CONDITIONS OF IMMEDIATE INFERENCE
Sec. 1. The word Inference is used in two different senses, which are often
confused but should be carefully distinguished. In the first sense, it
means a process of thought or reasoning by which the mind passes from
facts or statements presented, to some opinion or expectation. The data
may be very vague and slight, prompting no more than a guess or surmise;
as when we look up at the sky and form some expectation about the
weather, or from the trick of a man's face entertain some prejudice as
to his character. Or the data may be important and strongly significant,
like the footprint that frightened Crusoe into thinking of cannibals, or
as when news of war makes the city expect that Consols will fall. These
are examples of the act of inferring, or of inference as a process; and
with inference in this sense Logic has nothing to do; it belongs to
Psychology to explain how it is that our minds pass from one perception
or thought to another thought, and how we come to conjecture, conclude
and believe (_cf._ chap. i. Sec. 6).
In the second sense, 'inference' means not this process of guessing or
opining, but the result of it; the surmise, opinion, or belief when
formed; in a word, the conclusion: and it is in this sense that
Inference is treated of in Logic. The subject-matter of Logic is an
inference, judgment or conclusion concerning facts, embodied in a
proposition, which is to be examined in relation to the evidence that
may be adduced for it, in order to determine whether, or how far, the
evidence amounts to proof. Logic is the science of Reasoning in the
sense in which 'reasoning' means giving reasons, for it shows what sort
of reasons are good. Whilst Psychology explains how the mind goes
forward from data to conclusions, Logic takes a conclusion and goes back
to the data, inquiring whether those data, together with any other
evidence (facts or principles) that can be collected, are of a nature to
warrant the conclusion. If we think that the night will be stormy, that
John Doe is of an amiable disposition, that water expands in freezing,
or that one means to national prosperity is popular education, and wish
to know whether we have evidence sufficient to justify us in holding
these opinions, Logic can tell us what form the evidence should assume
in order to be conclusive. What _form_ the evidence should assume: Logic
cannot tell us what kinds of fact are proper evidence in any of these
cases; that is a question for the man of special experience in life, or
in science, or in business. But whatever facts constitute the evidence,
they must, in order to prove the point, admit of being stated in
conformity with certain principles or conditions; and of these
principles or conditions Logic is the science. It deals, then, not with
the subjective process of inferring, but with the objective grounds that
justify or discredit the inference.
Sec. 2. Inferences, in the Logical sense, are divided into two great
classes, the Immediate and the Mediate, according to the character of
the evidence offered in proof of them. Strictly, to speak of inferences,
in the sense of conclusions, as immediate or mediate, is an abuse of
language, derived from times before the distinction between inference as
process and inference as result was generally felt. No doubt we ought
rather to speak of Immediate and Mediate Evidence; but it is of little
use to attempt to alter the traditional expressions of the science.
An Immediate Inference, then, is one that depends for its proof upon
only one other proposition, which has the same, or more extensive,
terms (or matter). Thus that _one means to national prosperity is
popular education_ is an immediate inference, if the evidence for it is
no more than the admission that _popular education is a means to
national prosperity:_ Similarly, it is an immediate inference that _Some
authors are vain_, if it be granted that _All authors are vain_.
An Immediate Inference may seem to be little else than a verbal
transformation; some Logicians dispute its claims to be called an
inference at all, on the ground that it is identical with the pretended
evidence. If we attend to the meaning, say they, an immediate inference
does not really express any new judgment; the fact expressed by it is
either the same as its evidence, or is even less significant. If from
_No men are gods_ we prove that _No gods are men_, this is nugatory; if
we prove from it that _Some men are not gods_, this is to emasculate the
sense, to waste valuable information, to lose the commanding sweep of
our universal proposition.
Still, in Logic, it is often found that an immediate inference expresses
our knowledge in a more convenient form than that of the evidentiary
proposition, as will appear in the chapter on Syllogisms and elsewhere.
And by transforming an universal into a particular proposition, as _No
men are gods_, therefore, _Some men are not gods_,--we get a statement
which, though weaker, is far more easily proved; since a single instance
suffices. Moreover, by drawing all possible immediate inferences from a
given proposition, we see it in all its aspects, and learn all that is
implied in it.
A Mediate Inference, on the other hand, depends for its evidence upon a
plurality of other propositions (two or more) which are connected
together on logical principles. If we argue--
No men are gods;
Alexander the Great is a man;
.'. Alexander the Great is not a god:
this is a Mediate Inference. The evidence consists of two propositions
connected by the term 'man,' which is common to both (a Middle Term),
mediating between 'gods' and 'Alexander.' Mediate Inferences comprise
Syllogisms with their developments, and Inductions; and to discuss them
further at present would be to anticipate future chapters. We must now
deal with the principles or conditions on which Immediate Inferences are
valid: commonly called the "Laws of Thought."
Sec. 3. The Laws of Thought are conditions of the logical statement and
criticism of all sorts of evidence; but as to Immediate Inference, they
may be regarded as the only conditions it need satisfy. They are often
expressed thus: (1) The principle of Identity--'_Whatever is, is_'; (2)
The principle of Contradiction--'_It is impossible for the same thing to
be and not be_'; (3) The principle of Excluded Middle--'_Anything must
either be or not be_.' These principles are manifestly not 'laws' of
thought in the sense in which 'law' is used in Psychology; they do not
profess to describe the actual mental processes that take place in
judgment or reasoning, as the 'laws of association of ideas' account for
memory and recollection. They are not natural laws of thought; but, in
relation to thought, can only be regarded as laws when stated as
precepts, the observance of which (consciously or not) is necessary to
clear and consistent thinking: e.g., Never assume that the same thing
can both be and not be.
However, treating Logic as the science of thought only as embodied in
propositions, in respect of which evidence is to be adduced, or which
are to be used as evidence of other propositions, the above laws or
principles must be restated as the conditions of consistent argument in
such terms as to be directly applicable to propositions. It was shown in
the chapter on the connotation of terms, that terms are assumed by
Logicians to be capable of definite meaning, and of being used
univocally in the same context; if, or in so far as, this is not the
case, we cannot understand one another's reasons nor even pursue in
solitary meditation any coherent train of argument. We saw, too, that
the meanings of terms were related to one another: some being full
correlatives; others partially inclusive one of another, as species of
genus; others mutually incompatible, as contraries; or alternatively
predicable, as contradictories. We now assume that propositions are
capable of definite meaning according to the meaning of their component
terms and of the relation between them; that the meaning, the fact
asserted or denied, is what we are really concerned to prove or
disprove; that a mere change in the words that constitute our terms, or
of construction, does not affect the truth of a proposition as long as
the meaning is not altered, or (rather) as long as no fresh meaning is
introduced; and that if the meaning of any proposition is true, any
other proposition that denies it is false. This postulate is plainly
necessary to consistency of statement and discourse; and consistency is
necessary, if our thought or speech is to correspond with the unity and
coherence of Nature and experience; and the Laws of Thought or
Conditions of Immediate Inference are an analysis of this postulate.
Sec. 4. The principle of Identity is usually written symbolically thus: _A
is A; not-A is not-A_. It assumes that there is something that may be
represented by a term; and it requires that, in any discussion, _every
relevant term, once used in a definite sense, shall keep that meaning
throughout_. Socrates in his father's workshop, at the battle of Delium,
and in prison, is assumed to be the same man denotable by the same name;
and similarly, 'elephant,' or 'justice,' or 'fairy,' in the same
context, is to be understood of the same thing under the same
_suppositio_.
But, further, it is assumed that of a given term another term may be
predicated again and again in the same sense under the same conditions;
that is, we may speak of the identity of meaning in a proposition as
well as in a term. To symbolise this we ought to alter the usual
formula for Identity and write it thus: _If B is A, B is A; if B is
not-A, B is not-A_. If Socrates is wise, he is wise; if fairies frequent
the moonlight, they do; if Justice is not of this world, it is not.
_Whatever affirmation or denial we make concerning any subject, we are
bound to adhere to it for the purposes of the current argument or
investigation._ Of course, if our assertion turns out to be false, we
must not adhere to it; but then we must repudiate all that we formerly
deduced from it.
Again, _whatever is true or false in one form of words is true or false
in any other_: this is undeniable, for the important thing is identity
of meaning; but in Formal Logic it is not very convenient. If Socrates
is wise, is it an identity to say 'Therefore the master of Plato is
wise'; or, further that he 'takes enlightened views of life'? If _Every
man is fallible_, is it an identical proposition that _Every man is
liable to error_? It seems pedantic to demand a separate proposition
that _Fallible is liable to error_. But, on the other hand, the
insidious substitution of one term for another speciously identical, is
a chief occasion of fallacy. How if we go on to argue: therefore, _Every
man is apt to blunder, prone to confusion of thought, inured to
self-contradiction_? Practically, the substitution of identities must be
left to candour and good-sense; and may they increase among us. Formal
Logic is, no doubt, safest with symbols; should, perhaps, content itself
with A and B; or, at least, hardly venture beyond Y and Z.
Sec. 5. The principle of Contradiction is usually written symbolically,
thus: _A is not not-A_. But, since this formula seems to be adapted to a
single term, whereas we want one that is applicable to propositions, it
may be better to write it thus: _B is not both A and not-A_. That is to
say: _if any term may be affirmed of a subject, the contradictory term
may, in the same relation, be denied of it_. A leaf that is green on one
side of it may be not-green on the other; but it is not both green and
not-green on the same surface, at the same time, and in the same light.
If a stick is straight, it is false that it is at the same time
not-straight: having granted that two angles are equal, we must deny
that they are unequal.
But is it necessarily false that the stick is 'crooked'; must we deny
that either angle is 'greater or less' than the other? How far is it
permissible to substitute any other term for the formal contradictory?
Clearly, the principle of Contradiction takes for granted the principle
of Identity, and is subject to the same difficulties in its practical
application. As a matter of fact and common sense, if we affirm any term
of a Subject, we are bound to deny of that Subject, in the same
relation, not only the contradictory but all synonyms for this, and also
all contraries and opposites; which, of course, are included in the
contradictory. But who shall determine what these are? Without an
authoritative Logical Dictionary to refer to, where all contradictories,
synonyms, and contraries may be found on record, Formal Logic will
hardly sanction the free play of common sense.
The principle of Excluded Middle may be written: _B is either A or
not-A_; that is, _if any term be denied of a subject, the contradictory
term may, in the same relation, be affirmed_. Of course, we may deny
that a leaf is green on one side without being bound to affirm that it
is not-green on the other. But in the same relation a leaf is either
green or not-green; at the same time, a stick is either bent or
not-bent. If we deny that A is greater than B, we must affirm that it is
not-greater than B.
Whilst, then, the principle of Contradiction (that 'of contradictory
predicates, one being affirmed, the other is denied ') might seem to
leave open a third or middle course, the denying of both
contradictories, the principle of Excluded Middle derives its name from
the excluding of this middle course, by declaring that the one or the
other must be affirmed. Hence the principle of Excluded Middle does not
hold good of mere contrary terms. If we deny that a leaf is green, we
are not bound to affirm it to be yellow; for it may be red; and then we
may deny both contraries, yellow and green. In fact, two contraries do
not between them cover the whole predicable area, but contradictories
do: the form of their expression is such that (within the _suppositio_)
each includes all that the other excludes; so that the subject (if
brought within the _suppositio_) must fall under the one or the other.
It may seem absurd to say that Mont Blanc is either wise or not-wise;
but how comes any mind so ill-organised as to introduce Mont Blanc into
this strange company? Being there, however, the principle is inexorable:
Mont Blanc is not-wise.
In fact, the principles of Contradiction and Excluded Middle are
inseparable; they are implicit in all distinct experience, and may be
regarded as indicating the two aspects of Negation. The principle of
Contradiction says: _B is not both A and not-A_, as if _not-A_ might be
nothing at all; this is abstract negation. But the principle of Excluded
Middle says: _Granting that B is not A, it is still something_--namely,
_not-A_; thus bringing us back to the concrete experience of a continuum
in which the absence of one thing implies the presence of something
else. Symbolically: to deny that B is A is to affirm that B is not A,
and this only differs by a hyphen from B is not-A.
These principles, which were necessarily to some extent anticipated in
chap. iv. Sec. 7, the next chapter will further illustrate.
Sec. 6. But first we must draw attention to a maxim (also already
mentioned), which is strictly applicable to Immediate Inferences, though
(as we shall see) in other kinds of proof it may be only a formal
condition: this is the general caution _not to go beyond the evidence_.
An immediate inference ought to contain nothing that is not contained
(or formally implied) in the proposition by which it is proved. With
respect to quantity in denotation, this caution is embodied in the rule
'not to distribute any term that is not given distributed.' Thus, if
there is a predication concerning 'Some S,' or 'Some men,' as in the
forms I. and O., we cannot infer anything concerning 'All S.' or 'All
men'; and, as we have seen, if a term is given us preindesignate, we are
generally to take it as of particular quantity. Similarly, in the case
of affirmative propositions, we saw that this rule requires us to assume
that their predicates are undistributed.
As to the grounds of this maxim, not to go beyond the evidence, not to
distribute a term that is given as undistributed, it is one of the
things so plain that to try to justify is only to obscure them. Still,
we must here state explicitly what Formal Logic assumes to be contained
or implied in the evidence afforded by any proposition, such as 'All S
is P.' If we remember that in chap. iv. Sec. 7, it was assumed that every
term may have a contradictory; and if we bear in mind the principles of
Contradiction and Excluded Middle, it will appear that such a
proposition as 'All S is P' tells us something not only about the
relations of 'S' and 'P,' but also of their relations to 'not-S' and
'not-P'; as, for example, that 'S is not not-P,' and that 'not-P is
not-S.' It will be shown in the next chapter how Logicians have
developed these implications in series of Immediate Inferences.
If it be asked whether it is true that every term, itself significant,
has a significant contradictory, and not merely a formal contradictory,
generated by force of the word 'not,' it is difficult to give any better
answer than was indicated in Sec.Sec. 3-5, without venturing further into
Metaphysics. I shall merely say, therefore, that, granting that some
such term as 'Universe' or 'Being' may have no significant
contradictory, if it stand for 'whatever can be perceived or thought
of'; yet every term that stands for less than 'Universe' or 'Being' has,
of course, a contradictory which denotes the rest of the universe. And
since every argument or train of thought is carried on within a special
'universe of discourse,' or under a certain _suppositio_, we may say
that _within the given suppositio every term has a contradictory_, and
that every predication concerning a term implies some predication
concerning its contradictory. But the name of the _suppositio_ itself
has no contradictory, except with reference to a wider and inclusive
_suppositio_.
The difficulty of actual reasoning, not with symbols, but about matters
of fact, does not arise from the principles of Logic, but sometimes from
the obscurity or complexity of the facts, sometimes from the ambiguity
or clumsiness of language, sometimes from the deficiency of our own
minds in penetration, tenacity and lucidity. One must do one's best to
study the facts, and not be too easily discouraged.