448 This case is noticed by Lotze, Logic, § 99.
A second obvious case of circulus in probando is where we seek to establish one of the premisses of a syllogism by means of another syllogism in which the ultimate conclusion itself appears as a premiss. For example,—All M is P (for all S is P, and all M is S); and all S is M ; therefore, all S is P.
A third case, which for our immediate purpose is more important than either of the above, is where the major premiss is an enumerative universal, summing up a number of individual instances each one of which has been separately considered. For example, All the apostles were Jews; Peter was an apostle; therefore, Peter was a Jew. A universal proposition relating to a limited class, such as the apostles, is usually established by considering the members individually; and if the truth of a universal proposition could be established in this manner only, then the charge that syllogistic reasoning necessarily involves petitio principii would not admit of refutation. This appears to be assumed in the argument of Sextus Empiricus quoted above. It is also assumed in the following dilemma, which has been given as summing up Mill’s doctrine: “If all the facts of the major premiss of any syllogism have been examined, the syllogism is needless; and if some of them have not been examined, it is a petitio principii. But either all have been examined or some have not. Therefore, the syllogism is either useless or fallacious,” Mill’s own argument may also be quoted: “We cannot be assured of the mortality of all men, unless we are already certain of the mortality of every individual man” (Logic, ii. 3, § 2).449
449 Bain (Logic, Deduction, p. 208) taking as an example the syllogism, “All men are mortal, All kings are men, therefore, All kings are mortal,” asks “Supposing there were any doubt as to the conclusion that kings are mortal, by what right do we proclaim, in the major, that all men are mortal, kings included?” He then continues, “In order to say, ‘All men are mortal,’ we must have found in some other way that all kings and all people are mortal. So that the conclusion first contributes its quota to the major premiss, and then takes it back again.” The reply to Bain’s challenge is that if we are in doubt as to whether kings are mortal, we may resolve our doubt by shewing that kings belong to a class the mortality of which is admitted. The question then resolves itself into whether it is possible to establish the mortality of mankind in general without any explicit consideration of the particular case of kings.
427 It cannot, however, for a moment be allowed that universal propositions admit of proof only by enumeration. Propositions that do admit of such proof are indeed generally speaking of little importance. The syllogism is chiefly of value inferentially where the major premiss is universal in the fullest and most unlimited sense, that is, unconditionally universal, expressing a general law dependent on qualitative relations. The true character and value of such a premiss, though ordinarily written in the form All S is P, would be better brought out by the use of one of the forms Any S is P, Whatever is S is also P, It is the nature of S to be P, If anything is S it is P.450
450 Sigwart holds that, in order properly to understand the value of the syllogism, we should take as our type the conditional (or, as he expresses it, the hypothetical), rather than the categorical, syllogism. We need, he says, but glance at any mathematical or physical text-book to assure ourselves that by far the greater number of propositions which are used as major premisses are hypothetical in nature, if not in expression. “Propositions such as ‘two circles which intersect have no common centre’ are hypothetical in nature; the proposition states the condition upon which the predicate is denied.… It is the same with the formulae of analytical mechanics; these and others of the same description are hypothetical judgments, and inferences are made in accordance with them by substituting definite values for the general symbols” (Logic, § 55). Sigwart perhaps attaches undue importance to the mere question of form. If our major premiss is unconditionally universal, and is understood to be so, it does not affect the character of the reasoning whether we adopt the categorical mode of expression or the conditional. Sigwart’s reason for dwelling on the hypothetical force of the major premiss is to be found largely in the trivial nature of the examples that it has been customary to give of the purely categorical syllogism.
The following may be noted as typical cases in which the grounds for accepting the truth of the premisses of a syllogism are quite independent of any explicit knowledge of the truth of the conclusion.
(1) The major premiss may itself be accepted as axiomatic, or it may be deducible (without the assistance of the conclusion) from more ultimate principles that are accepted as axiomatic. It has indeed been argued that a self-evident maxim cannot be used, or is at any rate superfluous, as a proof, because any conclusion that it might be employed to establish would be itself equally self-evident.451 A consideration of ordinary 428 geometrical proofs will, however, at once shew that this is not necessarily the case, and that by the aid of self-evident premisses conclusions may be reached that are certainly not themselves self-evident.
451 Compare Bailey, Theory of Reasoning, p. 74.
(2) The major premiss may be based on authority, or may be accepted on testimony; or it may be the expression of a civil law, or of a command, or of a rule of conduct;452 and in none of these cases can it be in any degree grounded upon the conclusion.
452 “We find,” says Sigwart, “a wide field for our inferences in the application of general laws which have their origin in our will and are meant to regulate that will. In laying down a general rule of conduct, our will determines that there shall be a universally valid connexion between certain conditions and certain modes of action. If we will the general law, it is logically necessary that we should will the particular actions prescribed by the law, if our will is to be constant and consistent, and valid for everyone who agrees in willing the general law. All penal codes in imposing a penalty of imprisonment for theft, of capital punishment for murder, lay down a series of hypothetical judgments which establish a universal connexion between committing the crime and incurring the penalty. These judgments, moreover, may also be regarded as theoretical propositions in so far as they express the general obligation of the judge to give sentence in accordance with the law” (Logic, i. p. 337).
(3) The major premiss may be an imperfect induction, based on evidence that does not include the conclusion. As an example, we may take the reasoning involved in testing the nature of a given substance in practical chemistry. In a reasoning of this kind our immediate starting point is general knowledge of the properties of chemical substances. This knowledge has been inductively obtained, but it is impossible that it should in the slightest degree depend on any antecedent acquaintance with the properties of the particular substance which is now to be investigated for the first time. Or, again, we may take astronomical inferences based on the law of universal gravitation. That law is an induction based on particular observations, but it implies an infinite number of facts that form no part of the evidence on which it is accepted as true; and many of these facts are in the first instance brought to our notice as inferences from the law, not as data leading up to it. If it is affirmed that, in cases such as these, the major premisses cannot legitimately be established independently of the conclusions syllogistically derived from them, then 429 the validity of imperfect induction as a process of arriving at knowledge must be denied.
If asked to meet the argument contained in the preceding paragraph, Mill would doubtless refer to his doctrine of the function of the major premiss in a syllogism. The real proof of the conclusion of a syllogism, he would say, is to be found, not in the major premiss itself, but in the evidence on which the major premiss is based: the major premiss is nothing more than a memorandum of evidence from which the conclusion might be directly obtained: the intervention of the major premiss is often convenient, but it is not an essential link in the passage from the ultimate data to the conclusion. In reply, it may be said that there is at any rate a shifting of the ground here, and that Mill’s doctrine, even if accepted, fails to justify the charge that every syllogism involves petitio principii ; for it is admitted that the conclusion does not itself constitute any part of the data from which the major premiss is obtained. We must, however, go further and reject the doctrine on the ground that there are at any rate some cases in which the general law expressed by the major premiss is an absolutely necessary link in the argument. Thus, to take but one illustration, there are many consequences of the law of universal gravitation which it would be quite impossible to infer directly from the evidence lying behind that law without the intervention of the law itself.
Having regard then to instances such as those adduced above, we must reject the view that syllogistic reasoning essentially involves petitio principii, in the sense of circulus in probando. Any plausibility that the opposed view may possess depends upon some confusion between the statement that every syllogism is guilty of petitio principii in the above sense and the statement that in every syllogism the premisses presuppose the conclusion in the sense that they could not be true unless the conclusion were true.
The latter statement is applicable not only to syllogistic, but to all demonstrative, inference. The question may indeed be raised whether it is not applicable to all valid inference whatsoever. It is in fact one horn of the dilemma referred to in section 377.
430 At any rate it is a misuse of language to speak of a reasoning as involving petitio principii on this ground. By petitio principii is always understood a certain form of fallacy. But in making explicit what to begin with is merely implicit there is nothing that can by any stretch of language be termed fallacious. To say that all deductive science is nothing but a huge petitio principii is clearly an absurdity. The most that can be said is that in all demonstrative reasoning (so-called) there is really no inference from premisses but only the interpretation of premisses. So far as this is a mere question of language, it may suffice to note the paradoxical conclusions to which it leads; for example, that in the whole of Euclid there is no such thing as inference or proof. So far as it is not a mere question of language, it turns on points that we have already discussed, for instance, the possibility of there being an advance in knowledge subjectively considered although from the objective standpoint the conclusions reached contain nothing new. It is unnecessary to repeat the discussion with special reference to the syllogism.
CHAPTER X.
EXAMPLES OF ARGUMENTS AND FALLACIES.
382. In how many different
moods may the argument implied in the following proposition be stated?
“No one can maintain that all persecution is justifiable who
admits that persecution is sometimes ineffective.”
How would the formal correctness of the reasoning be affected by
reading “deny” for “maintain”? [V.]
383. No one can maintain
that all republics secure good government who bears in mind that good
government is inconsistent with a licentious press.
What premisses must be supplied to express the above reasoning in
Ferio, Festino and Ferison respectively? [V.]
384. Write the following
arguments in syllogistic form, and reduce them to the first
figure:—
(α) Falkland was a royalist and a patriot; therefore, some
royalists were patriots.
(β) All who are punished should be responsible for their
actions; therefore, if some lunatics are not responsible for their
actions, they should not be punished.
(γ) All who have passed the Little-Go have a knowledge of
Greek; hence A.B. cannot have passed the Little-Go, for he has
no knowledge of Greek. [K.]
385. “It is impossible
to maintain that the virtuous alone are happy, and at the same time
that selfishness is compatible with happiness but incompatible with
virtue.”
State the above argument syllogistically in as many different moods
as possible. [J.]
432 386. Give the technical name of the following argument:—Payment by results sounds extremely promising; but payment by results necessarily means payment for a minimum of knowledge; payment for a minimum of knowledge means teaching in view of a minimum of knowledge; teaching in view of a minimum of knowledge means bad teaching. [K.]
387. From P follows Q ; and from R follows S ; but Q and S cannot both be true; shew that P and R cannot both be true. [De Morgan.]
388. If (1) it is false that
whenever X is found Y is found with it, and (2) not less
untrue that X is sometimes found without the accompaniment of
Z, are you justified in denying that (3) whenever Z is
found there also you may be sure of finding Y? And, however
this may be, can you in the same circumstances judge anything about
Y in terms of Z? [R.]
389. Can the following
arguments be reduced to syllogistic form?
(1) The sun is a thing insensible;
The Persians worship the sun;
Therefore, the Persians worship a thing insensible.
(2) The Divine law commands us to honour kings;
Louis XIV. is a king;
Therefore, the Divine law commands us to honour Louis XIV.
[Port Royal Logic.]
390. Examine the following
arguments; where they are valid, reduce them if you can to syllogistic
form; and where they are invalid, explain the nature of the
fallacy:—
(1) We ought to believe the Scripture;
Tradition is not Scripture;
Therefore, we ought not to believe tradition.
(2) Every good pastor is ready to give his life for his sheep;
Now, there are few pastors in the present day who are ready to give their
lives for their sheep;
Therefore, there are in the present day few good pastors.
(3) Those only who are friends of God are happy;
Now, there are rich men who are not friends of God;
Therefore, there are rich men who are not happy. 433
(4) The duty of a Christian is not to praise those who commit
criminal actions;
Now, those who engage in a duel commit a criminal action;
Therefore, it is the duty of a Christian not to praise those
who engage in duels.
(5) The gospel promises salvation to Christians;
Some wicked men are Christians;
Therefore, the gospel promises salvation to wicked men.
(6) He who says that you are an animal speaks truly;
He who says that you are a goose says that you are an animal;
Therefore, he who says that you are a goose speaks truly.
(7) You are not what I am;
I am a man;
Therefore, you are not a man.
(8) We can only be happy in this world by abandoning ourselves to
our passions, or by combating them;
If we abandon ourselves to them, this is an unhappy state, since it is disgraceful, and we could never be content with it;
If we combat them, this is also an unhappy state, since there is
nothing more painful than that inward war which we are continually
obliged to carry on with ourselves;
Therefore, we cannot have in this life true happiness.
(9) Either our soul perishes with the body, and thus, having no
feelings, we shall be incapable of any evil; or if the soul survives
the body, it will be more happy than it was in the body;
Therefore, death is not to be feared. [Port
Royal Logic.]
391. Examine the following
arguments:—
(1) “He that is of God heareth my words: ye therefore hear
them not, because ye are not of God.”
(2) All the fish that the net inclosed were an indiscriminate
mixture of various kinds: those that were set aside and saved as
valuable, were fish that the net inclosed: therefore, those that were
set aside and saved as valuable, were an indiscriminate mixture of
various kinds.
(3) Testimony is a kind of evidence which is very likely to be
false: the evidence on which most men believe that there are pyramids
in Egypt is testimony: therefore, the evidence on which most men
believe that there are pyramids in Egypt is very likely to be false.
434
(4) If Paley’s system is to be received, one who has no
knowledge of a future state has no means of distinguishing virtue and
vice: now one who has no means of distinguishing virtue and vice can
commit no sin: therefore, if Paley’s system is to be received,
one who has no knowledge of a future state can commit no sin.
(5) If Abraham were justified, it must have been either by faith or
by works: now he was not justified by faith (according to James), nor
by works (according to Paul): therefore, Abraham was not justified.
(6) For those who are bent on cultivating their minds by diligent
study, the incitement of academical honours is unnecessary; and it is
ineffectual, for the idle, and such as are indifferent to mental
improvement: therefore, the incitement of academical honours is either
unnecessary or ineffectual.
(7) He who is most hungry eats most; he who eats least is most
hungry: therefore, he who eats least eats most.
(8) A monopoly of the sugar-refining business is beneficial to
sugar-refiners: and of the corn-trade to corn-growers: and of the
silk-manufacture to silk-weavers, &c., &c.; and thus each
class of men are benefited by some restrictions. Now all these classes
of men make up the whole community: therefore, a system of
restrictions is beneficial to the community. [Whately, Logic.]
392. The following are a few
examples in which the reader can try his skill in detecting fallacies,
determining the peculiar form of syllogisms, and supplying the
suppressed premisses of enthymemes:
(1) None but those who are contented with their lot in life can
justly be considered happy. But the truly wise man will always make
himself contented with his lot in life, and, therefore, he may justly
be considered happy.
(2) All intelligible propositions must be either true or false. The
two propositions “Caesar is living still,” and
“Caesar is dead,” are both intelligible propositions;
therefore, they are both true, or both false.
(3) Many things are more difficult than to do nothing. Nothing is
more difficult to do than to walk on one’s head. Therefore, many
things are more difficult than to walk on one’s head.
(4) None but Whigs vote for Mr B. All who vote for Mr B. are
ten-pound householders. Therefore, none but Whigs are ten-pound
householders. 435
(5) If the Mosaic account of the cosmogony is strictly correct, the
sun was not created till the fourth day. And if the sun was not
created till the fourth day, it could not have been the cause of the
alternation of day and night for the first three days. But either the
word “day” is used in Scripture in a different sense to
that in which it is commonly accepted now, or else the sun must have
been the cause of the alternation of day and night for the first three
days. Hence it follows that either the Mosaic account of the cosmogony
is not strictly correct, or else the word “day” is used in
Scripture in a different sense to that in which it is commonly
accepted now.
(6) Suffering is a title to an excellent inheritance; for God
chastens every son whom he receives.
(7) It will certainly rain, for the sky looks very black. [Solly, Syllabus of Logic.]
393. Examine the following
arguments; so far as they are valid, reduce them to syllogistic form;
and where they are invalid, explain the nature of the fallacy
involved:—
(1) If you argue on a subject which you do not understand, you will
prove yourself a fool; for this is a mistake that fools always make.
(2) It is not the case that any metals are compounds, and it is
incorrect to say that every metal is heavy; it may, therefore, be
inferred that some elements are not heavy, and also that some heavy
substances are not elements.
(3) No young man is wise; for only experience can give wisdom, and
experience comes only with age. [K.]
394. Examine technically the
following argument:—
Everyone is either well informed of the facts or already convinced
on the subject; no one can be at the same time both already convinced
on the subject and amenable to argument: hence it follows that only
those who are well informed of the facts are amenable to argument.
[J.]
395. Dr Johnson remarked that “a man who sold a penknife was not necessarily an ironmonger.” Against what logical fallacy was this remark directed? [C.]
396. Examine the following
arguments, pointing out any fallacies that they contain: 436
(a) The more correct the logic, the more certainly will the
conclusion be wrong if the premisses are false. Therefore, where the
premisses are wholly uncertain the best logician is the least safe guide.
(b) The spread of education among the lower orders will make
them unfit for their work: for it has always had that effect on those
among them who happen to have acquired it in previous times.
(c) This pamphlet contains seditious doctrines. The spread
of seditious doctrines may be dangerous to the State. Therefore, this
pamphlet must be suppressed. [C.]
397. Examine the following
arguments:—
(1) A telescope with the eye-piece at one side of the tube is
probably a reflector; Lord Rosse’s telescope is a reflector;
therefore, Lord Rosse’s telescope probably has the eye-piece at
one side of the tube.
(2) Good workmen do not complain of their tools; my pupils do not
complain of their tools; therefore, my pupils are probably good
workmen.
(3) If, on the one hand, the heathen, through want of better
knowledge, cannot help breaking the Ten Commandments, then they do not
stand condemned; if, on the other hand, they are condemned, it is for
doing that which they well knew was wicked, and which they were well
able to refrain from doing; therefore, whatever happens to them,
justice is satisfied. [K.]
398. Discuss the nature of
the reasoning contained, or apparently intended, in the following
sentences:—
It is impossible to prove that persecution is justifiable if you
cannot prove that some non-effective measures are justifiable; for no
persecution has ever been effective.
This deed may be genuine though it is not stamped, for some
unstamped deeds are genuine. [C.]
399. State the following
arguments in logical form, and examine their validity:—
(1) Poetry must be either true or false: if the latter, it is
misleading; if the former, it is disguised history, and savours of
imposture as trying to pass itself off for more than it is. Some
philosophers have therefore wisely excluded poetry from the ideal
commonwealth. 437
(2) If we never find skins except as the teguments of animals, we
may safely conclude that animals cannot exist without skins. If colour
cannot exist by itself, it follows that neither can anything that is
coloured exist without colour. So if language without thought is
unreal, thought without language must also be so.
(3) Had an armistice been beneficial to France and Germany, it
would have been agreed upon by those powers; but such has not been the
case; it is plain therefore that an armistice would not have been
advantageous to either of the belligerents.
(4) If we are marked to die, we are enow
To do our country loss: and, if to live.
The fewer men, the greater share of honour. [O.]
400. Examine logically the
following arguments:—
(a) If truthfulness is never found save with scrupulousness,
and if truthfulness is incompatible with stupidity, it follows that
stupidity and scrupulousness can never be associated.
(b) You say that there is no rule without an exception. I
answer that, in that case, what you have just said must have an
exception, and so prove that you have contradicted yourself.
(c) Knowledge gives power; consequently, since power is
desirable, knowledge is desirable. [L.]
401. Examine the following
arguments, stating them in syllogistic form, and pointing out
fallacies, if any:—
(a) Some who are truly wise are not learned; but the
virtuous alone are truly wise; the learned, therefore, are not always
virtuous.
(b) If all the accused were innocent, some at least would
have been acquitted; we may infer, then, that none were innocent,
since none have been acquitted.
(c) Every statement of fact deserves belief; many
statements, not unworthy of belief, are asserted in a manner which is
anything but strong; we may infer, therefore, that some statements not
strongly asserted are statements of fact.
(d) That many persons who commit errors are blameworthy is
proved by numerous instances in which the commission of errors arises
from gross carelessness. [M.]
402. Examine technically the
following arguments:—
(1) Those who hold that the insane should not be punished ought in
consistency to admit also that they should not be threatened; 438 for it is clearly
unjust to punish any one without previously threatening him.
(2) If he pleads that he did not steal the goods, why, I ask, did
he hide them, as no thief ever fails to do?
(3) Knavery and folly always go together; so, knowing him to be a
fool, I distrusted him.
(4) How can you deny that the infliction of pain is justifiable if
punishment is sometimes justifiable and yet always involves pain?
(5) If I deny that poverty and virtue are inconsistent, and you
deny that they are inseparable, we can at least agree that some poor
are virtuous. [V.]
403. Detect the fallacy in
the following argument:—
“A vacuum is impossible, for if there is nothing between two
bodies they must touch.” [N.]
404. Consider the following
argument:—
Granted that A is B, to prove that B is
A. B (like everything else) is either A or not
A. If B is not A then by our first premiss we
have the syllogism—A is B, B is not A, therefore,
A is not A, which is absurd. Hence it follows that B is
A. [Professor Jastrow, in the Journal of Education February, 1897.]
405. Examine the following
argument:—
It is impossible to prove that society can continue to exist
without competition unless you can also prove that the absence of
competition would not lead to the deterioration of individuals; for a
society whose members deteriorate cannot long continue to exist. [M.]
406. Express the following
propositions in their simplest logical form; examine their mutual
consistency or inconsistency, and the validity of the final
conclusion:—
Some of Mr N’s published views are new, and some true;
in fact, they are all one or the other; and, though it cannot be
maintained in general that a view that is not new is on that account
necessarily not true, yet it can be confidently asserted that every
possible false view on this subject was propounded by some one or
other before Mr N. wrote: from which it would appear that while
it may or may not be that Mr N.’s views are all new, it is
certain that they are all true. [J.]
439 407. Examine technically the following arguments:—
(a)
“’Tis only the present that pains,
And the
present will pass.”
(b) All legislative restraint is either unjust or
unnecessary; since, for the sake of a single man’s interests, to
restrain all the rest of the community is unjust, and to restrain the
man himself is unnecessary.
(c) Only Conservatives—and not all of them—are
Protectionists; only Liberals—and not all of them—are Home
Rulers; but both parties contain supporters of women’s
franchise. Hence only Unionists—and not all of them—are
Protectionists, while the supporters of women’s franchise
contain both Unionists and Free-traders.
(d) No school-boy can be expected to understand
Constitutional History, and none but school-boys can be expected to
remember dates; so that no one can be expected both to remember dates
and to understand Constitutional History.
(e) To be wealthy is not to be healthy; not to be healthy is
to be miserable; therefore, to be wealthy is to be miserable.
(f) Whatever any man desires is desirable; every man desires
his own happiness; therefore, the happiness of every man is desirable.
[J.]
408. Examine the validity of
the following arguments:—
(1) I knew he was a Bohemian, for he was a good musician, and
Bohemians are always good musicians.
(2) Bullies are always cowards, but not always liars; liars,
therefore, are not always cowards.
(3) If all the soldiers had been English, they would not all have
run away; but some did run away; and we may, therefore, infer that
some of them at least were not English.
(4) None but the good are really to be envied; all truly wise men
are good; therefore, all truly wise men are to be envied.
(5) You cannot affirm that all his acts were virtuous, for you deny
that they were all praiseworthy, and you allow that nothing that is
not praiseworthy is virtuous.
(6) Since the end of poetry is pleasure, that cannot be unpoetical
with which all are pleased.
(7) Most M is P, Most S is M, therefore, Some S is
P. 440
(8) Old Parr, healthy as the wild animals, attained to the age of
152 years; all men might be as healthy as the wild animals; therefore,
all men might attain to the age of 152 years.
(9) It is quite absurd to say “I would rather not exist than
be unhappy,” for he who says “I will this, rather than
that,” chooses something. Non-existence, however, is no
something, but nothing, and it is impossible to choose rationally when
the object to be chosen is nothing.
(10) Because the quality of having warm red blood belongs to all
known birds, it must be part of their specific nature; but unknown
birds have the same specific nature as known birds; therefore, the
quality of having warm red blood must belong to the unknown as well as
the known birds, i.e., be a universal and essential property of
the species. [K.]
APPENDIX A.
THE DOCTRINE OF DIVISION.
409. Logical Division.—The term division, as technically used in logic, may be defined as the setting forth of the smaller groups which are contained under the extension of a given term. It is also defined as the separation of a genus into its constituent species. These two definitions are practically equivalent to one another. Division is to be distinguished from the setting forth of the individual objects belonging to a species, which is technically described as enumeration.
In logical division, the larger class which is divided is called the totum divisum, the smaller classes into which it is divided being the membra dividentia (dividing members). By the ground or principle of division (fundamentum sive principium divisionis) is meant that attribute or characteristic of the totum divisum upon whose modifications the division is based. A given class may of course be divided in different ways according to the particular attribute or attributes whose variations are selected as differentiating its various species. Thus, having regard to the equality or inequality of the sides, triangles may be divided into equilateral, isosceles, and scalene; or, having regard to the size of the largest angle, into obtuse-angled, right-angled, and acute-angled. Again, propositions are divisible according to their truth or falsity, or according to their quantity, or their quality, and so on.
It is sometimes said that the principle of division must be present throughout the dividing members, though constantly varied. On the other hand, it is said that in division we invariably try to think of some attribute which is predicable of certain members of the group, but not of others. The former of these statements does not 442 very well apply when we simply divide a class according to the presence or absence of some attribute (for example, candidates for the Civil Service into successful and unsuccessful) or when the attribute in question may be entirely wanting in some instances whilst present in varying degrees in other instances. In other words, given the attribute whose variations constitute our principle of division, we may have to recognise a limiting case in which it is altogether absent; thus, in dividing undergraduates according to their colleges, we may have to recognise a class of non-collegiate students. The second statement is always true when we simply contrast any given species with all the remaining species, and it may be considered adequate where we have division by contradictories. In other cases, however, it is inadequate; as, for instance, when we divide candidates who are successful in the Indian Civil Service Examination according to the province to which they are assigned.
410. Physical Division, Metaphysical Division, and Verbal Division.—Following the older logicians, we may distinguish division as defined in the preceding paragraph, that is, logical division in the strict sense, from other senses in which the term is used.
The division of an individual thing into its separate parts is called physical division or physical definition (Whately, Logic, p. 143) or partition ; as, for example, if we divide a watch into case, hands, face, and works; or a book into leaves and binding. We have, on the other hand, a logical division if we divide watches into gold, silver, &c., or into English, Swiss, American, &c.; or if we divide books into folios, quartos, &c. Bain (Logic, II. p. 197) gives the analysis of a chemical compound as an instance of logical division. It is rather an instance of physical division. In logical division the totum divisum is always predicable of all the individuals belonging to each of the membra dividentia ; for example, All men are animals, All squares are rectangles. But this is not the case in chemical analysis. We cannot say that oxygen is water, or that sulphur is vitriol, or that sodium is salt.
Distinct both from logical division and from physical division is the mental division of a thing into its separate qualities. This is called metaphysical division. We have an example when we enumerate the separate qualities of a watch, its size, accuracy, the material of which its case is composed, &c.; or when we specify the size of a book, its thickness, colour, the material of its binding, the quality of the paper of which its leaves are composed, and so on. 443 A physical division can be actually made; a watch, for example, can be taken to pieces. A metaphysical division, on the other hand, is only possible mentally. It should be added that the metaphysical division of individual objects may be made the basis of a logical division of the class to which they belong.
One further kind of division may be noticed, namely, the division of an ambiguous or equivocal term into its several significations. This is called verbal division (Clarke, Logic, p. 331) or distinction (Mansel’s Aldrich, p. 37). For example, we have to distinguish between a watch in the sense of a vigil, in the sense of a guard, and in the sense of a time-piece.
411. Rules of Logical Division.—The fundamental rules of logical division are (1) that the members of the division shall be mutually exclusive; and (2) that collectively they shall be exactly coextensive with the class that is divided. Thus if the class X is correctly divided into XA, XB, XC, the following propositions must hold good, namely, No XA is B or C, No XB is C or A, No XC is A or B, Every X is A or B or C.
The two following rules are generally added: (3) Each distinct act of division should proceed throughout upon one and the same basis or principle; (4) If the division involves more than one step, it should proceed gradually and continuously from the highest genus to the lowest species, that is to say, it should not pass suddenly from a high genus to a low species.
It may be objected that (1) and (2) ought not in a strict sense to be described as rules, but rather as constituting between them a precise statement of what is implied when we speak of a logical division. They become rules, however, in the sense that a professed logical division which fails to satisfy either of them implies relations between the members of the division which do not as a matter of fact hold good. Rules (3) and (4) are of a different character. They are rules in the sense that they must be complied with if a division is to have practical utility.
Rule (3) is not intended to condemn the processes of sub-division and co-division. Having made a division upon one principle, we may proceed to subdivide the classes thus arrived at in accordance with another principle, and so on indefinitely. A scientific classification will always consist of a hierarchy of classes thus obtained. There is again no reason why the same class should not for different purposes be divided in accordance with two or more different 444 principles, so long as these are kept distinct from one another, and the members of the different resulting divisions not confused together.
It has been said that a breach of rule (1) necessarily involves a breach of rule (3), since there cannot be any overlapping of classes so long as a division proceeds correctly upon a single principle. This does not, however, always hold good unless we interpret the word “correctly” as implying that precautions are taken to avoid any overlapping, which of course begs the question. Thus, if we divide triangles into those which have (a) a right angle, (b) an obtuse angle, (c) an acute angle, we may be said to proceed upon one principle, and yet the resulting classes are not mutually exclusive. It may, again, be argued that the classes equilateral triangle, isosceles triangle, scalene triangle (which result from a division based upon a single principle) are not mutually exclusive, since all equilateral triangles are isosceles.
This argument can only be met by saying that, in the first case, we are not proceeding upon any clear principle unless we make our division into triangles whose largest angle is an obtuse angle, a right angle, or an acute angle, respectively; nor unless, in the second case, our principle is the maximum number of sides that are equal to one another, so that an isosceles triangle is defined as a triangle that has two and only two sides equal. Any overlapping of classes is then in each case provided against; but only, it may be argued, because special precautions have been taken to attain this end. By the adoption of similar precautions, a division which proceeds “correctly” upon a single principle will also be exhaustive.
Looking at the question from the other side we may note that a division which satisfies both rule (1) and rule (2) may nevertheless be a cross-division; for it may happen that two different principles of division yield coincident results. For example, an isosceles triangle being defined as a triangle that has two and only two sides equal, there is a cross-division, but no overlapping of classes, or omission of any class contained in the totum divisum, if we divide triangles into scalene, isosceles, and equiangular; or if we divide plants into acotyledons, monocotyledons, and exogens.
As regards rule (4), it is to be observed that a division which proceeds per saltum will usually be much less effective than one in which the intermediate steps are filled in. The worst violation of this rule occurs when the division is disparate, that is, when “one of the classes into which we divide is an immediate and proximate 445 class, while others are mediate and remote” (Clarke, Logic, p. 242); as, for example, if we divide animals into invertebrates, fishes, amphibians, reptiles, birds, elephants, horses, dogs, &c.
Another rule of division is sometimes added, namely, that “none of the dividing members must be equal in extent to the divided whole” (Clarke, Logic, p. 236). When this rule is broken, the division is said to become null and void, because one of the sub-divisions contains no members. From the formal point of view, however, the observance of this rule can hardly be insisted upon. We need not regard a division as necessarily implying the actual occurrence of all its members in the universe of discourse; and the rule in question would deprive the logician of the right to employ the powerful method of division by contradictories. It may be a different matter when we are considering scientific classification from the material standpoint.
412. Division by Dichotomy.—Division by dichotomy or, as it is sometimes called more distinctively, dichotomy by contradiction is the division of a class simply with reference to the presence or absence of a given attribute or set of attributes; as, for example, when X is divided into XA and Xa (where a = not-A). An illustration is afforded by the Tree of Porphyry or Ramean Tree, in which Substances are first divided into Corporeal Substances (Bodies) and Incorporeal Substances, Bodies being then divided into Animate Bodies (Living Beings) and Inanimate Bodies, Living Beings being next divided into Sensitive Living Beings (Animals) and Insensitive Living Beings, and Animals being in their turn divided into Rational Animals (Men) and Irrational Animals. At each step in this scheme we proceed by taking contradictories. It was in praise of dichotomal division that Jeremy Bentham, who is here quoted with approval by Jevons (Principles of Science, 30, § 12), spoke of “the matchless beauty of the Ramean Tree.” When this method is employed we ensure formally that the members of our division shall be mutually exclusive and collectively exhaustive. For, by the law of contradiction, No X is both A and a ; and, by the law of excluded middle, Every X is either A or a.
It is pointed out by Spalding (Logic, p. 146) and by Jevons (Principles of Science, 30, § 9) that all logically perfect division is ultimately reducible to dichotomy, usually with the implication that some of the sub-classes which are à priori possible are not as a matter of fact to be found in the universe of discourse. Thus, 446 if we take the class X and divide it into XA and XB we imply that in the class X, A and B are never found either both present or both absent. Hence the division is equivalent to the following dichotomal division:—
Any other division, however complicated in its character, may be reduced to dichotomy in a similar way. This is interesting and important and brings out the value of dichotomy as a method of testing divisions. It must be understood, however, that in speaking of all division as ultimately reducible to dichotomy, it is not intended to imply that dichotomy usually represents our actual procedure in making divisions. Each sub-class is usually arrived at immediately by reference to some positive modification of the fundamentum divisionis ; and the different sub-classes are co-ordinate with one another. Consider, for example, the division of conic sections into parabolas, hyperbolas, ellipses, circles, and pairs of straight lines. It must be added that from the material standpoint, pure division by dichotomy is of little scientific value, because of the indefinite character of the sub-classes which are determined negatively.
413. The place of the Doctrine of Division in Logic.—The doctrine of division, as treated by the older logicians, receives little recognition by some modern writers on two very different grounds: (1) by Mill, taking the material standpoint, it is regarded as too purely formal, and hence is merged in the doctrine of scientific classification; (2) by some writers belonging to the conceptualist school, e.g., Mansel, it is rejected as not being sufficiently formal.
(1) It is true that the rules of logical division lead us a very little way in practical science. They give certain conditions which must be complied with; but they neither help us towards making good divisions, nor provide us with a test which is capable of being formally applied. Leaving dichotomy on one side, we cannot, without the aid of material knowledge, even determine whether the members of a given division are mutually exclusive and collectively 447 exhaustive. When, however, we avowedly pass beyond purely formal considerations and take up a material standpoint, then the doctrine of division should rightly give place to a doctrine of classification, which is not content with such rules as those laid down above, but seeks to indicate the principles that should serve as a guide in the classification of objects scientifically.
In regard to the use of the terms division and classification, Miss Jones draws a distinction which is of value and to which it might be well systematically to adhere. “Division and classification are the same thing looked at from different points of view; any table presenting a division presents also a classification. A division starts with unity and differentiates it; a classification starts with multiplicity, and reduces it to unity, or at least to system” (Elements of Logic, p. 123).
(2) It remains to be considered how far any treatment of division whatever can properly fall under the consideration of formal logic. From this point of view division is usually contrasted with definition. The latter of these—using the phraseology of the conceptualist logicians—expounds the intension of a concept; the former expounds its extension. But the intension of a concept is said to be far more intrinsic to it than its extension. Given a concept its intension is necessarily given; but knowledge of its extension, such as may serve to determine its division, will require a fresh appeal to the subject-matter. “Division,” says Mansel, “is not, like definition, a mental analysis of given materials: the specific difference must be added to the given attributes of the genus; and to gain this additional material, it is necessary to go out of the act of thought, to seek for new empirical data” (Prolegomena Logica, p. 192). For example, the division of members of the University of Cambridge into those in statu pupillari and members of the Senate could not be obtained without something more being given than the mere conception of a member of the University. Moreover, unless we proceed by contradictories, we cannot, when we have got our division, formally determine whether it complies with our rules or not.
The above position may be accepted, if an exception is made for division by dichotomy. Mansel, however, and some other logicians will not even allow that division by dichotomy is a formal process; and here they lay themselves open to criticism. The grounds on which their view is based are twofold:—(i) It is not sufficient that 448 the genus to be divided be given; the principle of division must be given also. “Even in the case of dichotomy by contradiction the principle of division must be given, as an addition to the attributes comprehended in the concept, before the logician can take a single step” (Prolegomena Logica, p. 207). “The division of A into B and not-B is not strictly formal; for the dividing attribute, not being part of the comprehension of A, has to be sought for out of the mere act of thought, after A has been given” (Mansel’s Aldrich, p. 38). (ii) We cannot tell à priori that both the sub-classes obtained by dichotomy really exist. How, for example, can we divide A into B and not-B when for anything we know to the contrary all A may be B? “Logically, the division of animal into mortal and immortal is as good as that into rational and irrational” (Mansel’s Aldrich, p. 38). Both these arguments are summed up in the following quotation from Mr Monck: “It is alleged indeed that Logic enables us to divide all the B’s into the B’s which are C’s and the B’s which are not C’s…… But Logic does not supply us with the term C and after we have obtained this term there are two cases in which the proposed division fails, namely, where all the B’s are C’s and where none of them are so. In either of these events the class B remains as whole and undivided as before; and whether they have occurred or not cannot be ascertained by Logic. This Division by Dichotomy, as it is called, is as much outside the province of Logic as any other kind of division” (Logic, p. 174).
As regards the first of the above arguments, there is no reason why the principle of division (A) should not be assumed given as well as the totum divisum (X). The question is whether we can then formally divide X into XA and Xa. The fact that A must be given as well as X does not prevent the possibility of formal division by dichotomy, any more than the fact that the conclusion of a syllogism is not contained in one premiss alone prevents the syllogism from being a formal process.
The force of the second argument depends upon the implication that all the sub-classes obtained as the result of a division necessarily exist in the universe of discourse. If this implication is granted, then dichotomy is certainly not a formal process; but why need we assume the existence of all the sub-classes obtained by dichotomy? Without such an assumption, our division may not have much practical utility, but its formal validity will remain unaffected. 449 We have only to make it clear that we are dividing the extension of a term, not its denotation, in the sense in which extension and denotation have been already distinguished.453 This is in keeping with the general standpoint of formal logic, which can deal with classes without regarding their existence as necessarily guaranteed in any assigned universe of discourse. If we are not allowed to apply the principle of excluded middle in formal logic and say Every X is A or a, until we know that there actually exist both XA’s and Xa’s, we shall be exceedingly hampered, and can make but little progress, especially in the treatment of complex inferences. Some schemes of symbolic logic (e.g., Jevons’s) depend essentially and explicitly upon an antecedent scheme of dichotomal division.