CHAPTER 59.FACTORIAL FORMULA REVISION.
1.Adding Fractions to 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 fractionsf7+f8has four factors,F10,F13–F15, the identical-looking integerF10has 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 calledregression.
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 factorF6as an integer, but restored to the fractional formf5f7, its factors areF6,F11,F13,F15. The fraction (IOp), orf8, has the factorsF4,F7-8,F10,F12-15. The only factors these have in common areF13,F14. These equal the fractional formulaf5f7f8, or inductively the integerF13, 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 59.1Adding Fractions in Closed Systems.
Integer | Fractions | +f5= | +f6= | +f7= | +f8= |
F1 | f5f5 | F1 | F5 | F6 | F8 |
F2 | f6f6 | F5 | F2 | F9 | F7 |
F3 | f7f7 | F6 | F9 | F3 | F10 |
F4 | f8f8 | F8 | F7 | F10 | F4 |
F5 | f5f6 | F5 | F5 | F11 | F12 |
F6 | f5f7 | F6 | F11 | F6 | F13 |
F7 | f6f8 | F12 | F7 | F14 | F7 |
F8 | f5f8 | F8 | F12 | F13 | F8 |
F9 | f6f7 | F11 | F9 | F9 | F14 |
F10 | f7f8 | F13 | F14 | F10 | F10 |
F11 | f5f6f7 | F11 | F11 | F11 | F15 |
F12 | f5f6f8 | F12 | F12 | F15 | F12 |
F13 | f5f7f8 | F13 | F15 | F13 | F13 |
F14 | f6f7f8 | F15 | F14 | F14 | F14 |
F15 | f5f6f7f8 | F15 | F15 | F15 | F15 |
Note that the universal integersF1–F4are translated into their corresponding particular fractionsf5–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, namelyF11–F15for 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 fractionsf7–f12, taken from 2 to 6 at a time, are amplified by one of these six fractions, to yield an integer in the rangeF7–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.
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, ifF3andF11both emerge, then ‘F3orF11‘ is true; if now sayF11is judged more weighty, then by apodosisF3is 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 chooseF3.
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 fractionsF11–F15are made possible by harmonization, as by amplification, of integers.
Table 59.2Harmonization of Equal Closed-Systems Integers.
Int. | F1 | F2 | F3 | F4 | F5 | F6 | F7 | F8 | F9 | F10 | F11 | F12 | F13 | F14 | F15 |
F1 | F1 | ||||||||||||||
F2 | F5 | F2 | |||||||||||||
F3 | F6 | F9 | F3 | ||||||||||||
F4 | F8 | F7 | F10 | F4 | |||||||||||
F5 | F5 | F5 | F11 | F12 | F5 | ||||||||||
F6 | F6 | F11 | F6 | F13 | F11 | F6 | |||||||||
F7 | F12 | F7 | F14 | F7 | F12 | F15 | F7 | ||||||||
F8 | F8 | F12 | F13 | F8 | F12 | F13 | F12 | F8 | |||||||
F9 | F11 | F9 | F9 | F14 | F11 | F11 | F14 | F15 | F9 | ||||||
F10 | F13 | F14 | F10 | F10 | F15 | F13 | F14 | F13 | F14 | F10 | |||||
F11 | F11 | F11 | F11 | F15 | F11 | F11 | F15 | F15 | F11 | F15 | F11 | ||||
F12 | F12 | F12 | F15 | F12 | F12 | F15 | F12 | F12 | F15 | F15 | F15 | F12 | |||
F13 | F13 | F15 | F13 | F13 | F15 | F13 | F15 | F13 | F15 | F13 | F15 | F15 | F13 | ||
F14 | F15 | F14 | F14 | F14 | F15 | F15 | F14 | F15 | F14 | F14 | F15 | F15 | F15 | F14 | |
F15 | F15 | F15 | F15 | F15 | F15 | F15 | F15 | F15 | F15 | F15 | F15 | F15 | F15 | F15 | F15 |
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 integersF5andF9, say, make their appearance. Regressing them to their fractional equivalents, we getf5f6andf6f7. Factorization of these yieldsF5,F11–12,F15andF9,F11,F14–15. The common factors in these two series areF11,F15, and these are the factorial formula off5f6f7. By selection of the strongest factor, we concludeF11, 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–F10are unchanged, butF11,F12,F15becomeF5, andF13andF14becomeF8andF9, respectively, as shown in the column labeled ‘first generalization’. More radical revision is thus always required forF1–F10, and sometimes forF11–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 ofInOnis ‘Epand/orAp‘, which translates into three possible formulas:ApInO,IEpOn,ApIEpO, whose factorial counterparts are, respectively,F8,F13, andF9,F14, andF10. The strongest factor in these three sets areF8,F9, andF10. The strongest of these three in turn isF8. The first solution to test is thereforeF8; if the problem remains,F9is tried, and if this fails,F10should succeed; lastly, we can tryF13, thenF14. If none of these work, the problem must lie elsewhere.
Table 59.3aIndefinite Denial of Natural Integers.
Bad | Implied | 1st | Denials | Strongest | Out | |
Int. | Gross | Gen. | (and/or) | Gross | SF | of |
F1 | An | F1 | Op | AInOp | F6 | 1 |
F2 | En | F2 | Ip | IpEOn | F7 | 1 |
F3 | AEp | F3 | O, In | AInOp | F6 | 3 |
F4 | ApE | F4 | On, I | IpEOn | F7 | 3 |
F5 | InOn | F5 | Ep, Ap | ApInO | F8 | 3 |
F6 | AInOp | F6 | O, Ep, An | An | F1 | 4 |
F7 | IpEOn | F7 | En, I, Ap | En | F2 | 4 |
F8 | ApInO | F8 | On, Ep, A | AEp | F3 | 5 |
F9 | IEpOn | F9 | E, In, Ap | ApE | F4 | 5 |
F10 | ApIEpO | F10 | On, E, In, A | AEp | F3 | 7 |
F11 | InOn | F5 | Ep, Ap | ApInO | F8 | 3 |
F12 | InOn | F5 | Ep, Ap | ApInO | F8 | 3 |
F13 | ApInO | F8 | On, Ep, A | AEp | F3 | 5 |
F14 | IEpOn | F9 | E, In, Ap | ApE | F4 | 5 |
F15 | InOn | F5 | Ep, Ap | ApInO | F8 | 3 |
Table 59.3bOther Possible Strong Denials.
Int. | Gross | 2nd | SF | 3rd | SF | 4th | SF | 5th | SF | 6th | SF | 7th | SF |
F1 | An | ||||||||||||
F2 | En | ||||||||||||
F3 | AEp | ApInO | F8 | ApIEpO | F10 | ||||||||
F4 | ApE | IEpOn | F9 | ApIEpO | F10 | ||||||||
F5 | InOn | IEpOn | F9 | ApIEpO | F10 | ||||||||
F6 | AInOp | AEp | F3 | ApInO | F8 | ApIEpO | F10 | ||||||
F7 | IpEOn | ApE | F4 | IEpOn | F9 | ApIEpO | F10 | ||||||
F8 | ApInO | InOn | F5 | AInOp | F6 | IEpOn | F9 | ApIEpO | F10 | ||||
F9 | IEpOn | InOn | F5 | IpEOn | F7 | ApInO | F8 | ApIEpO | F10 | ||||
F10 | ApIEpO | ApE | F4 | InOn | F5 | AInOp | F6 | IpEOn | F7 | ApInO | F8 | IEpOn | F9 |
F11 | InOn | IEpOn | F9 | ApIEpO | F10 | ||||||||
F12 | InOn | IEpOn | F9 | ApIEpO | F10 | ||||||||
F13 | ApInO | InOn | F5 | AInOp | F6 | IEpOn | F9 | ApIEpO | F10 | ||||
F14 | IEpOn | InOn | F5 | IpEOn | F7 | ApInO | F8 | ApIEpO | F10 | ||||
F15 | InOn | IEpOn | F9 | ApIEpO | F10 |
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 areAEp,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, ifAInis false, thenOand/orEpmust be true, thenAEporInOorIEpOfollow. The strongest factors of these areF3,F5, andF9, 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 ofInOandIEpO, namelyF8,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.