www.TheLogician.net © Avi Sion - all rights reserved
© Avi Sion
All rights reserved
Avi Sion, 1990 (Rev. ed. 1996) All rights reserved.
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
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,
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
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
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
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
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.
Knowing that some S are P,
and not having found any S which are not P,
we may induce that 'All S are P'.
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 ->
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 ->
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
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
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.