ABBREVIATED AND COMPOUND ARGUMENTS
Sec. 1. In ordinary discussion, whether oral or written, it is but rarely
that the forms of Logic are closely adhered to. We often leave wide gaps
in the structure of our arguments, trusting the intelligence of those
addressed to bridge them over; or we invert the regular order of
propositions, beginning with the conclusion, and mentioning the
premises, perhaps, a good while after, confident that the sagacity of
our audience will make all smooth. Sometimes a full style, like
Macaulay's, may, by means of amplification and illustration, spread the
elements of a single syllogism over several pages--a pennyworth of logic
steeped in so much eloquence. These practices give a great advantage to
sophists; who would find it very inconvenient to state explicitly in
Mood and Figure the pretentious antilogies which they foist upon the
public; and, indeed, such licences of composition often prevent honest
men from detecting errors into which they themselves have unwittingly
fallen, and which, with the best intentions, they strive to communicate
to others: but we put up with these drawbacks to avoid the inelegance
and the tedium of a long discourse in accurate syllogisms.
Many departures from the strictly logical statement of reasonings
consist in the use of vague or figurative language, or in the
substitution for one another of expressions supposed to be equivalent,
though, in fact, dangerously discrepant. Against such occasions of error
the logician can provide no safeguard, except the advice to be careful
and discriminating in what you say or hear. But as to any derangement
of the elements of an argument, or the omission of them, Logic
effectually aids the task of restoration; for it has shown what the
elements are that enter into the explicit statement of most
ratiocinations, namely, the four forms of propositions and what that
connected order of propositions is which most easily and surely exposes
the validity or invalidity of reasoning, namely, the premises and
conclusion of the Syllogism. Logic has even gone so far as to name
certain abbreviated forms of proof, which may be regarded as general
types of those that actually occur in debate, in leading articles,
pamphlets and other persuasive or polemic writings--namely, the
Enthymeme, Epicheirema and Sorites.
Sec. 2. The Enthymeme, according to Aristotle, is the Syllogism of probable
reasoning about practical affairs and matters of opinion, in contrast
with the Syllogism of theoretical demonstration upon necessary grounds.
But, as now commonly treated, it is an argument with one of its elements
omitted; a Categorical Syllogism, having one or other of its premises,
or else its conclusion, suppressed. If the major premise be suppressed,
it is called an Enthymeme of the First Order; if the minor premise be
wanting, it is said to be of the Second Order; if the conclusion be left
to be understood, there is an Enthymeme of the Third Order.
Let the following be a complete Syllogism:
All free nations are enterprising;
The Dutch are a free nation:
.'. The Dutch are enterprising.
Reduced to Enthymemes, this argument may be put thus:
In the First Order:
The Dutch are a free nation:
.'. The Dutch are enterprising.
In the Second Order--
All free nations are enterprising;
.'. The Dutch are enterprising.
In the Third Order--
All free nations are enterprising;
And the Dutch are a free nation.
It is certainly very common to meet with arguments whose statement may
be represented by one or other of these three forms; indeed, the
Enthymeme is the natural substitute for a full syllogism in oratory:
whence the transition from Aristotle's to the modern meaning of the
term. The most unschooled of men readily apprehend its force; and a
student of Logic can easily supply the proposition that may be wanted in
any case to complete a syllogism, and thereby test the argument's formal
validity. In any Enthymeme of the Third Order, especially, to supply the
conclusion cannot present any difficulty at all; and hence it is a
favourite vehicle of innuendo, as in Hamilton's example:
Every liar is a coward;
And Caius is a liar.
The frankness of this statement and its reticence, together, make it a
biting sarcasm upon Caius.
The process of finding the missing premise in an Enthymeme of either the
First or the Second Order, so as to constitute a syllogism, is sometimes
called Reduction; and for this a simple rule may be given: Take that
term of the given premise which does not occur in the conclusion (and
which must therefore be the Middle), and combine it with that term of
the conclusion which does not occur in the given premise; the
proposition thus formed is the premise which was requisite to complete
the Syllogism. If the premise thus constituted contain the predicate of
the conclusion, the Enthymeme was of the First Order; if it contain the
subject of the conclusion, the Enthymeme was of the Second Order.
That a statement in the form of a Hypothetical Proposition may really be
an Enthymeme (as observed in chap. v. Sec. 4) can easily be shown by
recasting one of the above Enthymemes thus: _If all free nations are
enterprising, the Dutch are enterprising_. Such statements should be
treated according to their true nature.
To reduce the argument of any ordinary discourse to logical form, the
first care should be to make it clear to oneself what exactly the
conclusion is, and to state it adequately but as succinctly as possible.
Then look for the evidence. This may be of an inductive character,
consisting of instances, examples, analogies; and, if so, of course its
cogency must be evaluated by the principles of Induction, which we
shall presently investigate. But if the evidence be deductive, it will
probably consist of an Enthymeme, or of several Enthymemes one depending
on another. Each Enthymeme may be isolated and expanded into a
syllogism. And we may then inquire: (1) whether the syllogisms are
formally correct according to Barbara (or whatever the appropriate
Mood); (2) whether the premises, or the ultimate premises, are true in
fact.
Sec. 3. A Monosyllogism is a syllogism considered as standing alone or
without relation to other arguments. But, of course, a disputant may be
asking to prove the premises of any syllogism; in which case other
syllogisms may be advanced for that purpose. When the conclusion of one
syllogism is used to prove another, we have a chain-argument which,
stated at full length, is a Polysyllogism. In any Polysyllogism, again,
a syllogism whose conclusion is used as the premise of another, is
called in relation to that other a Prosyllogism; whilst a syllogism one
of whose premises is the conclusion of another syllogism, is in relation
to that other an Episyllogism. Two modes of abbreviating a
Polysyllogism, are usually discussed, the Epicheirema and the Sorites.
Sec. 4. An Epicheirema is a syllogism for one or both of whose premises a
reason is added; as--
All men are mortal, for they are animals;
Socrates is a man, for rational bipeds are men:
.'. Socrates is mortal.
The Epicheirema is called Single or Double, says Hamilton, according as
an "adscititious proposition" attaches to one or both of the premises.
The above example is of the double kind. The Single Epicheirema is said
to be of the First Order, if the adscititious proposition attach to the
major premise; if to the minor, of the Second Order. (Hamilton's
_Logic_: Lecture xix.)
An Epicheirema, then, is an abbreviated chain of reasoning, or
Polysyllogism, comprising an Episyllogism with one or two enthymematic
Prosyllogisms. The major premise in the above case, _All men are mortal,
for they are animals,_ is an Enthymeme of the First Order, suppressing
its own major premise, and may be restored thus:
All animals are mortal;
All men are animals:
.'. All men are mortal.
The minor premise, _Socrates is a man, for rational bipeds are men_, is
an Enthymeme of the Second Order, suppressing its own minor premise, and
may be restored thus:
All rational bipeds are men;
Socrates is a rational biped:
.'. Socrates is a man.
Sec. 5. The Sorites is a Polysyllogism in which the Conclusions, and even
some of the Premises, are suppressed until the arguments end. If the
chain of arguments were freed of its enthymematic character, the
suppressed conclusions would appear as premises of Episyllogisms.
Two varieties of Sorites are recognised, the Aristotelian (so called,
though not treated of by Aristotle), and the Goclenian (named after its
discoverer, Goclenius of Marburg, who flourished about 1600 A.D.). In
order to compare these two forms of argument, it will be convenient to
place side by side Hamilton's classical examples of them.
Aristotelian. Goclenian.
Bucephalus is a horse; An animal is a substance;
A horse is a quadruped; A quadruped is an animal;
A quadruped is an animal; A horse is a quadruped;
An animal is a substance: Bucephalus is a horse:
.'. Bucephalus is a substance. .'. Bucephalus is a substance.
The reader wonders what is the difference between these two forms. In
the Aristotelian Sorites the minor term occurs in the first premise, and
the major term in the last; whilst in the Goclenian the major term
occurs in the first premise, and the minor in the last. But since the
character of premises is fixed by their terms, not by the order in which
they are written, there cannot be a better example of a distinction
without a difference. At a first glance, indeed, there may seem to be a
more important point involved; the premises of the Aristotelian Sorites
seem to proceed in the order of Fig. IV. But if that were really so the
conclusion would be, _Some Substance is Bucephalus_. That, on the
contrary, every one writes the conclusion, _Bucephalus is a substance_,
proves that the logical order of the premises is in Fig. I. Logically,
therefore, there is absolutely no difference between these two forms,
and pure reason requires either that the "Aristotelian Sorites"
disappear from the text-books, or that it be regarded as in Fig. IV.,
and its conclusion converted. It is the shining merit of Goclenius to
have restored the premises of the Sorites to the usual order of Fig. I.:
whereby he has raised to himself a monument more durable than brass, and
secured indeed the very cheapest immortality.
The common Sorites, then, being in Fig. I., its rules follow from those
of Fig. I:
(1) Only one premise can be particular; and, if any, only that in which
the minor term occurs.
For, just as in Fig I., a particular premise anywhere else involves
undistributed Middle.
(2) Only one premise can be negative; and, if any, only that in which
the major term occurs.
For if there were two negative premises, at the point where the second
entered the chain of argument there must be a syllogism with two
negative premises, which is contrary to Rule 5; whilst if one premise be
negative it must be that which contains the major term, for the same
reason as in Fig. I., namely, that the conclusion will be negative, and
that therefore only a negative major premise can prevent illicit process
of the major term.
If we expand a Sorites into its constituent syllogisms, the conclusions
successively suppressed will reappear as major premises; thus:
(1) An animal is a substance;
A quadruped is an animal:
.'. A quadruped is a substance.
(2) A quadruped is a substance;
A horse is a quadruped:
.'. A horse is a substance.
(3) A horse is a substance:
Bucephalus is a horse:
.'. Bucephalus is a substance.
This suffices to show that the Protosyllogism of a Goclenian Sorites is
an Enthymeme of the Third Order; after which the argument is a chain of
Enthymemes of the First Order, or of the First and Third combined, since
the conclusions as well as the major premises are omitted, except in the
last one.
Lest it should be thought that the Sorites is only good for arguments so
frivolous as the above, I subjoin an example collected from various
parts of Mill's _Political Economy_:--
The cost of labour depends on the efficiency of labour;
The rate of profits depends on the cost of labour;
The investment of capital depends on the rate of profits;
Wages depend on the investment of capital:
.'. Wages depend on the efficiency of labour.
Had it occurred to Mill to construct this Sorites, he would have
modified his doctrine of the wages-fund, and would have spared many
critics the malignant joy of refuting him.
Sec. 6. The Antinomy is a combination of arguments by which contradictory
attributes are proved to be predicable of the same subject. In symbols,
thus:
All M is P; All N is p;
All S is M: All S is N:
.'. All S is P. .'. All S is p.
Now, by the principle of Contradiction, S cannot be P and p (not-P):
therefore, if both of the above syllogisms are sound, S, as the subject
of contradictory attributes, is logically an impossible thing. The
contradictory conclusions are called, respectively, Thesis and
Antithesis.
To come to particulars, we may argue: (1) that a constitution which is
at once a monarchy, an aristocracy and a democracy, must comprise the
best elements of all three forms; and must, therefore, be the best of
all forms of government: the British Constitution is, therefore, the
best of all. But (2) such a constitution must also comprise the worst
elements of monarchy, aristocracy and democracy; and, therefore, must be
the worst of all forms. Are we, then, driven to conclude that the
British Constitution, thus proved to be both the best and worst, does
not really exist at all, being logically impossible? The proofs seem
equally cogent; but perhaps neither the best nor the worst elements of
the simpler constitutions need be present in our own in sufficient force
to make it either good or bad.
Again:
(1) Every being who is responsible for his actions is free;
Man is responsible for his actions:
.'. Man is free.
(2) Every being whose actions enter into the course of nature is not free;
Man is such a being:
.'. Man is not free.
Does it, then, follow that 'Man,' as the subject of contradictory
attributes, is a nonentity? This doctrine, or something like it, has
been seriously entertained; but if to any reader it seem extravagant (as
it certainly does to me), he will no doubt find an error in the above
arguments. Perhaps the major term is ambiguous.
For other examples it is enough to refer to the _Critique of Pure
Reason_, where Kant sets out the Antinomies of Rational Cosmology. But
even if we do not agree with Kant that the human understanding, in
attempting to deal with certain subjects beyond its reach, inevitably
falls into such contradictory reasonings; yet it can hardly be doubted
that we not unfrequently hold opinions which, if logically developed,
result in Antinomies. And, accordingly, the Antinomy, if it cannot be
imputed to Reason herself, may be a very fair, and a very wholesome
_argumentum ad hominem_. It was the favourite weapon of the Pyrrhonists
against the dogmatic philosophies that flourished after the death of
Aristotle.