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FUTURE LOGIC©
Avi Sion, 1990 (Rev. ed. 1996) All rights reserved.
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.
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:
In the mixed modality system, an individual has 6 alternative states of
being:
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:
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:
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.
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.
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.
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.
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