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FUTURE LOGIC©
Avi Sion, 1990 (Rev. ed. 1996) All rights reserved.
CHAPTER 56.
APPLIED FACTOR SELECTION.
We will, to begin with, deal with the closed system of natural modality,
first listing the results of factor selection, then analyzing and justifying our
proposals. As usual, all the results obtained can by analogy be replicated for
the closed system of temporal modality. The corresponding results for the more
bulky open system of mixed modality will be presented later.
The following table shows the proposed preferred (natural) factors for
natural gross formulas, selected on the basis of the uniformity principle.
Deductive cases, those with a single factor on formal grounds, are included for
completeness.
The information in the elementary or compound premise is always assumed
to be all available data on the subject to predicate relation concerned. If more
data makes its appearance, then we are faced with another premise, and the
conclusion may accordingly be different.
The column 'NF' indicates the original number of factors, the next column lists
them in sequence, and the column 'SF'
shows the selected factor among them, which is our proposed conclusion..
Table 56.1 Factor Selection in Natural Modality.
The above table shows that, given a particular and/or potential (or even
actual) proposition, we are unable to decide which way and how far to generalize
it, without reference to the whole gross formula. If the gross formula consists
of a single element, the conclusion is easy; it is the universal necessary
proposition of like polarity. But if the gross formula is a compound, then the
inductive path of any element in it depends on which other elements are
involved. This is important to keep in mind.
We see that in some cases a particular proposition has become general,
without change of modality; in other cases, the modality is raised, without
change of quantity; in others still, both quantity and modality are affected.
Also, two particular elements of a gross premise may emerge in the factorial
conclusion as overlapping, or they may be separated.
Effectively, we have obtained the valid moods of natural modal induction
(and by extension, those for temporal modality). They are not as numerous as
appears, for we can distinguish 11 groups of valid moods among them, each
defined by the best conclusion yielded. The conclusions being F1-F10
and F3-4.
The groupings together include 13 primary valid moods, each of which has
a number of subalterns. A primary mood in any group is one yielding the highest
conclusion from the lowest premise. Subaltern moods are of two kinds.
The secondary premise may be higher than the primary one, yet yield a
no-better conclusion, so that in effect the induction proper occurs after
eduction of the lower premise. For example, ApI
first implies Ip, from which An is
thereafter induced. Or the secondary conclusion may be lower than the primary
one, in which case it is in effect educed from the higher conclusion after the
induction proper. For example, Ip
yields An by induction, and then AIn,
say, is inferred, since implied by An.
However, note well that subaltern moods are more certain than their
corresponding primaries, because the number of factors they eliminate is lesser.
Thus, for instance, In to An only eliminates
7 factors, whereas Ip to An
eliminates 13 factors. The movement is more cautious, and therefore more likely
to turn out to be correct in the long run.
The generalization from I to A, or from O to E,
found in the closed system of actuals, can in this wider system of modal
induction be viewed as a partial generalization. We move from a formula of 12
factors to one of 3 factors. We have not narrowed our position down to a single
integer, but have nevertheless diminished the area of doubt considerably. Such
limited generalizations are always permissible, of course, if they suffice for
the needs of a specific inquiry.
The rules of generalization clarify the various aspects of the uniformity
principle. They are presented here, prior to detailed analysis of the valid
moods, to facilitate the reader's understanding of the discussion, but in fact
they simply summarize the insights accumulated in the course of case by case
examination.
The uniformity principle for factor selection, has a variety of
implications. Some of these emerge in the paradigm of actual induction, but
others become apparent only in modal logic. The rules of generalization serve to
expose the variety of considerations which arise, and provide us with more
specific guidelines than the basic principle.
The various vectors of uniformity often interfere with each other, in
such a way that satisfying the requirements of the one, frustrates the demands
of the other. This is because different factors stress different things. For
instance, one factor may stress quantitative generalization, another may stress
modality generalization. Case study of such conflicts of interest gradually
clarified the order of importance of the different tendencies. The rules of
generalization thus have an order of priority.
a.
Polarity.
First in line is the requirement that the conclusion resemble the premise in
polarity. If there is but one polarity in the premise, the same will remain
solitary in the conclusion. If the premise is a bipolar compound, so must the
conclusion be. One cannot induce a different or supplementary polarity. Such
innovation has no basis in the uniformity principle, and can only occur with
factual justification. Many factor selections, seeming to involve change of
quantity or modality, rather stem from this inertia of polarity.
b.
Quantity.
Next in line is increase in quantity, as far as consistent. This is the prime
change induction seeks to effect. This is because a universal proposition is
most open to testing, by drawing its consequences through deductive logic.
Maximal extensional generalization is to be favored over improvements in
modality or other uniformities, wherever possible. It is the paradigm of the
uniformity principle, an assumption that properties tend to relate to classes,
rather than being scattered accidentally.
c.
Modality.
Uniformity implies an overall preference, not only for the more general
alternative, but also for the factor of higher modality. However, modality
generalization is only next in importance to that of quantity. But it is still
this high on the list, for similar reasons: practically, because the higher the
category, the more testable the result; metaphysically, because we assume a
stable substratum beneath the changes we perceive.
Within either closed system, necessity is preferred to actuality, and
actuality to possibility. In the open system, mixing modality types, natural
necessity should be favored over constancy, and temporariness over potentiality,
whenever the prior guidelines allow it. This is obvious from the relative
positions of these various categories on the modality continuum.
d.
Symmetry.
If the premise consists of elements of opposite polarity which are identical in
both quantity and modality, the conclusion must have the same evenness. There
would have to be factual basis for one side or the other to grow in quantity or
modality more than the other; the uniformity principle does not justify such
loss of symmetry. This is why the conclusion in a few cases cannot be a single
factor, but a disjunction of two.
If on the other hand, the compound premise gives one or the other
polarity a higher quantity or modality, the conclusion may or may not favor the
one over the other: it depends on other considerations. Many subaltern moods
have the unevenness of their premise in this way removed by the conclusion.
e.
Overlap.
If some elements of a compound premise are known to converge, at least that same
degree of overlap must reappear in the conclusion. Overlap cannot be lost by
induction.
On the other hand, it may be gained. If overlap is not at all assured
originally, it may be assumed, provided no prior considerations are put in
jeopardy. Where there is a question as to whether two separately discovered
particulars overlap or not, the uniformity principle would seem to suggest that
they be applied to each other's extensions, so that both be maximally
generalized.
However, if overlap is open to doubt, and making its assumption would
cause problems in other respects, the adoption of the divergence hypothesis is
acceptable. Overlap is of less importance than other issues, because it is
conceptually derived from them.
f.
Simplicity.
Lastly, but still significant, is the concern with fragmentation. In a choice
between a factor with few fractions and another with many, both of which satisfy
the prior guidelines, the former is preferable. We should not fragment the
extension beyond the minimum feasible, always preferring the simplest
alternative. This is an aspect of uniformity, in that it opposes diversity
between the members of the class concerned. If indeed the more complex
alternative is true, it will eventually impose itself through particularization.
The applications of these rules of generalization will now be seen
through specific examples.
Let us now review each primary valid mood of natural induction in some
detail. In every case, to repeat, the gross premise, be it elementary or
compound, is assumed to represent all available information on the subject to
predicate relation concerned.
a.
From any premise of single polarity, may be induced a universal necessary
of same polarity. This is the most obvious application of the uniformity
principle: there is no basis for presuming the other polarity at all possible.
The primary moods in this group involve increase in both quantity and modality.
They are:
Ip ->
An
Given solely that Some S are P,
we may induce that All S must be P.
Op ->
En
Given solely that Some S are not P,
we may induce that No S can be P.
The subaltern premises to Ip ->
An
are: I, In, Ap, ApI,
ApIn, A, AIn,
An. The case An to An
is of course deductive, even tautologous, and only listed to show the
continuity. The subaltern (elementary) conclusions to Ip
are: A, Ap,
In, I; to I: A,
Ap, In;
to In: A, Ap;
to Ap: A, In,
I; to ApI: A,
In; to ApIn:
A; to A: In;
and to AIn: none. Similarly, Op -> En
has some 16 subaltern inductive moods (not counting compound conclusions).
b.
From a conjunction of particular premises of different polarity, one of
which is actual and the other potential, the best inductive conclusion is a
similar conjunction of universal premises. Here, the uniformity principle leads
us to assume the particulars to fully overlap, and to generalize quantity only
(not modality), to obtain a result with the original bipolarity.
IOp ->
AEp
Given solely that Some S are P and some S can not-be P,
we may induce that All S are P though all can not-be P.
IpO ->
ApE
Given solely that Some S are not P and some S can P,
we may induce that No S are P though all can be P.
It is clear that this induction occurs in stages. Consider the mood IOp
-> AEp.
First the elements of IOp are made to
converge into the fraction (IOp),
dropping 3 factors, then this particular fraction is generalized into its
universal equivalent (AEp), dropping
a further 7 factors. Effectively, I
has been generalized to A, and Op to Ep.
Alternative conclusions, though formally conceivable, seem less
justifiable. For instance, (In)(On), by assuming nonoverlap, would cause baseless fragmentation of
the extension, and result in a modal equality between the poles which was
originally lacking. Whereas, say, (In)(IOp),
while granting partial overlap and uneven modality, would fragment the extension
without specific reason. Furthermore, a general conclusion is always to be
preferred to a particular one, even one of stronger modality, because it is more
readily tested.
The 4 subaltern premises ApIOp,
IEp, AOp, ApIEp
yield the same result. In their case, a partial overlap, meaning the fraction (IOp), is already implied, since one of the elements of the compound
is universal already. In each case, consequently, less generalization is
involved than in the primary mood, and the result is somewhat more trustworthy.
All the same comments can be made concerning the mood IpO
-> ApE
and its subalterns.
c.
When the premise is a compound of two particular potentials of different
polarity, an imperfect conclusion may be drawn, diminishing the number of
factors to two universal compounds in disjunction. Here, the original modal
symmetry inhibits a more definite result, which would strengthen one side more
than the other. But there is still an improvement in specificity, a guarantee of
overlap and generalization of quantity having been achieved. The disjunctive
result can be used in dilemmatic arguments.
IpOp ->
'(AEp) or (ApE)'
Given that Some S can be P and some S can not-be P,
we may induce that
either 'All S are P, though all can not-be P'
or 'All S are notP, though all can be P'.
The subaltern premises IpEp, ApOp, and ApEp have the
same result. Note that the conclusion is not simply ApEp, which would allow the factor (IOp)(IpO) as an
alternative. Precisely for this reason, ApEp
-> '(AEp)
or (ApE)' is not a deductive inference, as those from AEp
to (AEp) or from ApE to (ApE)
were, but an induction diminishing the number of factors from 3 to 2. Even
eliminating the fragmentation inherent in (IOp)(IpO) makes the effort worthwhile.
These moods may be viewed as to some extent subsidiary to the preceding
group, tending toward the same sort of conclusion, but not quite succeeding. The
elimination of particularistic alternatives, such as (In)(On),
is based on similar argument.
d.
From two particular actuals of opposite polarity, we induce two
particular necessaries with corresponding polarities. Here, we may not in any
case generalize quantity, for the four universal factors are deductively
inconceivable, anyway; none of them would be compatible with the premise; they
are not among the available factors. Thus, only modality, the next best thing,
is increased as far as it goes, up to necessary; thusly, for both poles, to
retain the original symmetry.
IO ->
(In)(On)
Given solely that Some S are P and some S are not P,
we may induce that Some S must be P and some cannot.
Note that in this special case, the uniformity principle causes
divergence, rather than overlap, for the sake of obtaining a higher modality,
while retaining the original evenness in modality. Although the compound IO
implies I+Ip+O+Op,
so that we might induce (IOp)(IpO)
to achieve maximum overlap, the proposed conclusion is preferable, because it
involves necessity instead of mere actuality and effectively no greater
fragmentation of the extension. As for (In)(On)(IOp)(IpO),
though equally conceivable in principle, and involving both necessity and
overlap advantages to some extent, it is rejected, because it introduces an
excessive fragmentation, for which no argument is forthcoming.
The premises IOn, InO, and InOn
may be viewed as subalterns to IO, as
well as to the primaries considered next.
e.
From the conjunction of two particulars of opposite polarity, one of
which is necessary and the other potential, a conjunction of two particular
necessaries of opposite polarity is induced. Here, the original asymmetry and
the conceivable partial overlap, are sacrificed to improvement in modality. Any
universal conclusion is again out of the question, on formal grounds.
InOp ->
(In)(On)
Given solely that Some S must be P and some can not-be,
we may induce that Some S must be P and some cannot be.
IpOn ->
(In)(On)
Given solely that Some S can be P and some cannot be P,
we may induce that Some S must be P and some cannot be.
These two moods are independent primaries, and not subalterns to IO
->
(In)(On), note well, since neither InOp
nor IpOn formally implies IO.
They are, however, closely related, having in common the same conclusion, and
the same subaltern premises IOn, InO, InOn.
Note well, incidentally, that InOn
->
(In)(On) is indeed an inductive argument, and not a deductive one, since
InOn has 4 factors originally, 3 of which are then eliminated, for reasons of
asymmetry or excessive fragmentation, as our table shows.
f.
Two more groups of valid moods are distinguished by their more complex
primary premises and conclusions. They are the following.
ApInOp -> (In)(IOp)
Given that All S can be P, some S being necessarily P,
and others potentially not P,
we may induce that the latter S are actually P.
IpEpOn -> (On)(IpO)
Given that All S can not-be P, some S being necessarily not P,
and others potentially P,
we may induce that the latter S are actually not P.
In the positive case, we first separate the (In)
fraction from the remainder IpOp,
which we know to overlap since Ap is
general and given; then we favor the (IOp)
outcome, generalizing Ip to I,
on the basis that I is already
implicit in In. In comparison, the (IpO)
eventuality, though conceivable, would require a move from Op
to O, for which no specific basis is
found, so that it may be inductively eliminated. The mood AInOp yields the (In)(IOp)
conclusion deductively, not inductively, since this is its only factor. Similar
comments can be made with regard to the parallel negative cases.
ApIO ->
(In)(IpO)
Given that All S can be P, some S being actually not P,
and others being actually P,
we may induce that the latter S must be P.
IEpO ->
(On)(IOp)
Given that All S can not-be P, some S being actually P,
and others being actually not P,
we may induce that the latter S cannot be P.
Here again, in the positive case, we first separate the (IpO)
fraction, on the grounds that Ap is
general and that I and O cannot overlap; then we generalize the remaining I
segment of the extension to In. The (IOp) eventuality, though conceivable since O implies Op, is rejected
on the basis that it involves a weaker category of modality compared to (In);
as for the conjunction of both (In)
and (IOp), this would introduce a needless additional fragmentation into
the equation. The subaltern premise AInO
yields the same inductive conclusion, by elimination of only the latter
eventuality, for the same reason. Similar comments can be made with regard to
the parallel negative cases.
g.
The inference from ApIEpO to (IOp)(IpO) is deductive, as we saw in factorial analysis.
On the other hand the move from the gross conjunction of the two
particular fractions (IOp) and (IpO) as a
premise, to the integer (IOp)(IpO)
is inductive, not deductive. For the common factors of the fractions are not
only F10, but also F13, F14,
F15. The latter three, which involve the conjunction of (In)
or (On) or (In)(On)
to (IOp)(IpO), are formally
conceivable, but in this context rejected, on the basis that they introduce new
fragments without specific justification.
The other gross conjunctions of fractions, in twos or threes, similarly
yield their integral counterparts, F11-F14, by induction. In the case of four fractions, the F15
conclusion is deductive.
All that has been said for natural factor selection, could be repeated
for temporal factor selection. The two closed systems behave identically.
We shall now list the valid moods of open system induction, with a
minimum of comments, for the record. The reader is encouraged to review the
valid moods, with reference to the rules of generalization, to justify the
selection of this or that factor rather than any other, in each case.
We saw, in earlier chapters, that when natural and temporal modality are
considered together, 63 integers (see table 52.2) and 195 gross formulas (see
table 51.1) may be generated. In an appendix, we developed a table showing the
factorial analysis of all gross formulas. The factorial analysis of the
particular fractions, on the other hand, may be found in table 52.2 (reading it
downward).
The valid moods of open system induction, are easily extracted from these
sources of information. In accordance with the law of generalization, the factor
to select in induction is usually the first, the one with the lowest ordinal
number; though, in a few cases, we must select the first two factors in
disjunction to maintain symmetry. This is so, simply because I numbered the
factors that way, in order of generality, necessity, and simplicity.
Be careful not to confuse the closed system factors with the open system
factors; the symbols F1-F15 have mostly
different meanings in each context. Also remember not to equate the four
compound particular fractions, (IcOp),
(IpOc), (IOt), (ItO) to their
gross equivalents. Each of the former has 32 factors, whereas IcOp
and IpOc have 47 each, and IOt
and ItO 53 each.
The table below shows the selected factors for all gross formulas in the
mixed modality system. Premises with the same inductive conclusion are grouped
together, and their common result is given. The number of factors for each
formula is listed under the heading 'NF'.
There are, we see, 23 groups of valid moods, with numbers lying between F1
and F21. A total of 33 of the moods
are primary; these are indicated by 3 asterixes (***). The remaining moods are subalterns of these.
Note that 11 moods are in fact deductive, rather than inductive, since
they were found to have only one factor when analyzed; one of these is the sole
listed representative of Group F21. These are included for completeness.
While the individual fractions are also included in our table, the
various gross conjunctions of two to six particular fractions have been ignored,
to avoid excessive volume; these obviously yield their integral counterparts, F7-F63, as inductive
results. Table
56.2 Factor
Selection in the Open System. (Click
to see table.)
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