FUTURE LOGIC

CHAPTER 58.GROSS FORMULA REVISION.

1.Amplification.

2.Harmonization.

3.Unequal Gross Formulas.

4.Equal Gross Formulas.

5.Applications.

1.Amplification.

In gross formula amplification, we merely add the new data to the old, usually to obtain a more complex gross formula, of greater informational specificity. The tables of opposition tell us what elements are compatible; so that, as long as all the elements involved may coexist, they can be conjoined together consistently. The revised formula should then be analyzed factorially, and its strongest factor selected in accordance with the law of generalization. The new result may or may not differ from the original. Let us consider some examples.

The easiest case is when the additional data is already implicit in the original, as inA+I=AorInOp+I=InOp. Here there has effectively been no change of formula, though one can say that the original formula is further confirmed. Any generalization made from the original remains the same.

Next, we have the case of a formula of single polarity amplified by data of the same polarity not implicit in it, as inA+In=AIn. Here, there is a change of formula, but this has still no effect on eventual generalization, except to render it more certain by diminishing the number of factors dropped by it.

Thirdly, we have the case of a formula being so affected by additional, though consistent, data, that its inductive results may change. For instance, takeApIn+Op=ApInOp: the original conclusion fromApInwould be (An); after addition ofOp, the gross formula becomesApInOp, and its conclusion is (In)(IOp), quite different. Or again, takeApInOp+O=ApInO: here, instead of concluding (In)(IOp), we are now forced to conclude (In)(IpO).

There is no need to list all possible cases of gross formula amplification, for they are already effectively spelled out in the chapter on factor selection. Such situations may occur either prior to generalization, when we are still dealing with gross formulas, or at a more advanced stage of the proceedings, provided we are able to retrace our past course.

In the event of a conflict between two formulas, if we are able to remember the consistent gross formulas which gave rise to these generalizations, or these deductive consequences of generalization, which are now in conflict, we can resolve the conflict by merging the root propositions into a new, amplified formula, which henceforth will serve as our inductive premise. But such radical revision is not always possible.

All that has been said so far is applicable equally to naturals, to temporals, and to mixed modality gross formulas.

2.Harmonization.

Harmonization, or reconciliation, concerns situations where the original and additional data are in conflict. In a looser sense, we might view even amplification as involving a kind of ‘conflict’: that between the assumption that the original thesis told the whole story, and the discovery through the new input that it did not. But conflict here means specifically oppositional incompatibility.

Some of the elements are contradictory or contrary, and the issue is how to resolve such discrepancy once it is noticed. Something has to ‘give’, on one side or the other or on both sides, for the controversy to be defused. An adaptation of sorts is required.

Harmonization follows the familiardialecticalpattern:thesis, antithesis, and synthesis.

Conflicts arise because, somewhere along the line, there has been an over-generalization. We exaggerated the impact of certain observations, insights, or narrow inferences, and, sooner or later, this was bound to lead to inconsistency, whether with these same terms, or in consequent propositions involving other terms. However, when the inconsistency arises, it may not be immediately clear which side to blame for the error.

The way conflicts are to be resolved depends, firstly, on the relative weights of the original and new data.

The degree of credibility of a thesis is a function of how clearly it is formulated, how tightly it is argued, how much connection it has to the rest of knowledge, how well established its conceptual and logical sources are, how empirical it is, how often it is confirmed, how dependent we are on it for practical purposes. Credibility is thus a variable appearance, a phenomenon by which the many forces affecting our trust in a thesis are summed up at any given point in the development of knowledge.

The credibility of a thesis is also a function of the existence of alternative ideas or theories, and their relative credibility. A thesis may be credible on its own, yet, when viewed in perspective, in comparison to others, it may seem less likely than some other. When two or more theses are in conflict, an evaluation of their relative weights must be made, and an order of priority assigned to them.

Note that very often, we assign greater weight to an older thesis, one to which we are more accustomed; such conservatism is somewhat justified on the basis that the older thesis is time tested. But it can happen that a new thesis may be shown, by thorough examination, to be in all respects of equal weight or better; from a scientific point of view, age is ultimately accidental, and not an over-riding cause for inertia.

3.Unequal Gross Formulas.

The simplest case of conflict resolution is when either the original or the new data, whichever, has greater weight. In such case, the dominant thesis remains untouched by the conflict, and its submissive antithesis alone must bear the burden of adjustment.

Each thesis may be an element or compound; and it may be positive, negative, or of mixed polarity. Obviously, however, only elements of opposite polarity may be inconsistent with each other; though of course, if elements of opposite polarity are low enough on the scale of quantity and modality, they may not create a conflict.

The general rule of harmonization of theses of unequal weight is, any element(s) explicit or implicit in the submissive thesis, which are contradictory or contrary to some element(s) in the dominant thesis, must be diminished in quantity and modality until compatible, or even, if needful, totally denied.

Some examples. If the dominant isAn, then all negative theses fromOpon up must be rejected. If the dominant isAIn, then onlyOpmay coexist, sinceEpwould contradictIn, andOwould contradictA, and perforce any statement still stronger than those would be contrary. If the dominant isApI, then onlyOand/orEpare acceptable, sinceOnor more would conflict withAp, andEor more would clash withI. If the dominant isApInO, thenA,Ep,On, and all their subalternants are impossible.

Thus, when one thesis is given more weight, the outcome is easy enough to predict, using the rules of opposition. However, harmonization does not end there. We must now consider the significance of our rejection of certain elements. If, say, the submissive thesis wasAn, and we were forced to particularize it toApI, then we must find out if any of our other beliefs include or implyAnorA, and correct these too in turn; also, if any deductive inferences were drawn fromAnorA, they are now rendered less sure (though not automatically deniable).

Furthermore, such weighted harmonization usually, though not always, leaves a remainder. If, for instance,Anis pitted againstEn, with the latter dominant, the result is simplyEn, since evenIpwould maintain conflict. But ifAis pitted againstE, again with the latter dominant, onlyAis rejected, while its subalternApremains in force, since there is no reason to surrender it, and is to be added toE, forming the compoundApE.

In this way, we do not push the harmonization beyond the necessary minimum, but subtract only those levels of quantity or modality which cause the interference. Such conjunction of remainder (of the submissive) to the dominant follows the pattern already described under the heading of amplification.

Harmonization may change our original inductive conclusion, or leave it unaffected. For instance, if we start with thesisApIOp, and findEfrom other sources, and deciding the former stronger, resolve the conflict toApIEpO, the inductive conclusion is changed from (AEp) to (IOp)(IpO). However, ifAOpwas the original thesis, andEarose in submissive conflict with it, the resolutionAEpdoes not affect our initial induction of (AEp).

Again, all that has been said so far is applicable as well to naturals, to temporals, and to mixed modality gross formulas.

4.Equal Gross Formulas.

By far more complex, and interesting, is harmonization between gross formulas of equal weight which are in contention. We know of no reason to favor one thesis over the other, so that their credibilities are in equilibrium, and we seek their common grounds, their points of agreement. We are supposedly unable to retrace the course which led up to them, or pin-point where we may have erred, and so must deal with them as we find them, narrowing their scopes to acceptable levels, down to where they can coexist.

More often than not, there may be more than one way to resolve the conflict. This is where factor selection comes into play, providing us with a convincing basis for arbitration. Without this beautiful instrument, such formula revision would have been pure guesswork.

Conflicts between gross formulas of equal weight are resolved step by step, as follows:

a.Fuse the original two gross formulas, whatever they be, into oneinconsistentcompound of mixed polarity; we can do this, because all elements have equal weight. Then, separate the positive and negative elements or compounds from each other, to form two gross formulas each of which has uniform polarity.

b.Now, try out the hypothesis of harmonization of quantity. This means, firstly, diminish the quantity of all the elements in both formulas to particular; we shall call the resulting compound the ‘quota’. Secondly, find the ‘remainder’ of this operation, if any; that is, what explicit or implicit elements in the two formulas are compatible with the quota. Thirdly, fuse quota and remainder into one compound, and identify its strongest factor.

c.Next, similarly try out the hypothesis of harmonization of modality. Thus, firstly, diminish the modality of all the elements in both formulas to potential; the resulting compound being the ‘quota’. Secondly, find the ‘remainder’ of this operation, if any; that is, what explicit or implicit elements in the two formulas are compatible with the quota. Thirdly, fuse quota and remainder into one compound, and identify its strongest factor.

d.Lastly, compare the strongest factors of these two alternative conclusions. Whichever has the stronger strongest factor is the revised gross formula. The selected revised formula is, alone, the conclusion of the harmonization process, the resolution of the original conflict, the goal we pursued. Its strongest factor is accordingly the optional inductive conclusion, if we choose to generalize at this point.

Thus, the decision as to whether to revise the formulas by reference to quantity or to modality is made for us by factor selection. It is not subjective, but systematic, controlled by what would be the inductive result in either case. There is but one universal law for both inductive processes.

Our results demonstrate that, had we sought guidance through some separate ‘law of particularization’, we would have made many mistakes, as will be seen. Some conflicts are resolved on the side of contraction of quantity, others are resolved through modality contraction, others still yield the same result on either side; there is no general rule in that respect. Particularization has no special basis, but is entirely determined by the same law as generalization: namely, selection of the strongest factor.

5.Applications.

The following table shows the results of conflict resolution obtained, by the method described above, for gross formulas in the closed system of natural modality. As usual, similar results could be worked out for temporal modality.

The positive and negative elements of the conflicting theses are grouped separately, and listed under the heading ‘given conflict’. The resolutions by quantity and modality, and their respective strongest factor (SF) are listed in adjacent columns. In each case, the quota is displayed first, and the remainder if any is added on to it with a ‘+‘ sign. The corresponding strongest factor, note well, concerns the complete compound of quota and remainder (this compressed version is not shown in the table, being easily constructed). We know the selected factor in each case from our prior inquiry on this topic (the reader is referred to the chapter on factor selection).

Finally, the inductive conclusion is given: notice that this is the selected factor with the lowest ordinal number of the two. The data has been classified by similarity of inductive conclusions, to show the inherent continuities. The range isF3–F10. The revised gross formula is the one whose strongest factor is selected as the inductive conclusion. Observe the uniformities found in each grouping.

Note the inclusion in this table, for the sake of completeness, of cases where the positive and negative sides are not in conflict (these are marked ‘okay’). Excluding these, there are in all 50 valid moods. Among them, 18 have been earmarked with an asterisk (*): these may be viewed as the main moods, in that they present the strongest conflicts for each distinct revised formula. Examples are given after the table.

Table 58.1 Conflict Resolutions for Equal Gross Naturals.

Some examples are in order.

·ApInversusEp. Contracting their quantities, we obtainInOp; we can still keep theApexplicit inApIn, since it is not in conflict withEp; hence, we getApInOp(strongest factorF6), under this hypothesis. Alternatively, contract their modalities, to obtainApEp; this allows forI; so,ApIEp(strongest factorF3) is yielded by this hypothesis. Comparing the two, the latter is found stronger, and so the revised gross formula isApIEp, andF3is its eventual generalization. Effectively, the harmonization particularizedIntoI, necessity to actuality, in this case.

·AnversusEn.Quantity harmonization isInOn; there is no remainder since evenApis excluded byOnandEpbyIn. Modality harmonization isApEp(AEbeing impossible); with remainderIO(rather thanAorE, which would be asymmetrical).InOnhasF5as its strongest factor, whereasApIEpOhasF10, so the former wins. In this case, the harmonization particularizedAntoInandEntoOn, that is, quantity instead of modality.

·AnversusOp. This is a case where both directions lead to the same conclusion. Whether our quota and remainder areInOpandA, orApOpandAIn, the result is stillAInOp(SF =F6). Here, particularization has downgraded both the quantity (AntoIn) and the modality (AntoA).

An especially interesting case is that ofAversusE. We saw that, within the closed system of actual propositions, this conflict yields conclusionIO, or (I)(O). However, here, in the wider context of natural modality, we see that the conclusion by any means is, more fully,ApIEpO, or (IOp)(IpO). Thus, we agree concerningIO, but we note thatApEpwere previously hidden in the traditional system. In any case, this shows that our broadened theory of induction is consistent and continuous with the classical.

A similar approach may be used for conflict resolution among gross formulas in the open system of mixed modality. Since the results stem directly from tables of opposition and factor selection, they will not be listed in this treatise, to avoid excessive volume. The processes involved are now firmly established and clearly exemplified, and that is sufficient for our present purposes.