The theorems given for practice in logic books are useful dialectic material, but they do not fully illustrate all the categories. Logicians have no definite categories, and in selecting examples they are unconsciously biassed in favour of those that can be most easily interpreted to signify classification. The really generalistic examples are rare; the most are judgments of inherence, admitted in virtue of the assumption that inherent properties can—when it is needful to preserve the traditional notion of predication—be considered class-ideas. Theorems in perspection and concretion we do not expect to find in logic books, for these, in so far as they are distinct from association, are categories peculiar to the Berkeleyan philosophy.

Whately has the following example in association—'Lias lies above red sandstone; red sandstone lies above coal; therefore lias lies above coal.' No doubt Whately would, in syllogising this, have changed the propositions to 'Red sandstone is lying,' &c., and have assumed that 'lying above coal' is a class to which red sandstone belongs.

Here are examples of arguments in inherence—

A hot skin, quick pulse, intense thirst have invariably in my experience coexisted with fever; the person now examined exhibits these symptoms, so I infer that he has a fever.

Great width of skull between the ears is invariably found united with a destructive temperament; this animal's skull is very wide between the ears; hence it may be concluded that he has a destructive temperament.

Cloven feet belong universally—i.e. as far as our experience goes—to horned animals; we may conclude that this fossil animal, since it appears to have had cloven feet, was horned.

I.
Cloven feet	inhere with horns
Fossilanimalappearsto have had cloven feet	it is probable he had horns

When an architect, contemplating the fragments of a building, restores it in imagination after the analogy of similar buildings, we have an argument in inherence. Such speculations are generally too long and complex for analysis, but an instructive example occurs in Canon Rawlinson's Seventh Oriental Monarchy, which I will venture to quote, marking the phrases that introduce or express the rational idea. Observe the difference of style between this, which is real practical reasoning, and the trivial certainties of Syllogistic.

'What remains of this massive erection [the Takht-i-Khosru, or palace of ChosroËs Anushirwan, at Ctesiphon] is a mere fragment, which, to judge from the other extant Sassanian ruins, cannot have formed so much as one fourth part of the original edifice. Nothing has come down to our day but a single vaulted hall on the grandest scale, together with the mere outer wall of what no doubt constituted the main facade of the building. The apartments, which, according to all analogy, must have existed at the two sides, and in the rear, of the great hall, some of which should have been vaulted, have wholly perished. Imagination may supply them from the Firuzabad, or the Mashita palace; but not a trace, even of their foundations, is extant; and the details consequently are uncertain, though the general plan can scarcely be doubted. At each side of the great hall were probably two lateral ones, communicating with each other, and capable of being entered either from the hall or from the outer air. Beyond the great hall was probably a domed chamber equalling it in width, and opening upon a court, round which were a number of moderate-sized apartments. The entire building was no doubt an oblong square, of which the shorter sides seem to have measured 370 feet. It had at least three, and may not improbably have had a larger number of entrances, since it belongs to tranquil times and a secure locality.'

The most notable argument in the category of concretion is undoubtedly the inference as to the sphericity of the earth. Next is the sub-inference by Columbus that China could be reached by sailing westward from Portugal. If the syllogistic opinion were valid—that a conclusion must be absolutely true or absolutely false—the expedition of Columbus was based on a fallacy. Most people think it was eminently rational.

No one has seen the north or the south poles, and the conviction that they could be realised, if certain difficulties of transport were overcome, is a sub-inference of the same character.

Here is a common type of inference in perspection—

III.
That church	is 100 yards off
A man appears on the roof of the ch.	he is 100 ys. off

And this—

III.
That distant house	is 60 ft. high
Itappearstobescaffolded toathirdofitsheight	the scaffold is about 20 feet high

In these cases we have not seen the man or the scaffolding before, and have not measured the latter or the distance to the former: the conclusions are imaginary judgments fairly drawn from known premises.

The deciphering of hieroglyphics, cuneiform inscriptions, and remains of other dead and forgotten languages, is argument in causation. Examples cannot conveniently be quoted even in a condensed form, but this kind of reasoning is most interesting dialectically from the slightness of the analogies that are nevertheless found to give valid conclusions.

This is considered argument by Whately—

I.
Louis	is a good king
ThegovernorofFrance is Louis	thereforetheg.ofF.is a good king

The supposed case is a verbal proposition, serving to rename the subject of precedent. There is no reasoning. If we already know that Louis is a good king and is also the governor of France (the given matters of fact), there is no rational imagination involved in rearranging these data as in the proposed conclusion.

'He who calls you a man speaks truly; he who calls you a fool calls you a man; therefore he who calls you a fool speaks truly.'—A fallacy of cross reasoning, and the predicate is a class.

I.
All fools	are men
Youareaman	N. C.

'Nothing is heavier than platina; feathers are heavier than nothing; therefore feathers are heavier than platina.'—A trivial equivoque.

The following is more subtle. 'Theft is a crime; theft was encouraged by the laws of Sparta; therefore the laws of Sparta encouraged crime.'—At most the laws of Sparta encouraged one crime; but there is a fallacy of equivocation. Taking things surreptitiously from the person in whose possession they may be, is not a crime—is not theft—in a society so communistic as the Spartan. There it was encouraged as an exercise in adroitness. This example shows the necessity of knowing the matter of the argument.

'Warm countries alone produce wine; Spain is a warm country; therefore Spain produces wine.'

V.
Wine	isp.inw.countries
Spainisawarmc.	N. C.

'Meat and drink are necessaries of life; the revenues of Vitellius were spent in meat and drink; therefore the revenues of Vitellius were spent on the necessaries of life.'—Fallacy of composition: meat and drink in moderate quantities are necessaries of one life, but not food of every kind and in excessive quantities.

'He who is most hungry eats most; he who eats least is most hungry; therefore he who eats least eats most.'—A fallacy of accident: he who eats least does not at the same time eat most.

'Whatever body is in motion must move either in the place where it is, or in the place where it is not; neither of these is possible; therefore there is no such thing as motion.'—It is an abuse of reason to attempt to disprove matters of fact. The conclusion of an argument being always problematical, it can have no force against actual experience. We experience motion, therefore it cannot be disproved.

'A wise lawgiver must either recognise the rewards and punishments of a future state, or he must be able to appeal to an extraordinary Providence, dispensing them regularly in this life; Moses did not do the former, therefore he must have done the latter'—(Warburton, from Whately).—The reasoner omitted to establish that Moses was a wise lawgiver, so that the precedent does not apply to his case, except by courtesy.

'That man is independent of the caprices of fortune who places his chief happiness in moral and intellectual excellence; a true philosopher is independent of the caprices of fortune; therefore a true philosopher is one who places his chief happiness in moral and intellectual excellence.' An instance of cross reasoning.

I.
He who places	isindependent
Philosopherisindependent	N. C.

'For those who are bent on cultivating their minds by diligent study, the incitement of academical honours is unnecessary; and for the idle it is ineffectual, for such are indifferent to mental improvement; therefore the incitement of academical honours is either unnecessary or ineffectual.'

A fallacy of doubtful precedent: because two kinds of students are not benefited by the hope of honours it is prematurely concluded that no others exist who may be so benefited.

'He who bears arms at the command of the magistrate does what is lawful for a Christian; the Swiss in the French service, and the British in the American service, bore arms at the command of the magistrate; therefore they did what is lawful for a Christian.'

The conclusion is valid so far as the information given enables us to judge. If we know from other sources that the Swiss and British who are referred to, committed atrocities at the command of the magistrate, the conclusion is a fallacy of accident. In general it is lawful to obey a magistrate, but there may be particular cases when it is not.

'Anyone who is candid will refrain from condemning a book without reading it; some reviewers do not refrain from this; therefore some reviewers are not candid.'—This is cross reasoning and invalid. It is one thing to say that the uncandid do not refrain, and another that all who do not refrain are uncandid. The conclusion is taken from the latter proposition, which is not asserted.

'Everyone desires happiness; virtue is happiness; therefore everyone desires virtue.'

I.
Whoeverdesiresaneffect	desiresthecauseofthateffect
Everyonedesiresthehappiness whichiscausedbyvirtue	everyone desires virtue

The case is manifestly untrue.

'He who has a confirmed habit of any kind of action exercises no self-denial in the practice of that action; a good man has a confirmed habit of virtue; therefore he who exercises self-denial in the practice of virtue is not a good man.'—(Arist. Eth. Bk. II., from Whately.)

VI.		I.
He who has a habit, &c.	exercises no self-denial	Hewhoexercisesno self-denialinthe practiceofvirtue	is good
A good man has this habit with respect to virtue	Heexercisesno self-denialwith respecttovirtue	He who does exercise &c.	N.C.

The conclusion drawn is fallacious, the second theorem being based on contrast.

'According to theologians, a man must possess faith to be acceptable to the Deity; now he who believes all the fables of heathen mythology must possess faith; therefore such a one must, according to theologians, be acceptable to the Deity.'

'Faith' is ambiguous, meaning in the precedent, spiritual aspiration, and in the case ignorant credulity.

'No evil should be allowed that good may come of it; all punishment is an evil; therefore no punishment should be allowed that good may come of it.'—'Evil' is ambiguous, meaning wrong-doing in the precedent and pain in the case; the conclusion is therefore fallacious.

'The principles of justice are variable; the appointments of nature are invariable; therefore the principles of justice are no appointments of nature.'—(Arist. Eth. Bk. V., from Whately.) The terms 'principles of justice' and 'nature' require to be defined. It might be said that justice is one principle, everywhere and always the same, and that only its embodiments in law and custom are variable.

'What happens every day is not improbable; some things, against which the chances are many thousands to one, happen every day; therefore some things against which the chances are many thousands to one, are not improbable.'—A fallacy of division: that improbable things in general happen every day does not render the occurrence of any one a probable event.

'Protection from punishment is plainly due to the innocent; therefore, as you maintain that this person ought not to be punished, it appears that you are convinced of his innocence.'—A fallacy of cross reasoning.

I.
Innocent persons	deserveprotection
This person deserves protection	N. C.

'He who cannot possibly act otherwise than he does, has neither merit nor demerit in his action; a liberal and benevolent man cannot possibly act otherwise than he does in relieving the poor; therefore such a man has neither merit nor demerit in his action.'—To 'have merit in an action' is scarcely intelligible. A man's merit is in his character, and his actions are effects and signs of character.

'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.'

An instance of the fallacy of division: what is true of the whole contents of the net is not necessarily true of a portion of the contents.

'A desire to gain by another's loss is a violation of the tenth commandment; all gaming, therefore, since it implies a desire to profit at the expense of another, involves a breach of the tenth commandment.'—A valid argument in inherence.

All desire to gain by, &c.	violates
Gaminginvolvesthiskindofdesire	it violates

'He that destroys a man who usurps despotic power in a free country, deserves well of his countrymen; Brutus destroyed Caesar, who usurped despotic power in Rome; therefore he deserved well of the Romans.'—If Rome was 'a free country' the conclusion is valid.

'No fish suckles its young; the whale suckles its young; the whale is therefore no fish.'

What suckles	is no-fish
Whale suckles	it is no-fish

'This explosion must have been occasioned by gunpowder, for nothing else would have possessed sufficient force.'

V.
Explosions of a certain destructiveness	can only be occasioned by gunpowder
Thisexplosionhasthat destructiveness	it must have been occasioned by gunpowder

'Every man should be moderate, for excess will cause disease.'

V.
To avoid disease	moderationisrequisite
Everymanshouldavoid disease	every man should be moderate

'Blessed are the merciful, for they shall obtain mercy.'

V.
To obtain mercy	is blessed
Thosewhoshowmercy obtain mercy	they must be considered blessed

'Some speculative men are unworthy of trust; for they are unwise, and no unwise man can be trusted.'

I.
Unwise men	are not to be trusted
Some speculative men are unwise	theyarenottobetrusted

'No idle person can be a successful writer of history; therefore Hume, Macaulay, Hallam and Grote must have been industrious.'

I.
Successful historians	are not idle persons
Humeandtherestwere successful historians	theycannothavebeen idle persons

'Lithium is an element; for it is an alkali-producing substance, which is a metal, which is an element.'—Fallacy of no-application.

Every alk. prod. subst.	is a metal	Every metal	is an element
L. is alk. p. subst.	itisametal	L. is a metal	itisanelement

'Rational beings are accountable for their actions; brutes not being rational, are therefore exempt from responsibility.'

I.
Rational beings	are accountable
Brutesnotrational	N. C.

'Whatever tends to withdraw the mind from pursuits of a low nature deserves to be promoted; classical learning does this, since it gives us a taste for intellectual enjoyments; therefore it deserves to be promoted.'

V.		I.
Whatever gives	tends	Whatever tends	deserves
Learning gives	ittends	Learning tends	itdeserves

'Bacon was a great lawyer and statesman; and as he was also a philosopher, we may infer that any philosopher may be a great lawyer and statesman.'—The theorem infers the general inherence of philosophy with eminence in law and politics, from the single instance of Bacon: it is evidently a fallacy of doubtful precedent.

'Snowdon is the highest mountain in England and Wales. Snowdon is not so high as Ben Nevis. Therefore the highest mountain in England and Wales is not so high as Ben Nevis.'

This means: 'the highest mountain in England and Wales is called Snowdon, and it is not so high as Ben Nevis.' The apparent conclusion merely repeats a part of the information given already. There is no case. The following is a theorem of the same kind—

'Lithium is the lightest metal known. Lithium is the metal indicated by one bright line in the spectrum. Therefore the lightest metal known is the metal indicated by a spectrum of one bright line.'

'If ye were Abraham's children, ye would do the works of Abraham.'

VI.
Abraham's children	do his works
If ye were his children	yewoulddohisworks

'Since all metals are elements, the most rare of all the metals must be the most rare of all the elements.'—There is a suppressed precedent to the effect that the most rare individual of a species must be the most rare of its genus, which may or may not be true.

'All vice is odious; but avarice is a vice; for it makes men slaves; therefore avarice is odious.'

V.		I.
Whateverenslaves	is a vice	All vice	is odious
Avarice enslaves	itisavice	Avarice is a vice	itisodious

'Bucephalus is a horse; a horse is a quadruped; a quadruped is an animal; an animal is a substance; therefore Bucephalus is a substance.'

All horses	are quads.	Quads.	animals	Animal	substance
Bu. is a h.	heisq.	B. is quad.	heisan.	Bu. is an.	heissubst.

This is what logicians call a Sorites. There may be a chain of valid arguments, in which the conclusion of one is precedent or case of the next; but the propositions just quoted do not make an argument, being merely a string of classifications. If we know what Bucephalus and substance mean, we know by perception that Bucephalus is a substance.

'Every being is then happy when it acquires the proper perfection of its nature; and consequently all vital beings are capable of receiving felicity that are capable of arriving at the perfection of their nature.'

I.
Everybeingthatacquires	is happy
Allvitalbeingscapable of acquiring	mustbecapableofreceiving felicity

This is perilously near tautology; it can be saved only by assuming that 'every being capable of happiness' is a more extensive class than 'all vital beings capable of arriving at the perfection of their nature.'

'The soul's debility is not owing to her lapse into matter; for as this lapse is voluntary, the soul must have sinned prior to her descent.'

VI.
Voluntary lapse	proves prior sin
Soul's lapse is v.	herdebilitymusthavebeen antecedent to lapse

Cogito ergo sum.—

I.
Whatever thinks	is
I think	ImustbelievethatIam

That we exist is the most certain fact we know: it cannot be strengthened by any argumentation. If we can doubt that we are, we can with better reason doubt that we think.

Here is Hamilton's example of a disjunctive syllogism, which he considered a valid argument—'The hope of immortality is either a rational expectation or an illusion; but the hope of immortality is a rational expectation; therefore the hope of immortality is not an illusion.' It is a flagrant tautologism.

'If man be not a morally responsible being, he must want either the power of recognising moral good (as an intelligent agent), or the power of willing it (as a free agent); but man wants neither of these powers; therefore man is a morally responsible being.'

Adopted by Hamilton from Krug and given as valid. It is first a fallacy of contrast, and if amended in this respect it would still be a fallacy of tautology.

I.
If m. lacked certain pp.	he would be irresp.
He does not lack these pp.	N. C.

'If Aeschines joined in the public rejoicings, he is inconsistent; if he did not, he is unpatriotic; but he either joined, or not, therefore he is either inconsistent or unpatriotic.'—An excellent specimen of logicians' logic: on a par with this—If it is fine weather, I go; if it rains, I stay; it must either rain or be fine, therefore I must either go or stay.

'If the world were eternal, the most useful arts, such as painting, &c. would be of unknown antiquity: and on the same supposition there would be records long prior to the Mosaic; and likewise the sea and land in all parts of the globe might be expected to maintain the same relative situations now as formerly: but none of these is the fact: therefore the world is not eternal.'

Ifsomethingsweredifferent from what they are	the w. would be eternal
They are not different	N. C.

'If the world existed from eternity, there would be records prior to the Mosaic; and if it were produced by chance, it would not bear marks of design: there are no records prior to the Mosaic, and the world does bear marks of design: therefore it neither existed from eternity, nor is it the work of chance.'

Two theorems are here mixed together, both fallacies of contrast—

Existence of records	would prove the w. etern.
Records do not exist	the w. is non-eternal

Non-existenceofmarks	wd. pr. w. made by chance
The marks exist	w.wasnotmadebychance

'If this man were wise, he would not speak irreverently of Scripture in jest; and if he were good he would not do so in earnest; but he does it, either in jest or earnest; therefore he is either not wise or not good.'

As it stands this is quite circular, but it might be rendered valid by generalisation:—

VI.
Tospeakirrev.ofScr.in jest or earnest	indicates that a man is not wise or not good
This man does it	wemustinferthatheisnot w. or not g.

'If virtue were a habit worth acquiring, it must ensure either power, or wealth, or honour, or pleasure; but virtue ensures none of these; therefore virtue is not a habit worth attaining.' Fallacy of contrast—

I.
What ensures	is worth
V.doesnotensure	N. C.

'If men are not likely to be influenced in the performance of a known duty by taking an oath to perform it, the oaths commonly administered are superfluous; if they are likely to be so influenced, everyone should be made to take an oath to behave rightly throughout his life; but one or other must be the case; therefore either the oaths commonly administered are superfluous, or everyone should be made to take an oath to behave rightly throughout his life.'—This will be more intelligible if contracted thus: If oaths fail to influence they are superfluous; if they influence they should be obligatory; but they either influence or do not; therefore they are either superfluous or should be obligatory. There is no argument; the alternative conclusions merely repeat the alternative precedents.

'If virtue is voluntary, vice is voluntary; but virtue is voluntary; therefore so is vice.' (Arist. Eth. Bk. III. quoted by Whately.) This is a circular way of saying that we believe it to be a fact that vice is voluntary. The argumentative form is probably supposed to give the assertion greater weight than it would have if expressed as a perceptual judgment.

This is valid argument, according to Hamilton—'If man were suited to live out of society, he would either be a god or a beast; but man is neither a god nor a beast; therefore he is not suited to live out of society.'—It has faults of contrast and tautology.

I.
Only gods and beasts	are suited
Manisneitherg.norb.	N. C.

'If iron is impure, it is brittle; but this iron is impure; therefore it is brittle.'—A valid dogmatic argument.

I.
Impure iron	is brittle
Thisironisimp.	it must be br.

'If the weather is fine, we shall go into the country; now the weather is fine, therefore we shall go into the country.'—We never get beyond the simple judgment that our going into the country is associated with fine weather.

The following is valid:—'As often as the weather is fine, my brother has a habit of going into the country; if the weather be fine to-morrow I infer that he will go into the country.' Here a particular hypothetical case is illustrated by reference to a general habit.

'As often as the weather is fine my brother goes into the country; if it be not fine to-morrow I conclude that he will not go into the country.'—A fallacy of contrast: we are not informed in the antecedents what the brother does on wet days.

'If there are sharpers in the company we ought not to gamble; but there are no sharpers in the company; therefore we ought to gamble.'

I.
Presence of sh.	forbids to gamble
Absence of sh.	N. C.

'Logic as it was cultivated by the schoolmen proved a fruitless study; therefore logic as it is cultivated at the present day must be a fruitless study likewise.'—We must take the conclusion as valid, until we know in what respects modern logic is superior to scholastic logic.

'Few treatises of science convey important truths, without any admixture of error, in a perspicuous and interesting form: therefore though a treatise would deserve much attention which should possess such excellence, it is plain that few treatises of science do deserve much attention.'

This means no more than that treatises of a certain excellence would deserve attention, and that there are few of them. There is no argument.

'Some objects of great beauty answer no other purpose but to gratify the sight: many flowers have great beauty; and many of them accordingly answer no other purpose than to gratify the sight.'

I.
Someobj.whichanswer	are beautiful
Manyflowersarebeaut.	N. C.

'None but Whites are civilised; the ancient Germans were Whites; therefore they were civilised.'

I.
Allcivilisednations	are Whites
Anc.Ger.wereWh.	N. C.

'Wilkes was a favourite with the populace; he who is a favourite with the populace must understand how to manage them; he who understands how to manage them must be well acquainted with their character; he who is well acquainted with their character must hold them in contempt; therefore Wilkes must have held the populace in contempt.'

Favourites	must kn. how to manage	Hewhokn.how to manage	must be acquainted	He who is acq.	must despise
W. was a fav.	heknewhowtoman.	W. knew	hewasacq.	W.wasacq.	hemusthavedesp.

'Something has existed from eternity. For since something now is, it is manifest that something always was. Otherwise the things that now are must have risen out of nothing, absolutely and without cause. Which is a plain contradiction in terms. For to say a thing is produced, and yet that there is no cause at all of that production, is to say that something is effected when it is effected by nothing, that is, at the same time when it is not effected at all. Whatever exists has a cause of its existence, either in the necessity of its own nature, and then it must have been of itself eternal: or in the will of some other being, and then that other being must, at least in the order of nature and causality, have existed before it.'

In this theorem we have a case—'Something is'; and a conclusion—'Something has existed from eternity.' The reasoner seeks a credible or conceivable precedent by which to connect that conclusion with the case.

We are offered a choice of two theorems. The first is untenable, for we have never had the experience that is given as precedent; it is also tautological, as the 'something' of the case is the 'whatever' of the precedent.

V.
Whatever exists in the necessity of its own nature	has existed from eternity
'Something'existsinthenecessity of its own nature	ithasexistedfrometernity

It is not inconceivable that something should be self-existent, but we know nothing as to its being eternal. We are not familiar enough with self-existent things and eternal things to warrant us in asserting dogmatically that where the first quality is, there also must be the second.

The next theorem is that everything must be caused, and that causation involves a regressum ad infinitum. On this principle there must have been things for an eternity backwards. According to the theory of causation given in section xxvii, a true beginning is reached when we discover the motive, design and power that produced an effect. It is not necessary to ask next what caused that motive, design and power. The infinite regress is applicable only to material sequence, in which there is no proper beginning or end. The author of the above argument seems to be trying to combine the notion of causation by will with that of infinite regress. But his language is too obscure to make it certain what he means exactly.

The three following theorems—in a diluted form—occur in an otherwise excellent work on the politics and social life of the ancient Greeks.

'The Athenians who opposed the union of Greece and Macedonia were old men, and the result was mischievous; other similar instances are found in history; therefore the government of old men is always mischievous.'—A fallacy of false generality. Everybody knows that some old men have been wise governors. Cicero, from his experience, drew the opposite conclusion—that the only safe rulers were old men.

'All old political leaders are mischievous; Gladstone is old; therefore he is to be considered politically mischievous.'—Even were the precedent not false the argument is superfluous, for the effect of Gladstone's politics is now matter of fact or history.

'Gladstone is politically mischievous; he advocates Home Rule for Ireland; therefore Irish Home Rule must be mischievous.'—A fallacy of division: a political leader might on the whole be mischievous, but his measures need not on that account be each and every one mischievous.

If dialectic were taught generally and on a rational method, a responsible author would avoid bad reasoning of this sort as carefully as he avoids bad grammar, vulgar imagery, or faulty arithmetic.

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