CHAPTER 50. ACTUAL INDUCTION.
Induction is the branch of Logic concerned with determining how general propositions — and, more broadly, how necessary propositions — are established as true, from particular or potential data.
By ‘actual induction’, I mean induction of actual propositions; by ‘modal induction’, I mean induction of modal propositions (referring to de-re modality).
We saw, in the analysis of Deductive processes, that although we can infer a general or particular proposition from other general propositions, through opposition, eduction or syllogism, it seems impossible to deductively infer general truths from particular ones only.
Indeed, it is even, according to the rules of syllogism, just about impossible to deduce a particular proposition from particular premises only: there has to be a general premise; the only exceptions to this rule are found in eduction, and in a limited number of third figure syllogisms, which allow us to obtain particular conclusions without use of a general premise: but these are too special to be claimed as important sources.
If, then, virtually all deduction presupposes the prior possession of general premises, where do these first general premises originate, or more precisely, how are they themselves shown to be true? Obviously, if such first premises, whatever their content, are open to doubt and of little credibility, then all subsequent deduction from them, however formally trustworthy, may be looked upon with healthy skepticism. As computer programmers say, “Garbage in, garbage out”. Conclusions drawn from spurious premises could nonetheless be true, but it would be mere chance, not proof.
Furthermore, these ‘first general premises’ we mentioned are not few in number. We are not talking here of a few First Principles, like the axioms of logic, from which exclusively all knowledge is to be derived. We require an extremely large number of first general premises, with all sorts of contents, to be able to develop a faithful image of our actual knowledge base. While mathematical sciences, like arithmetic, algebra or geometry, can seemingly be reduced to a very limited number of axioms, this is a feat not easy to duplicate in sciences like physics or psychology, or in everyday thinking.
If, now, we introspect, and observe our actual thinking processes as individuals, and analyze the actual historical development of Science, the accumulation of knowledge by humankind as a whole, we see clearly that, although deduction plays a large and important role, it is not our only source of knowledge. Even axioms in mathematics have been identified over time, and been subject to improvement or change. In practise, however faultless our deductions, our knowledge is clearly an evolving, flexible, thing. Ideas previously ignored, eventually make their appearance in our body of knowledge; thoughts once considered certain, turn out to be incorrect, and are modified or abandoned.
The primary source of knowledge is not deduction, but observation. This term is to be understood here is its broadest, and most neutral, sense, including both passive experiences and those experimentally generated.
Observation is to be understood as in itself a neutral event. It is consciousness, awareness, of appearances, phenomena, such as they present themselves, without judgement as to their ultimate meaning or value in the full scheme of things. Observation concerns the given, in its most brutal, unordered, unprocessed form.
Any interpretation that we attach to an observation, is to be regarded as a separate phenomenon; the distinction between these two is not always easy to make, nevertheless. Interpretation, in contrast to observation, attempts to relate phenomena, to place them in a supposed order of things, to evaluate their credibility and real significance in the widest possible context. It is a relatively complex mental process, and more subject to error. Its purpose is to tell us whether, all things considered, an experience was illusory or real.
In this treatise, I will evolve an original theory of induction, in considerable detail, with reference to categorical propositions: first for actuals, then more broadly for modals. I will not here deal with natural, temporal, or extensional conditionals, at all, but it will become obvious that the same methods and principles can be extended to those forms as well, though the formulas involved are bound to be enormously more complex; I leave the task to future logicians with my compliments!
The first step in induction is formulation of particular propositions on the basis of observation. This is a more complicated process than we might at first sight suppose. It does not merely consist in observation of a perceptible phenomenon, but includes the conceptual factor of abstraction of ‘universals’, the similarities on which we base our verbalization of terms, copula, and particular quantity. Pure observation forms no judgement; it is meditation on, simple consciousness of, the object at hand. The moment a thought is expressed, even a particular proposition, we have interpretation, conceptual correlation. The question of truth or falsehood is yet a separate judgement.
It follows, in passing, that a particular proposition based on observation of concrete phenomena, cannot be viewed as extremely superior in value to one based on observation of abstract phenomena. Both involve abstraction of sorts and verbalization. Their difference is only in the qualitative character of object involved, in the relative accessibility of the evidence.
Now, all observation concerns primarily individual instances. We have seen that singular propositions point to a single specific individual under consideration (referred to by ‘this’), whereas particular propositions are quantitatively indefinite and need not specify the individuals they concern (we just say ‘some’). A plural but specific proposition, involving the quantity ‘these’, is essentially singular in nature, or a conjunction of singulars; it differs from a genuine particular, which is more broadly intended. We have seen, too, that singulars imply particulars, by formal opposition.
Normally, unless the subject is a namable individual person or animal, a uniquely complex entity we deal with on a regular basis, our singular propositions are only temporary furniture in our knowledge base. I may say to you “look, this rose, unlike the others in my garden, is blue’ or “this particle swerved to the left in our experiment”, but ultimately, the individual is ignored or forgotten, and only an indefinite particular proposition is retained in the record. Furthermore, although a particular can be inferred from one singular, it is more often based on a plurality of observations.
In any case, induction of a particular proposition is free of generalization. It is observed that some S are P, so we say ‘Some S are P’. If some S are scrutinized and observed not to be P, we say ‘Some S are not P’. If no observation has been carried out, our faculties being shut off to the question, or the objects concerned being inaccessible to direct observation, or indirect observation (experiment through instruments), no inductive conclusion is drawn. We may still infer this or that particular deductively, of course.
The induction of general propositions, however, occurs by generalization. This obviously does not concern special cases where full enumeration is possible, as in ‘all these S are P’, or in cases where the subject class is very prescribed so that ‘all S’ is an accessible number of instances; here, the general proposition can be viewed as effectively singular in nature. Normally, a general proposition is open-ended, and the number of instances involved extremely large (e.g. all the insects in the world), and inaccessible to observation (for example, having existed in the past, or yet to be born). Here, we tend to extrapolate from known instances, to the unknown. We predict many other phenomena, from a limited number of observed phenomena.
The basic principle of generalization is to assume observed, particular uniformities to be applicable generally, until and unless we have reason to think otherwise. A particular proposition arrived at by deductive means can also of course be used as a basis for generalization. The reliability of a generalization is variable, depending on certain factors.
Observation is itself not always a simple process of perception. It may involve research or experiment with certain prior assumptions, methodological or factual, which may require review and testing. The validity of the final generalization depends on the reliability of such prior factors. As well, if a research or experiment process is easily duplicated by other people, socially accessible, it is granted more credence, than a one-time, esoteric observation. Even so, ad hominem arguments count in this domain; a person of known honesty and intelligence may be allowed considerable leeway, in comparison to a habitual liar or scatterbrain.
The degree of effort and ingenuity involved in making the observations in question, also affects the reliability of the generalization. If we observe a limited number of instances and then generalize, and thereafter make no effort to, periodically or in new situations, check our result, it is less reliable obviously than if we remain open-minded, vigilant, and actively research possible deviations from our initial assumption.
The generalization should be reviewed whenever the surrounding context of knowledge has been modified in any way which might conceivably affect it. Comparison of the assumed generality to new information as it comes up, serves not only to verify it but to further confirm it if it stands the test. Here, deductive logic plays its crucial role, guiding us in verifying consistency, by opposition or uncovering implications, helping us to interconnect all our knowledge.
The more alike in nature, the simpler, the phenomena in question are known to be, the more credible and trustworthy our generalization. A generalization concerning, say, gold nuggets, is more reliable than one concerning living cells, because the instances of the former differ in little more than time and space, whereas instances of the latter, though exhibiting some considerable uniformities, are more often found to have individual differences.
The following might be presented as the valid moods of generalization from particular propositions, whether obtained by induction or deduction, to illustrate its basic method.
I -> A
Knowing that some S are P,
and not having found any S which are not P,
we may induce that ‘All S are P’.
O -> E
Knowing that some S are not P,
and not having found any S which are P,
we may induce that ‘No S is P’.
I + O, knowing some S to be P and some not to be P, inhibits generalization.
Lastly, not having found any S which are P or any S which are not P, strictly leaves us with nothing to say.
However, in practise, if research was made, we might tentatively induce that ‘No S are P’ or ‘All S are P’, preferring the E conclusion if P is in content a positive quality, or the A conclusion if P is in content a negative quality. A distinction is here made between presence and absence of something, which cannot be expressed in formal terms, but is comprehensible. Such generalization concerns, not so much the subject-matter of our propositions, but the process of observation itself.
The reverse process of particularization, is also noteworthy. We start with a general proposition, obtained by generalization or deduction, and a new observation which contradicts it; granting that the latter and its sources more credible than the former, we scale it down for consistency. Thus:
A + O -> IO
Having supposed that all S are P,
but finding some S not to be P,
we conclude that ‘only some S are P’.
E + I -> IO
Having supposed that no S are P,
but finding some S to be P,
we conclude that ‘only some S are not P’.
In practise, faced with such a situation, we might try to mitigate the result, by reformulating the original general thesis, so that we retain a generality. In the above, this would mean altering the subject, by delineating exceptions to it or substituting a narrower subcategory of it, and/or altering the predicate, by widening it (in positive cases) or narrowing it (in negative cases). Thus, suppose S1 and S2 are subspecies of S, and suppose P’ is a genus embracing P among others, and that P1 and P2 are subspecies of P, then:
in A + O -> IO, we may review the initial All S are P, to:
· All S1 are P (and No S2 is P), or to:
· All S are P’ (though only some S are P).
Here, we narrow the subject or widen the predicate.
in E + I -> IO: we may review the initial No S is P, to:
· No S1 is P (and All S2 are P), or to:
· No S is P1 (though some S are P2).
Here, we narrow the subject or narrow the predicate.
A pitfall in generalization is selection of too broad a subject-concept, or too wide or narrow a predicate-concept, when formulating the initial observation.
When particular entities are observed as having a certain property, the question arises are they so qua being of some species classification (like crocus, say), or qua belonging to some genus (like flowers, say). If we are tempted at the outset to adopt the genus as our subject, we may soon be disappointed, and have to later retract, and particularize the property down to the species, as above. Alternatively, we may be cautious, and adopt the species as subject, and later, finding the wider statement true, would generalize as follows:
All S1 and all S2 are P,
S1 and S2 are all the species of S,
therefore, All S are P.
Here, we broaden the subject.
Likewise, we may initially select a too limited predicate (e.g. blue) or a too vague one (e.g. colored), and later be obliged to qualify our assumption, as shown above.
Either way, in the long run, the correct subject and predicate should impose themselves, assuming the pursuit of knowledge is continued. So the process is not in itself flawed, but induction proceeds by gradual evolution.
It should be obvious that the above ‘inductive arguments’, and those presented further on, involve a premises-conclusion relationship, of a logical modality other than that found in ‘deductive argument’. Here, we are concerned with inductive implication, which boasts a connection only of logical probability; it is less binding than the logical necessity which characterizes deductive implication.
The validity of man’s inductions, his observations and generalizations, as such, cannot be consistently denied. One can deny this or that specific case to be justified, by adducing evidence to the contrary, but the processes themselves cannot be in principle doubted. For the simple reason that, in so doing, the skeptic is himself formulating a general statement, and so bringing about its own demise. A self-contradictory statement simply has no logical standing. It is automatically and irretrievably false. There are no loop-holes in this reasoning.
The fact that knowledge is contextual, does not imply that it is entirely problematic. The appearances involved in observation and generalization must be taken at their face value, and recognized as indubitably valid, until and unless some specific cause for doubt is brought to the fore, which itself stands the tests of inductive and deductive logic. If that doubt turns out to be indeed justified, the initial observation or generalization is admitted, ex-post-facto, to have been mistaken, and modified or abandoned to restore consistency.
Our ignorance of a great variety of epistemological and ontological descriptive facts, such as the nature of consciousness, the workings of our sensory perception or conceptualization, the nature of universals, and all related issues, in no way constitutes a credible reason for doubt. We are well protected by the axioms of logic. We may be humbly aware of our limitations, know with certainty that some of the beliefs we even now may cherish most are bound to turn out to be spurious as the adventure of knowledge progresses, but we may rest assured that not all will be overturned. It is logically impossible, inconceivable to suppose otherwise.
Man does not need to be omniscient to know. Our faculties are effective instruments of knowledge. Knowledge is a continuously evolving, flexible entity. Like a living organism, it changes and shifts, but somehow endures. We have not been endowed with a finished product, but we have been blessed with the means to gradually progress towards that distant goal. Knowledge is essentially functional, a biological tool of survival; as the need for information presents itself, so normally does the opportunity for its procurement. Knowledge is also a spiritual value, one to be attained by effort.
The important thing is to tailor one’s judgements to fit the facts. So long as one’s assumptions and beliefs are up to date, and continuously updated by new data as it appears, they remain reliable and useful.
To trust in one’s judgements does not abrogate one’s right to investigate alternatives and implications; indeed, it is responsible behavior. Certainty and open-mindedness, certainty and verification, are quite compatible. However, there is also a limit to how much one may toy with new ideas, without good reason and rigorous thought.
More broadly, we can say that cognition, like volition, has an ethic, including virtues and vices. Among the virtues are: reasonableness, honesty, making an effort, facing facts, courage, willingness to debate an idea one considers outrageous. Among the vices are: irrationalism, dishonesty, lethargy, evasion, fear of opposition or change, autism. This topic borders on psychology, and could be the subject of a whole treatise by itself.