FUTURE LOGIC

CHAPTER 56.APPLIED FACTOR SELECTION.

1.Closed Systems Results.

2.Some Overall Comments.

3.Rules of Generalization.

4.Review of Valid Moods.

5.Open System Results.

1.Closed Systems Results.

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.

A similar table can be drawn up for temporal modality, substituting the suffixesc,tforn,p.

2.Some Overall Comments.

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 beingF1–F10andF3-4.

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,ApIfirst impliesIp, from whichAnis 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,IpyieldsAnby induction, and thenAIn, say, is inferred, since implied byAn.

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,IntoAnonly eliminates 7 factors, whereasIptoAneliminates 13 factors. The movement is more cautious, and therefore more likely to turn out to be correct in the long run.

The generalization fromItoA, or fromOtoE, 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.

3.Rules of Generalization.

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.

4.Review of Valid Moods.

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 toIp->Anare:I,In,Ap,ApI,ApIn,A,AIn,An. The caseAntoAnis of course deductive, even tautologous, and only listed to show the continuity. The subaltern (elementary) conclusions toIpare:A,Ap,In,I; toI:A,Ap,In; toIn:A,Ap; toAp:A,In,I; toApI:A,In; toApIn:A; toA:In; and toAIn: none. Similarly,Op->Enhas 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 moodIOp->AEp. First the elements ofIOpare 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,Ihas been generalized toA, andOptoEp.

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 premisesApIOp,IEp,AOp,ApIEpyield 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 moodIpO->ApEand 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 premisesIpEp,ApOp, andApEphave the same result. Note that the conclusion is not simplyApEp, 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 fromAEpto (AEp) or fromApEto (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 compoundIOimpliesI+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 premisesIOn,InO, andInOnmay be viewed as subalterns toIO, 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 toIO->(In)(On), note well, since neitherInOpnorIpOnformally impliesIO. They are, however, closely related, having in common the same conclusion, and the same subaltern premisesIOn,InO,InOn.

Note well, incidentally, thatInOn->(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 remainderIpOp, which we know to overlap sinceApis general and given; then we favor the (IOp) outcome, generalizingIptoI, on the basis thatIis already implicit inIn. In comparison, the (IpO) eventuality, though conceivable, would require a move fromOptoO, for which no specific basis is found, so that it may be inductively eliminated. The moodAInOpyields 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 thatApis general and thatIandOcannot overlap; then we generalize the remainingIsegment of the extension toIn. The (IOp) eventuality, though conceivable sinceOimpliesOp, 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 premiseAInOyields 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 fromApIEpOto (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 onlyF10, but alsoF13,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, theF15conclusion is deductive.

All that has been said for natural factor selection, could be repeated for temporal factor selection. The two closed systems behave identically.

5.Open System Results.

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 symbolsF1–F15have 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, whereasIcOpandIpOchave 47 each, andIOtandItO53 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 betweenF1andF21. 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 ofGroupF21. 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 56.2Factor Selection in the Open System.

Arguments are Grouped according to their Conclusion (always an integer).