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The Logician © Avi Sion All rights reserved |
FUTURE LOGIC©
Avi Sion, 1990 (Rev. ed. 1996) All rights reserved.
CHAPTER 54.
MODAL INDUCTION.
4.
Generalization vs.
Particularization.
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.
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.
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.
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.
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. |
