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 a disjunction of 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, and F6 are the factors of 'A'. 'A' is thus to be read as 'F1 or F3 or F6', 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.

F10

F11

F12

F13

F14

F15

Elements

yes