CHAPTER 53.FACTORIAL ANALYSIS.
We are now in a position to analyze the precise content of gross formulas, restating them as factorial formulas. These consist in complex propositions identifying the alternative integers with which the given gross formula is consistent. In this context, integers will be referred to as factors, whether one or more of them are involved. The process may then be viewed as factorization or factorial analysis of information.
By restructuring information in factorial terms, we are able to recognize more clearly how close to, or far from, full knowledge we are with regard to the subject and predicate in question. We know that reality must fall under one or the other of the various integers in any case. Full knowledge implies that we can pinpoint one integer as the right one. Total ignorance implies that all the integers are equally likely outcomes for us. In between lies a mass of possibilities, where we know that certain of the integers are excluded, but we are still left with more than one integer to choose from.
Even in situations where we do have full knowledge, restating a gross formula as an integer composed of fractions, permits us to trace or express more precisely the way the extension is fragmented into different particular relations. But let us proceed, and the importance of this approach will become clear.
Note in passing that since singular statements are reducible to disjunctions of plurals, they can also be factorized, though this is not done below.
Let us concentrate again on natural modality as a closed system. Whatever is found true for natural modality can as usual be duplicated for temporal modality. Mixed modality will be dealt with further on.
The following table interprets the 49 gross formulas in factorial terms. A factorial formula is expressed as adisjunctionof one or more of the 15 integers (those marked ‘yes‘: the number of factors = ‘NF’); such disjunctives exclude all other integers (those left blank).
For example, as the table reveals,F1,F3, andF6are the factors of ‘A‘. ‘A‘ is thus to be read as ‘F1orF3orF6‘, i.e. as ‘(An) or (AEp) or (In)(IOp)’: these are the only 3 eventualities conceivable given that ‘A‘ is true, the rest being impossible outcomes. The disjunctive proposition is the factorial equivalent of ‘A‘, then.
Table 53.1 Factorial Analysis of Natural Gross Formulas.
NF | F1 | F2 | F3 | F4 | F5 | F6 | F7 | F8 | F9 | F10 | F11 | F12 | F13 | F14 | F15 | |
Elements | ||||||||||||||||
1 | An | yes | ||||||||||||||
1 | En | yes | ||||||||||||||
3 | A | yes | yes | yes | ||||||||||||
3 | E | yes | yes | yes | ||||||||||||
7 | Ap | yes | yes | yes | yes | yes | yes | yes | ||||||||
7 | Ep | yes | yes | yes | yes | yes | yes | yes | ||||||||
8 | In | yes | yes | yes | yes | yes | yes | yes | yes | |||||||
8 | On | yes | yes | yes | yes | yes | yes | yes | yes | |||||||
12 | I | yes | yes | yes | yes | yes | yes | yes | yes | yes | yes | yes | yes | |||
12 | O | yes | yes | yes | yes | yes | yes | yes | yes | yes | yes | yes | yes | |||
14 | Ip | yes | yes | yes | yes | yes | yes | yes | yes | yes | yes | yes | yes | yes | yes | |
14 | Op | yes | yes | yes | yes | yes | yes | yes | yes | yes | yes | yes | yes | yes | yes | |
NF | F1 | F2 | F3 | F4 | F5 | F6 | F7 | F8 | F9 | F10 | F11 | F12 | F13 | F14 | F15 | |
Compounds | ||||||||||||||||
1 | AEp | yes | ||||||||||||||
1 | ApE | yes | ||||||||||||||
1 | AInOp | yes | ||||||||||||||
1 | EOnIp | yes | ||||||||||||||
1 | ApEpIO | yes | ||||||||||||||
2 | AIn | yes | yes | |||||||||||||
2 | EOn | yes | yes | |||||||||||||
2 | AOp | yes | yes | |||||||||||||
2 | EIp | yes | yes | |||||||||||||
2 | ApIEp | yes | yes | |||||||||||||
2 | EpOAp | yes | yes | |||||||||||||
2 | ApInO | yes | yes | |||||||||||||
2 | EpOnI | yes | yes | |||||||||||||
3 | ApEp | yes | yes | yes | ||||||||||||
3 | ApInOp | yes | yes | yes | ||||||||||||
3 | EpOnIp | yes | yes | yes | ||||||||||||
3 | ApIO | yes | yes | yes | ||||||||||||
3 | EpOI | yes | yes | yes | ||||||||||||
4 | ApIn | yes | yes | yes | yes | |||||||||||
4 | EpOn | yes | yes | yes | yes | |||||||||||
4 | IEp | yes | yes | yes | yes | |||||||||||
4 | OAp | yes | yes | yes | yes | |||||||||||
4 | InOn | yes | yes | yes | yes | |||||||||||
5 | ApIOp | yes | yes | yes | yes | yes | ||||||||||
5 | EpOIp | yes | yes | yes | yes | yes | ||||||||||
6 | ApI | yes | yes | yes | yes | yes | yes | |||||||||
6 | EpO | yes | yes | yes | yes | yes | yes | |||||||||
6 | ApOp | yes | yes | yes | yes | yes | yes | |||||||||
6 | EpIp | yes | yes | yes | yes | yes | yes | |||||||||
6 | InO | yes | yes | yes | yes | yes | yes | |||||||||
6 | OnI | yes | yes | yes | yes | yes | yes | |||||||||
7 | InOp | yes | yes | yes | yes | yes | yes | yes | ||||||||
7 | OnIp | yes | yes | yes | yes | yes | yes | yes | ||||||||
9 | IO | yes | yes | yes | yes | yes | yes | yes | yes | yes | ||||||
11 | IOp | yes | yes | yes | yes | yes | yes | yes | yes | yes | yes | yes | ||||
11 | OIp | yes | yes | yes | yes | yes | yes | yes | yes | yes | yes | yes | ||||
13 | IpOp | yes | yes | yes | yes | yes | yes | yes | yes | yes | yes | yes | yes | yes |
This table is drawn up as follows. First we deal with elementary propositions, identifying which of them are logically implied by each factor (downward). Thus for instances,F5= (In)(On) impliesIn,On, and all their subalterns;F6= (In)(IOp) impliesA,In,Op, and all their subalterns. Note well that if a particular element is implicit in all the fractions, a universal element is implied by them together.
Then we derive the factorization of compounds from that of their elements, as follows. When two or more disjunctions involving incompatible alternatives are conjoined, the result is a disjunction of their common alternatives (if they have only one in common, that is the result; if they have none in common they are not conjoinable). Thus, to factorize a compound gross formula, we simply find the common factors of its elements. (See chapter 28.3, on ‘conjunctive multiplication’.)
For example: SinceE=F2orF4orF7, andIp= all factors butF2, thenEIp=F4orF7. This is because no two distinct factors, being mutually exclusive integers, can consistently be conjoined; while a factor conjoined with itself, yields itself as result.
Table 52.1, used to define the 15 integers as conjunctions of fractions, was read across. If it is read downward, it serves to factorize the 8 fractions; the fractions being then viewed as disjunctions of the integers which include them. If we compare the results so obtained to those in table 53.1, we see that they are the same in all cases except two, note well.
Whereas (IOp) = the fractionf7has factorsF3,F6,F9–F11,F13–F15,IOp= the gross formula has additionally factorsF5,F8,F12. This is because in (IOp) the extensions ofIandOpare known to fully coincide, while inIOpwe do not know whether the extensions ofIandOpoverlap wholly, partly, or not at all. Similarly, the fraction (IpO) and the gross formulaIpOare not factorially the same; the former being a disjunction ofF4,F7–F8,F10,F12–F15, whereas the latter further allows the alternativesF5,F9,F11.
This issue of overlap of particular elements is found in other gross formulas. Thus, most obviously, inIpOp, it is not clear whetherIpandOpoverlap, or if they do, to what degree. In the case ofIO, it would at first sight seem that they cannot overlap at all, since no instance of their extension can simultaneously fall under both; but as the factorial analysis ofIOasF5,F8–F15makes clear, overlap may indeed occur involving subalterns ofIand/orO. That is, the gross formulaIOis merely an abbreviation ofI+Ip+O+Op, so that (IpO) or (IOp) are conceivable fractions within it, though concealed.
The same issue can be raised for most other gross formulas involving particular elements. Only in 3 cases, mentioned in the next section, is the issue resolved unambiguously. This shows the inadequacy of gross formulas, their capacity to mislead, and it shows the value of factorial analysis.
When a gross formula involves a mix of universal element(s) and particular one(s), the ambiguity concerning overlap is lessened, though rarely removed. Thus for instance, inAOp, we can be sure that part of the extension ofAis the whole extension ofOp, and so we could make a ‘partial factorization’ toI+(IOp). But though this approach may improve our understanding of the situation somewhat, factorially speaking information is lost. For whereasA+Op=F3orF6, the formulaI+(IOp) equals no more than (IOp) alone, namelyF3orF6orF9–F11orF13–F15. The result is vaguer and lesser, because (IOp) implies I in any case.
When a gross formula involves only universal elements, total overlap is of course assured. But factorial analysis is still relevant, since without it we might remain unaware thatApEpnot only has the possible outcomes (AEp) or (ApE), but also has (IOp)(IpO) as an alternative.
Notice in table 53.1 our separation of elements and compounds, and also the classification of formulas by number of factors. The less factors a formula has, the closer it is to full definition, and the more knowledge we have. Ignorance would be disjunction of all 15 factors.
It is interesting to note that only 7 gross formulas have a single factor, i.e. result in an integer as their factorial equivalent. They are:An,En,AEp,ApE,AInOp,EOnIp, andApEpIO. The latter three cases are worth stressing, since they provide us with novel immediate inferences. Without the above systematic approach, it might not seem evident that the whole extension of the universal element(s) of the gross formulas are necessarily covered by only the particular fractions shown in these corresponding factorials.
AInOp=F6= (In)(IOp), meaning: ‘All S are P and some S must be P and some S can not-be P’ equals ‘Some S must be P, and some other S are P but can not-be P’.
EOnIp=F7= (On)(IpO), meaning: ‘No S is P and some S cannot be P and some S can be P’ equals ‘Some S cannot be P, and some other S are not P but can be P’.
ApEpIO=F10= (IOp)(IpO), meaning: ‘All S can be P and all S can not-be P, and some S are P and some S are not P’ equals ‘Some S are P, but can not-be P, whereas some other S are not P, but can be P’
All other gross formulas yield factorial formulas with more than one alternative; they are deficient stages of knowledge. Nevertheless, they are very valuable in this form, as will be seen when we deal with modal induction.
It also interesting to note that the 49 gross formulas, which are an exhaustive list (within natural modality) as we showed, are able to express only 7 of the 15 integral states of being (within this closed system). More than half the integers are inaccessible to this gross manner of formulation, namelyF5,F8–F9,F11–F15!
This shows the importance of the concepts we have introduced in this study. Without them, Logic cannot fulfill its task. And indeed, in everyday thought and discourse we commonly use such complicated statements as ‘Some S are P, andthesecan not-be P, while someothersare always P’. Such statements try to clarify the deployment of the extension, and are efforts towards factorial analysis.
There are in fact a total of 2 to the 15th power, minus 1 = 32,767 ways to disjoin the 15 factorsF1–F15. Any combination of one, two, three… up to 15 of them, is conceivable. Fifteen of these ways are the integers in isolation: they represent full knowledge. One way, the disjunction of all 15 factors, represents total ignorance (equivalent to saying nothing, except the formal truth that one of the factors must be applicable eventually). In between, we are left with 32,751 states of knowledge involving varying degrees of ignorance.
The 49 gross formulas thus represent only a very small number of the possible states of knowledge. As we said, they include only 7 of the 15 integers, and therefore only 42 of the 32,751 intermediate states. They may be the most common states of knowledge, being simple of expression; but as a system, gross formulation is very defective. It is incapable of expressing the finer gradations, the nuances, of information accessible to factorial analysis.
As knowledge unfolds, we move from one formula to the other. There is a dynamic process. One may first discover, whether deductively or inductively, thatAis true, say; thenOpis found true, and our knowledge adds up toAOp; then perhapsOis seen true, contradictingA, so we may supposeIOtrue instead; and so on. As our opinions shift around, reflecting new observations, new insights, new efforts of reasoning, we adopt an alternative formula to express our contextual position.
Factorial formulas are simply more precise tools than gross formulas, for handling such changes in data. Knowing exactly how many and which integers are still open, and how many and which integers are already excluded, the pursuit of full knowledge becomes more efficient. The goal is to eliminate all alternative integers from the list of possibilities, until only one remains: then our knowledge concerning the subject-predicate in question is clear. We may at a later stage acquire new information, which once again puts us in doubt, but in any given context factorial analysis lets us know where we stand.
Note that it may be — I am just guessing — that some of the deficient states of knowledge are equivalent to fractions or integers composed of conditional propositions. This is uncharted territory, I have not looked into it; but it seems likely, since categoricals and conditionals are particles of intersecting continua.
The above concerns, remember, the closed system of natural modality; or, by analogy (substitutingc,tsuffixes forn,p), that of temporal modality. But these, though used independently, are relatively artificial. It is only when we consider the two types together, in an open system, that our results are absolute, realistic. This will now be done.
We saw that in mixed modality (limiting ourselves to consideration of plurals), there are 12 fractions, from which 63 integers can be constructed. It follows that, here, there are 2 to the 63rd power, minus 1 = 9.2233 X 10 to the 18th power factorial formulas. That is, 9 million million million distinct states of knowledge, concerning any given subject to predicate relation! Of these 63, the one-factor formulas, are full knowledge; one, the 63-factor formula, is total ignorance; and the remainder are intermediate states, involving 2 to 62 factors in different disjunctive combinations.
We identified 20 elementary propositions, yielding 175 compounds, a total of 195 gross formulas. These are the most common states of knowledge, worthy of special attention, but still only a minute part of the total picture. These are analyzed factorially inAppendix 1,for the record.Though the table is parked separately because of its bulk, it is important.It is split into several segments for reasons of space, but they should be read as one.
The method used to develop it is the same as for the closed systems. First, we identify the elements implied by each of the 63 integers, viewing its fractions (if more than one) both individually and collectively. The factorial formula corresponding to an elementary gross formula is then a disjunction of all the integer(s) which imply it, since the list of integers is complete. Thereafter, we derive the factor(s) of a compound, simply by identifying the common factor(s) of its constituent elements.
Similar comments to those made previously apply here, concerning overlap. Especially note the difference between the factors of the fractions (IOt) and (ItO), given in table 52.2 and numbering 32 each, and the factors of the gross formulasIOtandItO, given in the appendix and numbering 53 each (including the 32 of the corresponding fraction).
Note that here we have 11 gross formulas resulting in a single factor; namely the 6 universal factors (An), (En), (AcEp), (ApEc), (AEt), (AtE), plus the following 5 compounds:
AcInOp | = | F8 | = | (In)(IcOp) |
EcOnIp | = | F9 | = | (On)(IpOc) |
AIcEpOt | = | F17 | = | (IcOp)(IOt) |
EOcApIt | = | F18 | = | (IpOc)(ItO) |
AtIEtO | = | F21 | = | (IOt)(ItO). |
These equations can be interpreted like the corresponding closed system equations were. The reader should recite or write their explicit meaning, to see how unexpected these results are. Without factorial analysis, we would have great difficulty finding, understanding or proving these inferences.
Thus 52 integers remain unexpressed by gross formula statements, and can only be specified through fractional notation. All gross formulas other than these 11 have more than one factor; the maximum number of factors for elements being 62, forIpandOp, and for compounds being 61, forIpOp.
Note also that the fractions and integers specific to natural or temporal modality could be factorized in the open system. This is not done here, to avoid further complications and because the results are not needed in subsequent discussion. But the reader is invited to do it, as a significant exercise, with a warning not to ignore overlap issues.
Other possible developments include: factorial analysis of stereo integers, of transitive categoricals and fractions and integers of them, and of all types of conditionals.
Many of the factorial formulas resulting from such analysis, may intersect in meaning with those considered here; I mention this as a speculation, to suggest to future logicians that they look for eventual equations.