then a solution to the problem is
forthcoming. If the initially selected factor has turned out to be incorrect, it
is simply eliminated from the series of alternatives, and the next strongest
factor in line is selected in its stead. Thus, eventually, by successive
elimination, any of the unused factors in the series might conceivably appear as
inductive conclusions.
b.
If, however, we are unable to recapture the original factorial formula,
then we are forced to backtrack to gross formula levels, in successively more
radical retreats. Each further retreat should be tested, to find out if it is
sufficient to remove the inconsistency we set out to combat. This means,
generalizing from the attempted gross formula, and checking whether the new
integer still implies the inconsistency.
The two tables below show the main results of this approach.
The first retreat would consist in abandoning all fractionating, and
considering only the gross formula implied by the doubted integer. The results
for the natural integers, listed in the column on the left, are shown under the
heading 'implied gross'.
This gross formula is then again generalized. This changes nothing in the
simpler integers, which are the only or strongest factor of the implied gross,
anyway; but it does affect the more fragmented integers. Thus, in natural
modality, F1-F10 are unchanged, but
F11, F12, F15 become F5,
and F13 and F14 become F8
and F9, respectively, as shown in the column labeled 'first
generalization'. More radical revision is thus always required for F1-F10,
and sometimes for F11-F15
(when the first proposal fails to solve the problem).
The next step is to consider all the possible gross denials of each
implied gross formula. Then, in a preliminary wave of generalization, determine
the strongest factor of each these possible gross denials, by referring to table
58.1. Then, in a final wave of generalization, in accordance with the same law,
select the strongest of these strongest factors, as suggested integral denial.
This tells us both, what the generalization of each of the alternative
gross denials is, and which of them to prefer as integral denial. If this first
option of denials fails to solve the problem, then the next strongest is chosen;
and so forth, till the inconsistency disappears.
If none of the strongest integral denials solves the problem, there is in
fact still the possibility that one of the weaker factors of the various
alternative gross denials, which we eliminated in the preliminary wave of
generalization, manages to solve it. These hidden factors should therefore be
reactivated, ordered, and tried successively.
If we grant the beginning assumption that the problem inconsistency was
indeed to be attributed to our integer, then we are eventually bound to arrive
at a solution by this method, since our treatment is exhaustive. If none of all
the possible factors gets rid of the inconsistency, then we can be sure that it
must have been due to another source. The assumption we started with, that the
other sources were all more trustworthy than our integer, must be wrong, and
each of the other sources now must be subjected to the same kind of scrutiny.
For example, the denial of InOn
is 'Ep and/or Ap', which
translates into three possible formulas: ApInO,
IEpOn, ApIEpO, whose factorial counterparts are, respectively, F8,
F13, and F9, F14,
and F10. The strongest factor in these three sets are F8,
F9, and F10. The
strongest of these three in turn is F8.
The first solution to test is therefore F8;
if the problem remains, F9 is tried,
and if this fails, F10 should
succeed; lastly, we can try F13, then
F14. If none of these work, the
problem must lie elsewhere. Table 59.3a
Indefinite Denial of Natural Integers.
Table
59.3b Other
Possible Strong Denials.
Note well that all the information in these tables is derived from
previous work. Similar tables can be prepared for temporal modality, and mixed
modality.
To achieve a comprehensive treatment of formula revision we would still
need to consider the situations listed below. Though Logic must eventually deal
with all such situations in detail, I will not do it all in the present work, so
as not to swamp my main findings in excessive minutiae. Also, in any case the
results are derivative, and already effectively contained in the situations
already covered (viz. interactions between gross formulas, and interactions
between integers).
The method used to solve the problem of indefinite denial of integers,
can be applied to two other situations:
a.
Indefinite denial of any gross compound.
b.
Any interaction between integers and gross formulas.
Indefinite denial of integers is at the confluence of these two larger
issues, a specific case of each, and an example for both.
a.
Effectively, this example shows us how to deal with denial of certain
compounds, those which integers imply. In natural modality, these are AEp, ApE, InOn,
AInOp, IpEOn, ApInO,
IEpOn, ApIEpO. But we
know that there are quite a few more possible compounds. In natural modality,
there are another 29, to be exact. Each of these is deniable by the
contradictories of its two or more elements, and any consistent conjunction(s)
of these. And, therefore, each is deniable by two or more contrary compounds.
The method established for indefinite denial of the compounds implied by
integers may be applied to all other compounds. Namely, we select the strongest
factor of each alternative compound, and then in turn opt for the strongest of
these as our first solution; if this is not satisfactory, we go on, successively
choosing the factors in descending order of strength. This method is nothing
new: it is a special application of the law of generalization.
For example, if AIn is false, then O
and/or Ep must be true, then AEp
or InO or IEpO follow. The
strongest factors of these are F3, F5,
and F9, respectively, so our first solution will be (AEp),
our second (In)(On), and our third (On)(IOp). If these fail to work, we try the hidden weaker factors of InO
and IEpO, namely F8, F1O-15,
successively.
b.
Indefinite denial of integers is one of the concerns in the larger issue
of interaction between integers and gross formulas. It deals with the specific
case of integers versus the contradictories of their implied elements. This
still leaves us with the cases of amplification of integers by the compatibles
of their implied elements, and the cases of harmonization between integers and
the contraries of their implied elements.
Only when all combinations of integers and gross elements and compounds
are considered, is the topic of formula revision exhausted. However, we have
already established the overall method, there is only a need to apply it
further. The method here again is to simply find the gross formula implied by
each integer, and amplify it with the various compatible elements or compounds,
or harmonize it with the various contrary elements or compounds, then generalize
the results and select the strongest of all available factors.
This should be done for the closed systems of natural and temporal
modality, and the open system of mixed modality. In this manner, using the
methods of factorial analysis and factor selection, we can cover the whole field
of formula revision. 5.
Revision of Deficient Formulas.
Lastly, we should mention, for the record, the issue of revision of
deficient formulas, consisting of the various disjunctions of two or more
factors, not embraced by gross formulas. We saw that these rare, though
conceivable, stages of knowledge fall under the law of generalization, like any
other factorial formula. With regard to their interactions, likewise, the
methods are the same.
If two such deficient formulas with common factors are conjoined, the
result is a formula containing only their common factor(s), which may be
integral, gross, or deficient. This amplification may in turn be generalized, by
selection of the strongest factor, as usual.
If two such deficient have no common factor, their conjunction represents
a conflict to resolve. If they are of unequal weight, the weightier is
preferred. If they are of equal weight, their harmonization is worked out as
follows.
We know that the conjunction of two disjunctions, say 'p or q' and 'r or
s', results in a disjunction of all the pairings-off of alternatives, as in 'p
and r' or 'p and s' or 'q and r' or 'q and s'. Since we are here concerned with
factors, which are mutually exclusive, then in the case of factorial formulas
without common factors, each pairing will be inconsistent. Each pair, therefore,
should be treated as a case of harmonization between equal integers, and the
result will be an integer embracing all their fractions.
When all the individual pairs have been so harmonized, we are left with
disjunction of one or more factors, which is our revised formula. This, whether
integral, gross, or again deficient, may now be generalized by selection of the
strongest factor, as usual.
Thus, amplification or harmonization of such deficient formulas present
no special problem, but can be readily reduced to previously dealt with issues.
Likewise, the interactions between deficients and integers or gross formulas are
easily handled.
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