CHAPTER 52. FRACTIONS AND INTEGERS.
To achieve a fuller analysis of the states of knowledge, we must introduce certain tools, which we will call fractions and integers. These concepts relate, not primarily to states of knowledge, but to states of being. Whereas knowledge can be deficient, being must be definite, so that the possibilities it involves are more limited in number.
In view of the large amounts of data involved, we will develop these concepts in two stages. First, we will consider natural modality in isolation, as a closed system. Whatever results are obtained for this type, can be obtained by analogy for temporal modality taken by itself, by substituting the subscripts c and t for n and p throughout, as usual. Thereafter, we will broaden the perspective, and deal with both types of modality together, as a continuous, open system.
a. To begin with, consider singulars. Within natural modality, an individual subject’s relation to a predicate has only 4 possible states of being, whatever the state of our knowledge concerning it. Two of these are elementary, and two are compound. They are as follows. The significance of the brackets, which are not really needed for singulars, will become apparent as we proceed.
(Rn) | This S must be P |
(Gn) | This S cannot be P |
(RGp) | This S is P, though it can not-be P |
(RpG) | This S is not P, though it can be P. |
All four of these imply actuality. The latter two are of course singular extensional conditional propositions. They have in common the fact of contingency, but the compound RpGp is not a state of being since it does not tell us which of the two possibilities is in fact actualized. Still, RGp and RpG are close relatives to each other, insofar as the individual may switch from the one to the other state of being, whereas the two necessaries and contingency as such are immutable and may not replace each other over time.
Logically, then, these four states are not only exhaustive (one of them must be true, of any individual), but also mutually exclusive (only one may be true, at least at the same time).
Similarly within a closed system of temporal modality, an individual must have one of the following 4 states of being:
(Rc) | (Gc) | (RGt) | (RtG) |
In the mixed modality system, an individual has 6 alternative states of being:
(Rn) | (Gn) | (RcGp) | (RpGc) | (RGt) | (RtG) |
Note that only two states are carried over from each of the closed systems, and two are new contributions by the open system. Thus, note well, although (RGp) and (RpG) are recognized as states of being within natural modality, they lose this status in the wider perspective; likewise, (Rc) and (Gc), though so recognized within temporal modality, they are found deficient in full context.
Nevertheless, in practise we often do limit our thinking to one system or the other, so it is worth considering them also in isolation. The precise correlation between these closed system fractions, and open system fractions, is as follows (no other alternatives than those listed being applicable):
(RGp) = (RcGp) or (RGt), since both imply R and Gp.
(RpG) = (RpGc) or (RtG), since both imply Rp and G.
(Rc) = (Rn) or (RcGp), since both imply Rc.
(Gc) = (Gn) or (RpGc), since both imply Gc.
The states of being may be called integers, because they are fully defining of the actual relationship of subject to predicate. They are whole units of information, involving no ambiguity, vagueness or remaining questions. The concept of fractions becomes useful when we turn to plural propositions; with regard to singulars it is identical with that of integers.
b. Let us now quantify the above ideas. We will henceforth ignore singular propositions, so as to simplify treatment.
Within the closed system of natural modality, we may by analogy recognize 8 plural fractions. These could be assigned the symbols f1–f8, contextually. They are:
f1 | f2 | f3 | f4 | f5 | f6 | f7 | f8 |
(An) | (En) | (AEp) | (ApE) | (In) | (On) | (IOp) | (IpO) |
Note the similarity between the two quartets, f1-f4 and f5-f8, as well as their correspondence to the earlier mentioned singulars. As will be seen, the 4 universal fractions are also integers; but the 4 particular fractions are not integers, they are merely building blocks of integers.
Brackets are not really needed for the universal fractions, though used to maintain a uniform notation, since universals cover an identifiable extension. But in the case of particular fractions, they are essential, because the instances involved are not formally designated or enumerated.
We will adopt the convention that two elements enclosed in brackets, such as I and Op in (IOp), subsume exactly the same extension: for every instance of the subject in the one, there corresponds an instance in the other. Thus, (IOp) means ‘Some S are P, though these same S also can not-be P’, or more briefly, as ‘Some S both are P and can not-be P’; similarly for (IpO). Such propositions are of course particular extensional conditionals.
Brackets serve as well to stress separation between two parts of an extension. Thus for example, the conjunction of two fractions (In)(IpO) indicates that ‘Some S must be P, while some other S both can be but are not P’. Such relationships can be expressed in practise through the language of extensional conditioning.
Obviously, two fractions may be identical in appearance, but in fact concern distinct or only partly overlapping segments of the whole extension of the subject. Thus for example, though (On) and (On) are outwardly the same, they may happen to refer to different instances.
Nevertheless, in such case, they can be merged into one fraction, which simply covers a wider extension equal to the sum of the original two. The conjunction of two similar fractions results in one similar fraction. For example, (On)(On) equals (On).
Within the closed system of temporal modality, there are similarly 8 plural fractions:
f1 | f2 | f3 | f4 | f5 | f6 | f7 | f8 |
(Ac) | (Ec) | (AEt) | (AtE) | (Ic) | (Oc) | (IOt) | (ItO) |
In the open system, viewing natural and temporal modalities as a continuum, we may recognize the following 12 fractions. These could be assigned the symbols f1–f12, contextually.
f1 | f2 | f3 | f4 | f5 | f6 | f7 | f8 | f9 | f10 | f11 | f12 |
(An) | (En) | (AcEp) | (ApEc) | (AEt) | (AtE) | (In) | (On) | (IcOp) | (IpOc) | (IOt) | (ItO) |
Here again, note the differences between the open system and the two systems limited to one modality type. The open system is the factually true system, the others being artificial constructs to some extent.
Aristotelean logic considered syllogism as a deductive process applicable to elementary propositions. But we saw in the previous chapter that compound propositions have a logic of their own, so that there are also (derivative) valid moods which function by compounding premises and conclusions.
The following are the most significant samples of this in the present context. They involve a bipolar fractional premise and conclusion, and thus show the transmissibility of particular fractional subsumption.
a. With appropriate complementary major premise (A and An), the fraction (IOp) or (IpO) for the minor and middle terms, yields a similar conclusion for the minor and major terms, through a mix of first and second figure syllogism.
1/AII and 2/AnOpOp:
All M are P, and all P must be M,
Some S are M, though these S can not-be M,
so, Some S are P, though these S can not-be P.
2/AOO and 1/AnIpIp:
All P are M, and all M must be P,
Some S are not M, though these S can be M,
so, Some S are not P, though these S can be P.
b. With appropriate necessary minor premise (An, which implies A), the fraction (IOp) or (IpO) for the middle and minor terms, yields a similar conclusion for the minor and major terms, through a double third figure syllogism.
3/IAI and 3/OpAnOp:
Some M are P, though these M can not-be P,
All M are S, indeed all M must be S,
so, Some S are P, though these S can not-be P.
3/OAO and 3/IpAnIp:
Some M are not P, though these M can be P,
All M are S, indeed all M must be S,
so, Some S are not P, though these S can be P.
We note that, since the fractional premise (and conclusion) are bipolar, the other premise compound must be all affirmative. Similar valid moods can be obtained for temporal and mixed modality fractions, with the appropriate changes in the accompanying premise.
Integers represent the possible states of being. As we pointed out, reality has to materialize in some fully definite way, though knowledge of it may be lacking, only partial or complete. The integers are thus, as states of knowledge, the few cases of complete information. Knowing these clearly, we can use them as factors to predict the many and various possibilities of incomplete information.
Plural integers consist of fractions, either universal fractions taken individually, or some combination of particular fractions. In reality, of course, the integers are monoliths and come first, and the fractions are abstractions we draw out of them by observing their common characters; but we move in the reverse direction as we construct a logical system to represent them.
a. In the closed system of natural modality, there are exhaustively 15 integers; these are mutually exclusive by the laws of opposition. Four of them consist of universal fractions, and eleven consist of conjunctions of two to four particular fractions.
We shall adopt, when useful, the symbols F1–F15 for the 15 integers; note that the numbering of these symbols is applicable within the modal framework under discussion, the same ones may be used with different meaning in another context. But it is well not to get overly symbolic, and remain conscious of the underlying significance in terms of standard A, E, I, O notation.
The following table lists the 15 integers and shows what conjunctions of fractions constitute them. Cells marked ‘yes‘ signify which fractions are included in the corresponding integer. This list must be complete since, mathematically, the 4 particular fractions can only combine in 2 to the 4th power – 1 = 15 ways, 4 of which are the universal fractions.
Table 52.1 The Integers of Natural Modality.
Fractions | |||||||||
(An) | (En) | (AEp) | (ApE) | (In) | (On) | (IOp) | (IpO) | ||
f1 | f2 | f3 | f4 | f5 | f6 | f7 | f8 | ||
Integers | |||||||||
(An) | F1 | yes | yes | ||||||
(En) | F2 | yes | yes | ||||||
(AEp) | F3 | yes | yes | ||||||
(ApE) | F4 | yes | yes | ||||||
(In)(On) | F5 | yes | yes | ||||||
(In)(IOp) | F6 | yes | yes | ||||||
(On)(IpO) | F7 | yes | yes | ||||||
(In)(IpO) | F8 | yes | yes | ||||||
(On)(IOp) | F9 | yes | yes | ||||||
(IOp)(IpO) | F10 | yes | yes | ||||||
(In)(On)(IOp) | F11 | yes | yes | yes | |||||
(In)(On)(IpO) | F12 | yes | yes | yes | |||||
(In)(IOp)(IpO) | F13 | yes | yes | yes | |||||
(On)(IOp)(IpO) | F14 | yes | yes | yes | |||||
(In)(On)(IOp)(IpO) | F15 | yes | yes | yes | yes |
We may in passing mention the relations of the 4 singular integers to plurals. We can say that (Rn) = (An) or (In)(); (Gn) = (En) or (On)(); (RGp) = (AEp) or (IOp)(); and (RpG) = (ApE) or (IpO)(), with the empty brackets signifying any combination of other particular fractions.
A similar list of integers can be drawn up for the closed system of temporal modality.
b. With regard to the mixed modality system, since this involves 12 fractions, of which 6 are universal and 6 particular, we can expect it to yield 2 to the 6th power – 1 = 63 integers. These are listed below (fractions conjoined into integers being marked ‘yes’). Remember, not only is this list exhaustive, but the integers are mutually exclusive. These will be assigned the contextual symbols F1–F63, when useful.
Table 52.2 The Integers of Mixed Modality.
FRACTIONS | |||||||
(In) | (On) | (IcOp) | (IpOc) | (IOt) | (ItO) | ||
f7 | f8 | f9 | f10 | f11 | f12 | ||
INTEGERS | |||||||
(An) | F1 | yes | |||||
(En) | F2 | yes | |||||
(AcEp) | F3 | yes | |||||
(ApEc) | F4 | yes | |||||
(AEt) | F5 | yes | |||||
(AtE) | F6 | yes | |||||
(In)(On) | F7 | yes | yes | ||||
(In)(IcOp) | F8 | yes | yes | ||||
(On)(IpOc) | F9 | yes | yes | ||||
(In)(IpOc) | F10 | yes | yes | ||||
(On)(IcOp) | F11 | yes | yes | ||||
(In)(IOt) | F12 | yes | yes | ||||
(On)(ItO) | F13 | yes | yes | ||||
(In)(ItO) | F14 | yes | yes | ||||
(On)(IOt) | F15 | yes | yes | ||||
(IcOp)(IpOc) | F16 | yes | yes | ||||
(IcOp)(IOt) | F17 | yes | yes | ||||
(IpOc)(ItO) | F18 | yes | yes | ||||
(IcOp)(ItO) | F19 | yes | yes | ||||
(IpOc)(IOt) | F20 | yes | yes | ||||
(IOt)(ItO) | F21 | yes | yes | ||||
(In)(On)(IcOp) | F22 | yes | yes | yes | |||
(In)(On)(IpOc) | F23 | yes | yes | yes | |||
(In)(On)(IOt) | F24 | yes | yes | yes | |||
(In)(On)(ItO) | F25 | yes | yes | yes | |||
(In)(IcOp)(IpOc) | F26 | yes | yes | yes | |||
(On)(IcOp)(IpOc) | F27 | yes | yes | yes | |||
(In)(IcOp)(IOt) | F28 | yes | yes | yes | |||
(On)(IpOc)(ItO) | F29 | yes | yes | yes | |||
(In)(IcOp)(ItO) | F30 | yes | yes | yes | |||
(On)(IpOc)(IOt) | F31 | yes | yes | yes | |||
(In)(IpOc)(IOt) | F32 | yes | yes | yes | |||
(On)(IcOp)(ItO) | F33 | yes | yes | yes | |||
(In)(IpOc)(ItO) | F34 | yes | yes | yes | |||
(On)(IcOp)(IOt) | F35 | yes | yes | yes | |||
(In)(IOt)(ItO) | F36 | yes | yes | yes | |||
(On)(IOt)(ItO) | F37 | yes | yes | yes | |||
(IcOp)(IpOc)(IOt) | F38 | yes | yes | yes | |||
(IcOp)(IpOc)(ItO) | F39 | yes | yes | yes | |||
(IcOp)(IOt)(ItO) | F40 | yes | yes | yes | |||
(IpOc)(IOt)(ItO) | F41 | yes | yes | yes | |||
(In)(On)(IcOp)(IpOc) | F42 | yes | yes | yes | yes | ||
(In)(On)(IcOp)(IOt) | F43 | yes | yes | yes | yes | ||
(In)(On)(IpOc)(ItO) | F44 | yes | yes | yes | yes | ||
(In)(On)(IcOp)(ItO) | F45 | yes | yes | yes | yes | ||
(In)(On)(IpOc)(IOt) | F46 | yes | yes | yes | yes | ||
(In)(On)(IOt)(ItO) | F47 | yes | yes | yes | yes | ||
(In)(IcOp)(IpOc)(IOt) | F48 | yes | yes | yes | yes | ||
(On)(IcOp)(IpOc)(ItO) | F49 | yes | yes | yes | yes | ||
(In)(IcOp)(IpOc)(ItO) | F50 | yes | yes | yes | yes | ||
(On)(IcOp)(IpOc)(IOt) | F51 | yes | yes | yes | yes | ||
(In)(IcOp)(IOt)(ItO) | F52 | yes | yes | yes | yes | ||
(On)(IpOc)(IOt)(ItO) | F53 | yes | yes | yes | yes | ||
(In)(IpOc)(IOt)(ItO) | F54 | yes | yes | yes | yes | ||
(On)(IcOp)(IOt)(ItO) | F55 | yes | yes | yes | yes | ||
(IcOp)(IpOc)(IOt)(ItO) | F56 | yes | yes | yes | yes | ||
(In)(On)(IcOp)(IpOc)(IOt) | F57 | yes | yes | yes | yes | yes | |
(In)(On)(IcOp)(IpOc)(ItO) | F58 | yes | yes | yes | yes | yes | |
(In)(On)(IcOp)(IOt)(ItO) | F59 | yes | yes | yes | yes | yes | |
(In)(On)(IpOc)(IOt)(ItO) | F60 | yes | yes | yes | yes | yes | |
(In)(IcOp)(IpOc)(IOt)(ItO) | F61 | yes | yes | yes | yes | yes | |
(On)(IcOp)(IpOc)(IOt)(ItO) | F62 | yes | yes | yes | yes | yes | |
(In)(On)(IcOp)(IpOc)(IOt)(ItO) | F63 | yes | yes | yes | yes | yes | yes |
Note that 6 of the integers are universals, and 57 are particulars. For reasons of space, the universal fractions f1–f6 are not shown here, but it should be clear that they coincide with the integers F1–F6; these incidentally imply the lone fractions f7–f12, respectively.
Any subject and predicate must be related in one of these ways, and only one. If any fraction involved contains an actual proposition, the applicable integer may change over time, though only one will be applicable at any moment. If none of the fraction(s) involved consist of actual propositions, the integer is immutable.
These are all the possible states of being, in the open system including all modal types, but they do not cover all states of knowledge. We may to different degrees be ignorant as to which of these full realities to apply in a given case, having only partial or no information concerning it.
In passing, let us mention that, here again, singular integers can be reduced to a disjunction of the universal and particular integers which resemble them.
Just as in reality S-P gross formulas are incomplete, without specification of the reverse P-S side, so likewise the integers we have so far considered are deficient pictures of reality. Integers which are solely defined by S to P relations, are ‘flat’ — in the real world, every S and P relation also has a P to S facet. Thus, only ‘stereoscopic’ integers are really ‘integers’, in the ultimate sense of full expressions of a relationship.
The combination of flat integers into stereo integers resembles the combination of fractions into integers. For example, the S to P relation is ‘All S must be P’ and the P to S relationship is ‘some P must be S and some cannot be S’. This could be written symbolically as, say, SP:(An) + PS:(In)(On). Many such two-way conjunctions of integers are possible; but of course, some combinations are interdicted by the laws of conversion.
I will not, in the following chapters, develop a logic for such complex integers, because the topic is just too vast. I think that the innovations in inductive logic, presented in this work, will be best served by avoiding such further complications. The factorial approach is what I want to highlight, and it will be more clearly put across using the simpler medium which I have adopted.
The reader is asked, nevertheless, to keep in mind the avenues of further development here hinted at. The logic covered here concerns ‘flat integers’: that for ‘stereo integers’ is yet to be dealt with, in some future work or by other logicians. The truth of what is said in this treatise is not affected, it is only made more partial a truth than implied. A flat integer may be viewed as a genus including a number of possible stereo integers.
Incidentally, we can further expand the whole study by considering transitive relations, like ‘S can or must become P’, in various combinations with each other or with static subsumptives. I will not venture into this field here.
Another direction of development to take note of, is consideration of conditional propositions, to the same extent as categoricals are dealt with in this treatise.
I pointed out, in the part on de-re conditioning, that categoricals and conditionals are all particles of a large continuum of modal propositions.
They share many hierarchies of implication, like for instance: ‘All S must be P’ (a categorical) implies, among other things, that ‘When certain things are S, they must be P’ (a natural conditional), and that ‘Anything which can be S, can be P’ (an extensional conditional).
Also for instance, the premises ‘All S must be P, and all P must be Q’ may be viewed as forming a categorical syllogism yielding ‘All S must Q’, or a productive argument for ‘When any S is P, it must be Q’ (natural) or ‘Any S which must be P, must be Q’ (extensional). This shows that a one-predicate (categorical) form is the top of both natural and extensional hierarchies, of (conditional) forms with two predicates (or eventually more).
Similarly with temporal modality. And, in a still larger picture, all manners of disjunctive propositions (with any number of terms) can be included. The basis of any form used should always be kept in mind, of course.
It is clear that the whole doctrine of fractions and integers (including stereo as well as flat fractions and integers), and likewise all other aspects of factorial analysis and induction, can be expanded to include all de-re conditioning, of any form and modal type. Various and numerous compounds emerge from the combinations of all such propositions, whether of the same type and form, or of mixed type and/or form.
Particular fractions of categorical propositions may, as already mentioned, be expressed in conditional language. Similarly, compounds of such fractions, forming integers, may be clarified by conditionals.
For example: (In)(IOp) means {Some S must be P} and {Some S which are P, can not-be P}, but also takes for granted the formal truth that {No S which must be P, can not-be P}, and its contraposite {No S which can not-be P, must be P}’. As more fractions are conjoined, the interrelative statements become more complex, but are in any case expressible through extensional conditionals.
We may also interpose natural (or temporal conditionals) to express formal truths applicable within brackets, like (IOp) tacitly appeals to {When certain S are (as now) P, they cannot be nonP} and its contraposite {When certain S are not P, they cannot be P}, since the required bases are given in the categorical premises.
Thus, all types of conditioning are involved to some extent even in categorical fractionating and integration, fulfilling the role played by the brackets in symbolic descriptions. Our brackets are not artificial constructs, but shorthand notation for such implied conditionals, delimiting and separating extensions, and circumstances or times.
These examples are just some of the intersections of the different formal continua. A complete theory would have to be more systematic than that, and consider all conceivable conditionals rather than the few implied by categorical compounds.