FORMAL CONDITIONS OF MEDIATE INFERENCE
Sec. 1. A Mediate Inference is a proposition that depends for proof upon
two or more other propositions, so connected together by one or more
terms (which the evidentiary propositions, or each pair of them, have in
common) as to justify a certain conclusion, namely, the proposition in
question. The type or (more properly) the unit of all such modes of
proof, when of a strictly logical kind, is the Syllogism, to which we
shall see that all other modes are reducible. It may be exhibited
symbolically thus:
M is P;
S is M:
.'. S is P.
Syllogisms may be classified, as to quantity, into Universal or
Particular, according to the quantity of the conclusion; as to quality,
into Affirmative or Negative, according to the quality of the
conclusion; and, as to relation, into Categorical, Hypothetical and
Disjunctive, according as all their propositions are categorical, or one
(at least) of their evidentiary propositions is a hypothetical or a
disjunctive.
To begin with Categorical Syllogisms, of which the following is an
example:
All authors are vain;
Cicero is an author:
.'. Cicero is vain.
Here we may suppose that there are no direct means of knowing that
Cicero is vain; but we happen to know that all authors are vain and
that he is an author; and these two propositions, put together,
unmistakably imply that he is vain. In other words, we do not at first
know any relation between 'Cicero' and 'vanity'; but we know that these
two terms are severally related to a third term, 'author,' hence called
a Middle Term; and thus we perceive, by mediate evidence, that they are
related to one another. This sort of proof bears an obvious resemblance
(though the relations involved are not the same) to the mathematical
proof of equality between two quantities, that cannot be directly
compared, by showing the equality of each of them to some third
quantity: A = B = C .'. A = C. Here B is a middle term.
We have to inquire, then, what conditions must be satisfied in order
that a Syllogism may be formally conclusive or valid. A specious
Syllogism that is not really valid is called a Parasyllogism.
Sec. 2. General Canons of the Syllogism.
(1) A Syllogism contains three, and no more, distinct propositions.
(2) A Syllogism contains three, and no more, distinct univocal terms.
These two Canons imply one another. Three propositions with less than
three terms can only be connected in some of the modes of Immediate
Inference. Three propositions with more than three terms do not show
that connection of two terms by means of a third, which is requisite for
proving a Mediate Inference. If we write--
All authors are vain;
Cicero is a statesman--
there are four terms and no middle term, and therefore there is no
proof. Or if we write--
All authors are vain;
Cicero is an author:
.'. Cicero is a statesman--
here the term 'statesman' occurs without any voucher; it appears in the
inference but not in the evidence, and therefore violates the maxim of
all formal proof, 'not to go beyond the evidence.' It is true that if
any one argued--
All authors are vain;
Cicero wrote on philosophy:
.'. Cicero is vain--
this could not be called a bad argument or a material fallacy; but it
would be a needless departure from the form of expression in which the
connection between the evidence and the inference is most easily seen.
Still, a mere adherence to the same form of words in the expression of
terms is not enough: we must also attend to their meaning. For if the
same word be used ambiguously (as 'author' now for 'father' and anon for
'man of letters'), it becomes as to its meaning two terms; so that we
have four in all. Then, if the ambiguous term be the Middle, no
connection is shown between the other two; if either of the others be
ambiguous, something seems to be inferred which has never been really
given in evidence.
The above two Canons are, indeed, involved in the definition of a
categorical syllogism, which may be thus stated: A Categorical Syllogism
is a form of proof or reasoning (way of giving reasons) in which one
categorical proposition is established by comparing two others that
contain together only three terms, or that have one and only one term in
common.
The proposition established, derived, or inferred, is called the
Conclusion: the evidentiary propositions by which it is proved are
called the Premises.
The term common to the premises, by means of which the other terms are
compared, is called the Middle Term; the subject of the conclusion is
called the Minor Term; the predicate of the conclusion, the Major Term.
The premise in which the minor term occurs is called the Minor Premise;
that in which the major term occurs is called the Major Premise. And a
Syllogism is usually written thus:
Major Premise--All authors (Middle) are vain (Major);
Minor Premise--Cicero (Minor) is an author (Middle):
Conclusion--.'. Cicero (Minor) is vain (Major).
Here we have three propositions with three terms, each term occurring
twice. The minor and major terms are so called, because, when the
conclusion is an universal affirmative (which only occurs in Barbara;
see chap. x. Sec. 6), its subject and predicate are respectively the less
and the greater in extent or denotation; and the premises are called
after the peculiar terms they contain: the expressions 'major premise'
and 'minor premise' have nothing to do with the order in which the
premises are presented; though it is usual to place the major premise
first.
(3) No term must be distributed in the conclusion unless it is
distributed in the premises.
It is usual to give this as one of the General Canons of the Syllogism;
but we have seen (chap. vi. Sec. 6) that it is of wider application.
Indeed, 'not to go beyond the evidence' belongs to the definition of
formal proof. A breech of this rule in a syllogism is the fallacy of
Illicit Process of the Minor, or of the Major, according to which term
has been unwarrantably distributed. The following parasyllogism
illicitly distributes both terms of the conclusion:
All poets are pathetic;
Some orators are not poets:
.'. No orators are pathetic.
(4) The Middle Term must be distributed at least once in the premises
(in order to prove a conclusion in the given terms).
For the use of mediate evidence is to show the relation of terms that
cannot be directly compared; this is only possible if the middle term
furnishes the ground of comparison; and this (in Logic) requires that
the whole denotation of the middle should be either included or excluded
by one of the other terms; since if we only know that the other terms
are related to _some_ of the middle, their respective relations may not
be with the same part of it.
It is true that in what has been called the "numerically definite
syllogism," an inference may be drawn, though our canon seems to be
violated. Thus:
60 sheep in 100 are horned;
60 sheep in 100 are blackfaced:
.'. at least 20 blackfaced sheep in 100 are horned.
But such an argument, though it may be correct Arithmetic, is not Logic
at all; and when such numerical evidence is obtainable the comparatively
indefinite arguments of Logic are needless. Another apparent exception
is the following:
Most men are 5 feet high;
Most men are semi-rational:
.'. Some semi-rational things are 5 feet high.
Here the Middle Term (men) is distributed in neither premise, yet the
indisputable conclusion is a logical proposition. The premises, however,
are really arithmetical; for 'most' means 'more than half,' or more than
50 per cent.
Still, another apparent exception is entirely logical. Suppose we are
given, the premises--_All P is M_, and _All S is M_--the middle term is
undistributed. But take the obverse of the contrapositive of both
premises:
All m is p;
All m is s:
.'. Some s is p.
Here we have a conclusion legitimately obtained; but it is not in the
terms originally given.
For Mediate Inference depending on truly logical premises, then, it is
necessary that one premise should distribute the middle term; and the
reason of this may be illustrated even by the above supposed numerical
exceptions. For in them the premises are such that, though neither of
the two premises by itself distributes the Middle, yet they always
overlap upon it. If each premise dealt with exactly half the Middle,
thus barely distributing it between them, there would be no logical
proposition inferrible. We require that the middle term, as used in one
premise, should necessarily overlap the same term as used in the other,
so as to furnish common ground for comparing the other terms. Hence I
have defined the middle term as 'that term common to both premises by
means of which the other terms are compared.'
(5) One at least of the premises must be affirmative; or, from two
negative premises nothing can be inferred (in the given terms).
The fourth Canon required that the middle term should be given
distributed, or in its whole extent, at least once, in order to afford
sure ground of comparison for the others. But that such comparison may
be effected, something more is requisite; the relation of the other
terms to the Middle must be of a certain character. One at least of them
must be, as to its extent or denotation, partially or wholly identified
with the Middle; so that to that extent it may be known to bear to the
other term, whatever relation we are told that so much of the Middle
bears to that other term. Now, identity of denotation can only be
predicated in an affirmative proposition: one premise, then, must be
affirmative.
If both premises are negative, we only know that both the other terms
are partly or wholly excluded from the Middle, or are not identical with
it in denotation: where they lie, then, in relation to one another we
have no means of knowing. Similarly, in the mediate comparison of
quantities, if we are told that A and C are both of them unequal to B,
we can infer nothing as to the relation of C to A. Hence the premises--
No electors are sober;
No electors are independent--
however suggestive, do not formally justify us in inferring any
connection between sobriety and independence. Formally to draw a
conclusion, we must have affirmative grounds, such as in this case we
may obtain by obverting both premises:
All electors are not-sober;
All electors are not-independent:
.'. Some who are not-independent are not-sober.
But this conclusion is not in the given terms.
(6) (a) If one premise be negative, the conclusion must be negative: and
(b) to prove a negative conclusion, one premise must be negative.
(a) For we have seen that one premise must be affirmative, and that thus
one term must be partly (at least) identified with the Middle. If, then,
the other premise, being negative, predicates the exclusion of the
remaining term from the Middle, this remaining term must be excluded
from the first term, so far as we know the first to be identical with
the Middle: and this exclusion will be expressed by a negative
conclusion. The analogy of the mediate comparison of quantities may here
again be noticed: if A is equal to B, and B is unequal to C, A is
unequal to C.
(b) If both premises be affirmative, the relations to the Middle of both
the other terms are more or less inclusive, and therefore furnish no
ground for an exclusive inference. This also follows from the function
of the middle term.
For the more convenient application of these canons to the testing of
syllogisms, it is usual to derive from them three Corollaries:
(i) Two particular premises yield no conclusion.
For if both premises be affirmative, _all_ their terms are
undistributed, the subjects by predesignation, the predicates by
position; and therefore the middle term must be undistributed, and there
can be no conclusion.
If one premise be negative, its predicate is distributed by position:
the other terms remaining undistributed. But, by Canon 6, the conclusion
(if any be possible) must be negative; and therefore its predicate, the
major term, will be distributed. In the premises, therefore, both the
middle and the major terms should be distributed, which is impossible:
e.g.,
Some M is not P;
Some S is M:
.'. Some S is not P.
Here, indeed, the major term is legitimately distributed (though the
negative premise might have been the minor); but M, the middle term, is
distributed in neither premise, and therefore there can be no
conclusion.
Still, an exception may be made by admitting a bi-designate conclusion:
Some P is M;
Some S is not M:
.'. Some S is not some P.
(ii) If one premise be particular, so is the conclusion.
For, again, if both premises be affirmative, they only distribute one
term, the subject of the universal premise, and this must be the middle
term. The minor term, therefore, is undistributed, and the conclusion
must be particular.
If one premise be negative, the two premises together can distribute
only two terms, the subject of the universal and the predicate of the
negative (which may be the same premise). One of these terms must be the
middle; the other (since the conclusion is negative) must be the major.
The minor term, therefore, is undistributed, and the conclusion must be
particular.
(iii) From a particular major and a negative minor premise nothing can
be inferred.
For the minor premise being negative, the major premise must be
affirmative (5th Canon); and therefore, being particular, distributes
the major term neither in its subject nor in its predicate. But since
the conclusion must be negative (6th Canon), a distributed major term is
demanded, e.g.,
Some M is P;
No S is M:
.'. ------
Here the minor and the middle terms are both distributed, but not the
major (P); and, therefore, a negative conclusion is impossible.
Sec. 3. First Principle or Axiom of the Syllogism.--Hitherto in this
chapter we have been analysing the conditions of valid mediate
inference. We have seen that a single step of such inference, a
Syllogism, contains, when fully expressed in language, three
propositions and three terms, and that these terms must stand to one
another in the relations required by the fourth, fifth, and sixth
Canons. We now come to a principle which conveniently sums up these
conditions; it is called the _Dictum de omni et nullo_, and may be
stated thus:
Whatever is predicated (affirmatively or negatively) of a
term distributed,
With which term another term can be (partly or wholly)
identified,
May be predicated in like manner (affirmatively or
negatively) of the latter term (or part of it).
Thus stated (nearly as by Whately in the introduction to his _Logic_)
the _Dictum_ follows line by line the course of a Syllogism in the First
Figure (see chap. X. Sec. 2). To return to our former example: _All authors
are vain_ is the same as--Vanity is predicated of all authors; _Cicero
is an author_ is the same as--Cicero is identified as an author;
therefore _Cicero is vain_, or--Vanity may be predicated of Cicero. The
_Dictum_ then requires: (1) three propositions; (2) three terms; (3)
that the middle term be distributed; (4) that one premise be
affirmative, since only by an affirmative proposition can one term be
identified with another; (5) that if one premise be negative the
conclusion shall be so too, since whatever is predicated of the middle
term is predicated _in like manner_ of the minor.
Thus far, then, the _Dictum_ is wholly analytic or verbal, expressing no
more than is implied in the definitions of 'Syllogism' and 'Middle
Term'; since (as we have seen) all the General Canons (except the third,
which is a still more general condition of formal proof) are derivable
from those definitions. However, the _Dictum_ makes a further statement
of a synthetic or real character, namely, that _when these conditions
are fulfilled an inference is justified_; that then the major and minor
terms are brought into comparison through the middle, and that the major
term may be predicated affirmatively or negatively of all or part of the
minor. It is this real assertion that justifies us in calling the
_Dictum_ an Axiom.
Sec. 4. Whether the Laws of Thought may not fully explain the Syllogism
without the need of any synthetic principle has, however, been made a
question. Take such a syllogism as the following:
All domestic animals are useful;
All pugs are domestic animals:
.'. All pugs are useful.
Here (an ingenious man might urge), having once identified pugs with
domestic animals, that they are useful follows from the Law of Identity.
If we attend to the meaning, and remember that what is true in one form
of words is true in any other form, then, all domestic animals being
useful, of course pugs are. It is merely a case of subalternation: we
may put it in this way:
All domestic animals are useful:
.'. Some domestic animals (e.g., pugs) are useful.
The derivation of negative syllogisms from the Law of Contradiction (he
might add) may be shown in a similar manner.
But the force of this ingenious argument depends on the participial
clause--'having once identified pugs with domestic animals.' If this is
a distinct step of the reasoning, the above syllogism cannot be reduced
to one step, cannot be exhibited as mere subalternation, nor be brought
directly under the law of Identity. If 'pug,' 'domestic,' and 'useful'
are distinct terms; and if 'pug' and 'useful' are only known to be
connected because of their relations to 'domestic': this is something
more than the Laws of Thought provide for: it is not Immediate
Inference, but Mediate; and to justify it, scientific method requires
that its conditions be generalised. The _Dictum_, then, as we have seen,
does generalise these conditions, and declares that when such conditions
are satisfied a Mediate Inference is valid.
But, after all (to go back a little), consider again that proposition
_All pugs are domestic animals_: is it a distinct step of the reasoning;
that is to say, is it a Real Proposition? If, indeed, 'domestic' is no
part of the definition of 'pug,' the proposition is real, and is a
distinct part of the argument. But take such a case as this:
All dogs are useful;
All pugs are dogs.
Here we clearly have, in the minor premise, only a verbal proposition;
to be a dog is certainly part of the definition of 'pug.' But, if so,
the inference 'All pugs are useful' involves no real mediation, and the
argument is no more than this:
All dogs are useful;
.'. Some dogs (e.g., pugs) are useful.
Similarly, if the major premise be verbal, thus:
All men are rational;
Socrates is a man--
to conclude that 'Socrates is rational' is no Mediate Inference; for so
much was implied in the minor premise, 'Socrates is a man,' and the
major premise adds nothing to this.
Hence we may conclude (as anticipated in chap. vii. Sec. 3) that 'any
apparent syllogism, having one premise a verbal proposition, is really
an Immediate Inference'; but that, if both premises are real
propositions, the Inference is Mediate, and demands for its explanation
something more than the Laws of Thought.
The fact is that to prove the minor to be a case of the middle term may
be an exceedingly difficult operation (chap. xiii. Sec. 7). The difficulty
is disguised by ordinary examples, used for the sake of convenience.
Sec. 5. Other kinds of Mediate Inference exist, yielding valid conclusions,
without being truly syllogistic. Such are mathematical inferences of
Equality, as--
A = B = C .'. A = C.
Here, according to the usual logical analysis, there are strictly four
terms--(1) A, (2) equal to B, (3) B, (4) equal to C.
Similarly with the argument _a fortiori_,
A > B > C .'. (much more) A > C.
This also is said to contain four terms: (1) A, (2) greater than B, (3)
B, (4) greater than C. Such inferences are nevertheless intuitively
sound, may be verified by trial (within the limits of sense-perception),
and are generalised in appropriate axioms of their own, corresponding to
the _Dictum_ of the syllogism; as 'Things equal to the same thing are
equal to one another,' etc.
Now, surely, this is an erroneous application of the usual logical
analysis of propositions. Both Logic and Mathematics treat of the
_relations_ of terms; but whilst Mathematics employs the sign = for only
one kind of relation, and for that relation exclusive of the terms;
Logic employs the same signs (_is_ or _is not_) for all relations,
recognising only a difference of quality in predication, and treating
every other difference of relation as belonging to one of the terms
related. Thus Logicians read _A--is--equal to B_: as if _equal to B_
could possibly be a term co-relative with A. Whence it follows that the
argument _A = B = C .'. A = C_ contains four terms; though everybody sees
that there are only three.
In fact (as observed in chap. ii. Sec. 2) the sign of logical relation
(_is_ or _is not_), whilst usually adequate for class-reasoning
(coinherence) and sometimes extensible to causation (because a cause
implies a class of events), should never be stretched to include other
relations in such a way as to sacrifice intelligence to formalism. And,
besides mathematical or quantitative relations, there are others
(usually considered qualitative because indefinite) which cannot be
justly expressed by the logical copula. We ought to read propositions
expressing time-relations (and inferences drawn accordingly) thus:
B--is before--C;
A--is before--B:
.'. A--is before--C.
And in like manner _A--is simultaneous with--B; etc._ Such arguments (as
well as the mathematical) are intuitively sound and verifiable, and
might be generalised in axioms if it were worth while: but it is not,
because no method could be founded on such axioms.
The customary use of relative terms justifies some Mediate Inferences,
as, _The father of a father is a grand-father_.
Some cases, however, that at first seem obvious, are really delusive
unless further data be supplied. Thus _A co-exists with B, B with C; .'. A
with C_--is not sound unless _B_ is an instantaneous event; for where B
is perdurable, _A_ may co-exist with it at one time and _C_ at another.
Again: _A is to the left of B, B of C; .'. A of C_. This may pass; but it
is not a parallel argument that if _A is north of B and B west of C_,
then _A is north-west of C_: for suppose that A is a mile to the north
of B, and B a yard to the west of C, then A is practically north of C;
at least, its westward position cannot be expressed in terms of the
mariner's compass. In such a case we require to know not only the
directions but the distances of A and C from B; and then the exact
direction of A from C is an affair of mathematical calculation.
Qualitative reasoning concerning position is only applicable to things
in one dimension of space, or in time considered as having one
dimension. Under these conditions we may frame the following
generalisation concerning all Mediate Inferences: Two terms definitely
related to a third, and one of them positively, are related to one
another as the other term is related to the third (that is, positively
or negatively); provided that the relations given are of the same kind
(that is, of Time, or Coinherence, or Likeness, or Equality).
Thus, to illustrate by relations of Time--
B is simultaneous with C;
A is not simultaneous with B:
.'. A is not simultaneous with C.
Here the relations are of the same kind but of different logical
quality, and (as in the syllogism) a negative copula in the premises
leads to a negative conclusion.
An examination in detail of particular cases would show that the above
generalisation concerning all Mediate Inferences is subject to too many
qualifications to be called an Axiom; it stands to the real Axioms (the
_Dictum_, etc.) as the notion of the Uniformity of Nature does to the
definite principles of natural order (_cf._ chap. xiii. Sec. 9).