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© Avi Sion, 1990 (Rev. ed. 1996) All rights reserved.

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1. Knowability.

2. Equality of Status.

3. Stages of Induction.

4. Generalization vs. Particularization.

5. The Paradigm of Induction.

6. The Pursuit of Integers.

1. Knowability.

Some skeptical philosophers have attempted to write-off natural necessity, and potentiality, as unknowable, if not meaningless. We have shown the meaningfulness and importance of these concepts, in the preceding pages. Here, we will begin to show systematically how they may be induced.

At the outset, let us note that to assert that natural necessity cannot be known, is to claim knowledge of a naturally necessary phenomenon; this is implicit in the use of ‘cannot’ in such assertion. If the assertion were merely put as ‘man does not know natural necessity’, in an attempt to be consistent, we see that the statement would have no force; we could still ask ‘but can he?’ Thus, this concept is undeniable, and its attempted rejection untenable.

Furthermore, the formal link between natural necessity and potentiality, makes the latter also inevitable. They are two sides of the same coin, if either is admitted then the other logically follows by systematization: every concept must have a contradictory. The potentiality of something is merely negation of the natural necessity of its absence. Thus, the intrinsically concealed and invisible aspect of inactualized potentiality, is not an valid argument against its existence.

The induction of natural modality, and for that matter the more readily recognized temporal modality, follows the same patterns as those involved in the process of induction of extensional modality.

How are universal propositions induced? By a process of generalization, moderated by particularization. We consider it legitimate to move from empirically encountered instances to cases we have not yet come across, until the facts suggest otherwise. We do not regard our universal statements to cover no more than the perceived phenomena; but normally move beyond them into prediction.

Likewise, with constancy of conjunction, in the sense of temporal modality; this too involves an extrapolation from the known to the unknown, as everyone admits.

So ‘all’ and ‘always’ involve just as much assumption as ‘necessarily’ (in the sense of natural modality). They are all just as hard to establish. Why should we recognize the former and not the latter?

Further, the concepts of universality and constancy are ultimately just as mysterious, ontologically hard to define, as that of natural necessity, so the latter’s elusiveness cannot be a legitimate reason for singling it out.

If natural necessity is understood as one level higher (or deeper) than constancy, subject to all the usual laws of logic, generalization and particularization, it is seen to be equally empirical and pragmatic.

While the denial of natural necessity as such is unjustified, with regard to specific applications of the concept, we may of course in a given instance be wrong in our assumption that it is there. It is up to Logic to teach us proper procedures of induction and deduction, concerning such relationships. There is no problem in this viewpoint; belief in natural necessity as such does not obligate us to accept every eventual appearance of it as final.

As with any generalization, the movement from always to must, or from never to cannot, is legitimate, so long as it remains confirmed by experience. If ever a contradictory instance occurs, obviously our assumption is put in doubt and we correct our data-base accordingly, in the way of particularization.

2. Equality of Status.

We saw, in chapter 50, on induction of actuals, that induced particulars are based on the observation of singulars. Similarly, induction of temporaries or potentials is based on the observation of actuals. The same can be said of the bipolar particular fractions, which involve temporary or potential elements: they can be established by observation of the same instance of the subject being actually related to the predicate in different ways at different times or in different circumstances.

And just as not all particular actuals are induced, but some are arrived at by deductive means, so also temporary or potential knowledge is in practise not invariably inductive, but may derive from reasoning processes. Though ultimately, of course, some empirical basis is needed, in any case.

We additionally pointed out how, in the formation of particular propositions, there is also a large share of conceptual work. The same is true of other types of possibility. All statements involve concepts (the terms, the copula, the polarity, the qualifications of quantity or modality). They presuppose a mass of tacit understandings, relating to logical structure and mechanisms. Furthermore, there is always an evaluation process, placing the proposal in the broad context of current knowledge, to determine its fit and realism.

Thus, although pure observation is instrumental in the process, other mental efforts are involved. Abstraction and verbalization of possibility are not automatic consequences of awareness of singular actual events, and error is always a risk. This is equally true in all types of modality, whether extensional, temporal or natural. Thus, actual particulars cannot be claimed more plausible than temporaries or potentials.

And indeed, just as particularity is not superior in status to generality, so are the other types of possibility not intrinsically more credible than their corresponding necessaries. If we consider the controversies among philosophers to be resolved, and view the whole of Logic in perspective, we can say that all forms involve only some degree of observation, and a great deal of thought. Although the degree of empiricism admittedly varies, the amount of conceptualization is essentially identical.

This insight must not be construed to put knowledge in general in doubt, however. Such skepticism would be self-contradictory, being itself the pronouncement of a principle. That there is a process does not imply that its outcome is false. The process merely transports the data from its source to its destination, as it were; the data need not be affected on the way.

Rather, its significance is to put all forms on an equal plane, with regard to their initial logical value. Particulars are no better than universals; particulars are no better than temporaries, which in turn are no better than potentials; and the latter are no better than constants or natural necessaries. Every statement, whatever its form, has at the outset an equal chance of being true or false, and has to be judged as carefully.

3. Stages of Induction.

The classical theory of induction, we saw, describes two processes, generalization and particularization, as fundamental. If all we know is a particular proposition, I or O, we may assume the corresponding general proposition, A or E, true; unless or until we are forced by contradictory evidence to retract, and acknowledge the contingency IO.

Now, this description of the inductive process is adequate, when dealing with the closed system of actual propositions, because of the small number of forms it involves. In a broader context, when modal propositions of one or both types are taken into consideration, the need arises for a more refined description of the process.

This more complex theory brings out into the open, stages in or aspects of the process which were previously concealed. The ideas of generalization and particularization were basically correct, but their application under the more complicated conditions found in modal logic require further clarifications, which make reference to factorial analysis.

Needless to say, the new theory should be, and is, consistent in all its results with the old theory. It should be, and is, capable of embracing actual induction as a special case within a broader perspective which similarly guides, validates, and explains modal induction.

Our modified theory of induction, in the broadest sense, recognizes the following stages:

a. Preparation. The summary of current data in gross formulas, and their factorization. This is in itself a purely deductive process.

b. Generalization. Selection of the strongest factor in a factorial formula.

c. Drawing consequences, empirical testing, and comparing results to wider context. These include deductive work and observation.

d. Particularization. Revision of current formulas in the light of new data. This may necessitate weighting of information. Also, certain conflicts are resolved by factor selection, as in generalization.

e. Repeat previous steps as required.

Each of these processes requires detailed examination. The tasks of listing all conceivable gross formulas, and analyzing them factorially, as well as the tasks relating to deductive inference and comparison, have previously been dealt with. We now need to deal with the processes of factor selection and formula revision, which are the most characteristically inductive.

4. Generalization vs. Particularization.

We call generalization, those thought processes whose conclusions are higher than their premises; and we call particularization, those whose conclusions are lower. This refers to expansions and contractions on the scales of quantity and modality, essentially. As we move beyond the given, or its strictly deductive implications, into prediction, we are involved in induction of one kind or another.

The problem of generalization, which way and how far to advance and on what basis, is solved entirely by the method of factor selection. The problem of particularization, which way and how far to retreat and on what basis, is solved by the methods of formula revision, which may involve factor selection.

It will be seen that factor selection has a static component, which consists of the uniformity principle, which tells us which factor to select, and an active component, the practical carrying out of that decision. The act and basis of factor selection is technically identical, whether applied to generalization or to particularization.

The theory of factor selection makes clear that these processes do not consist of wild guesses, but proceed in a structured manner, requiring skill and precision.

We may view generalization as the positive force in induction, and particularization as the negative side. Generalization would often be too sweeping, if not kept in check by particularization. The function of the latter is to control the excesses of the former. Only the interplay of these two vectors results in proper induction. Induction is valid to the extent that it is a holistic application of both factor selection and formula revision.

In the pursuit of knowledge, laziness leads to error. An idea must be analyzed to the full, because its faults are sometimes concealed far down that course. The uncovering of a fault is a boon, allowing us to alter our idea, or take up a new one, and gain increased understanding and confidence.

The processes of generalization and particularization are going on in tandem all the time, in an active mind. Induction is not linear or pedestrian. Thoughts extend out tentatively, momentarily, like trial balloons, products of the imagination. But at the same time, verification is going on, unraveling the consequences of a suggestion, bringing other facts into focus from memory, or making new empirical inquiries, for comparison to the proposals made, and construction of a consistent idea. The wider the context brought into play, the greater the certainty that our course is realistic.

The role of Logic as a science is to provide the tools, which enable us to play this mental game with maximum efficiency and success. It is an art, but training and experience improve our performance of it.

5. The Paradigm of Induction.

Let us reconsider the paradigm of induction given by actual induction. By reviewing the closed system of actual propositions using factorial concepts, we can gain some insights into the stages and guiding assumptions of induction within any system.

There are only four plural actual forms: A, E, I, O. These are also the system’s fractions: (A), (E), (I), and (O). These in turn constitute three integers: (A), (E), and (I)(O), which are mutually exclusive and exhaustive. The 4 forms allow for 5 gross formulas: A, E, I, O, IO. These can be analyzed factorially using the integers: A = (A), E = (E), I = ‘(A) or (I)(O)’, O = ‘(E) or (I)(O)’, IO = (I)(O). But two disjunctions of factors remain unexpressed, namely: ‘(A) or (E)’, signifying incontingency, and ‘(A) or (E) or (I)(O)’, signifying no concrete information.

In this framework of factorial analysis, we can understand the induction of A from I, or of E from O, as a process involving factor selection, rather than solely as one of increase in quantity from some to all. The reverse process, of decrease in quantity, would also here be regarded differently, as primarily focusing on a new factorial situation.

Given I alone, we prefer the alternative outcome (A) to the deductively equally conceivable alternative (I)(O). Or, given O alone, we inductively anticipate the factor (E) as more likely than its alternative (I)(O). Our selection of one factor out of the available two is the dynamic aspect of the process. That we have specifically preferred the general alternative to the contingent one, is a second aspect; here, we take note of a principle that statically determines which of the alternative factors is selected.

If thereafter we find that our position must shift to IO, so well and good; in that case, only one integer is conceivable: (I)(O). In this case, we believed A to be true, then discovered O, or we assumed E then found I: the only available resolution of this conflict is by the compromise compound proposition IO; this is formula revision per se. Now, we analyze IO and find that it has only one factor (I)(O), so we can select it without doubt. However, had there been more than one conflict resolution or more than one factor (as occurs in wider systems), we would have had to again engage in factor selection.

Such an outlook seems somewhat forced and redundant within the closed system of actuals, but in the wider systems of modal propositions it becomes essential. It is only applied to actuals here for initial illustration purposes. For whereas with actuals, our choices are very limited numerically, when modality is introduced they are much more complicated, as will be seen.

In the wider systems, induction can usually take many paths, and has various possible limits. For instance, from Ip should we generalize in the direction of Ap or to In? Or again, from An should we particularize to Ap or to In? And how far up or down the scale may we go? Obviously, this depends on context, so when may Ip ascend to An, and when must it stop earlier, and when must An descend to Ip, and when may it stop earlier?

Such questions can only be answered scientifically and systematically by resorting to factorial analysis and related processes. This brief review of actual induction in such terms points the way to the solution of the problem.

6. The Pursuit of Integers.

The factor selection theory suggests that the goal of induction is to diminish the areas of doubt involved in deficient states of knowledge. Selecting a factor means eliminating a number of other factors, which, though they are formally logically conceivable alternatives, are intuitively thought to be less likely.

The ultimate result pursued by all induction is knowledge of integers, which does not necessarily mean a generality. Without integers, too many questions arise, and the mind cannot proceed. It is better to take up a working hypothesis, and keep testing it, than to passively await for an in any case unattainable absolute certainty. Knowledge is fed by action; it involves choices, decision-making.

The whole point of induction is to decide what integral proposition is most suggested by a given statement of deficient knowledge. We are to scrutinize its factorial equivalent and, on the basis of precise principle, select one factor as our inductive conclusion, or at least reduce the number of factors considerably. Deductively, all factors are equally likely outcomes, but inductively they can be narrowed down.

In certain cases, as factorial analysis showed, there is only one factor anyway; in such case, the conclusion is deductive, not inductive, and contextually certain. But in most cases, there are more than one factor, and selection is necessary. In some cases, we may for some purpose be satisfied with eliminating only some of the excess factors, and be left with a formula of two or more factors; the conclusion is not a single integer, but, still, less vague than previously, and might be expressed as a gross formula.