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is equal to

All cases of A not being B are cases of C being D. This is a case of C being D.

§ 769. So far as regards the consequent, the two species of complex reasoning hitherto discussed are identical both in appearance and reality. The apparent difference of procedure in the case of the antecedent, namely, that it is affirmed in the partly conjunctive, but denied in the disjunctive syllogism, is due merely to the fact that in the disjunctive proposition the truth of the consequent is involved in the falsity of the antecedent, so that the antecedent being necessarily negative, to deny it in appearance is in reality to assert it.


_The Disjunctive Syllogism regarded as an Immediate Inference_.

§ 770. If no stress be laid on the transition from disjunctive hypothesis to fact, the disjunctive syllogism will run with the same facility as its predecessor into the moulds of immediate inference.

§ 771.

_Denial of Antecedent_. Subalternation.

Either A is B or C is D, Every case of A not being B is a case of C being D. .’. A not being B, C is D. .’. Some case of A not being B is a case of C being D.

§ 772.

_Denial of Consequent_. Conversion by Contraposition + Subalternation.

Either A is B or C is D. All cases of A not being B are cases of C being D. .’. C not being D, A is B .’. All cases of C not being D are cases of A being B.
.’. Some case of C not being D is a case of A being B.

§ 773. Similarly the two invalid types of disjunctive syllogism will be found to coincide with fallacies of immediate inference.

§ 774.

_Affirmation of Antecedent_. Contraposition without Conversion.

Either A is B or C is D. All cases of A not being B are cases of C being D.
.’. A being B, C is not D .’. All cases of A being B are cases of C not being D.

§ 775. The affirmation of the antecedent thus comes under the formula–

All not-A is B,
.’. All A is not-B,

a form of inference which cannot hold except where A and B are known to be incompatible. Who, for instance, would assent to this?–

All non-boating men play cricket.
.’. All boating men are non-cricketers.

§ 776.

_Affirmation of Consequent_. Simple Conversion of A.

Either A is B or C is D. All cases of A not being B are cases of C being D.
.’.C being D, A is not B. .’. All cases of C being D are cases of A not being B.

§ 777. We may however argue in this way–

Conversion of A per accidens. Either A is B or C is D. All cases of A not being B are cases of C being D. .’. C being D, A is sometimes B. .’. Some cases of C being D are cases of A not being B.

The men who pass this examination must have either talent or industry. .’. Granting that they are industrious, they may be without talent.


_Of the Mixed Form of Complex Syllogism_.

§ 778. Under this head are included all syllogisms in which a conjunctive is combined with a disjunctive premiss. The best known form is

_The Dilemma_.

§ 779. The Dilemma may be defined as–

A complex syllogism, having for its major premiss a conjunctive proposition with more than one antecedent, or more than one consequent, or both, which (antecedent or consequent) the minor premiss disjunctively affirms or denies.

§ 780. It will facilitate the comprehension of the dilemma, if the following three points are borne in mind–

(1) that the dilemma conforms to the canon of the partly conjunctive syllogism, and therefore a valid conclusion can be obtained only by affirming the antecedent or denying the consequent;

(2) that the minor premiss must be disjunctive;

(3) that if only the antecedent be more than one, the conclusion will be a simple proposition; but if both antecedent and consequent be more than one, the conclusion will itself be disjunctive.

§ 781. The dilemma, it will be seen, differs from the partly conjunctive syllogism chiefly in the fact of having a disjunctive affirmation of the antecedent or denial of the consequent in the minor, instead of a simple one. It is this which constitutes the essence of the dilemma, and which determines its possible varieties. For if only the antecedent or only the consequent be more than one, we must, in order to obtain a disjunctive minor, affirm the antecedent or deny the consequent respectively; whereas, if there be more than one of both, it is open to us to take either course. This gives us four types of dilemma.

§ 782.

(1). _Simple Constructive._

If A is B or C is D, E is F.
Either A is B or C is D.
.’. E is F.

(2). _Simple Destructive._

If A is B, C is D and E is F.
Either C is not D or E is not F.
.’. A is not B.

(3). _Complex Constructive._

If A is B, C is D; and if E is F, G is H. Either A is B or E is F.
.’. Either C is D or G is H.

(4). _Complex Destructive_.

If A is B, C is D; and if E is F, G is H. Either C is not D or G is not H.
.’. Either A is not B or E is not F.

§ 783.

(1). _Simple Constructive_.

If she sinks or if she swims, there will be an end of her. She must either sink or swim.
.’. There will be an end of her.

(2). _Simple Destructive_.

If I go to Town, I must pay for my ticket and pay my hotel bill. Either I cannot pay for my ticket or I cannot pay my hotel bill. .’. I cannot go to Town.

(3). _Complex Constructive_.

If I stay in this room, I shall be burnt to death, and if I jump out of the window, I shall break my neck. I must either stay in the room or jump out of the window. .’. I must either be burnt to death or break my neck.

(4). _Complex Destructive_.

If he were clever, he would see his mistake; and if he were candid, he would acknowledge it. Either he does not see his mistake or he will not acknowledge it. .’. Either he is not clever or he is not candid.

§ 784. It must be noticed that the simple destructive dilemma would not admit of a disjunctive consequent. If we said,

If A is B, either C is D or E is F,
Either C is not D or E is not F,

we should not be denying the consequent. For ‘E is not F’ would make it true that C is D, and ‘C is not D’ would make it true that E is F; so that in either case we should have one of the alternatives true, which is just what the disjunctive form ‘Either C is D or E is F’ insists upon.

§ 785. In the case of the complex constructive dilemma the several members, instead of being distributively assigned to one another, may be connected together as a whole–thus–

If either A is B or E is F, either C is D or G is H. Either A is B or E is F.
.’. Either C is D or G is H.

In this shape the likeness of the dilemma to the partly conjunctive syllogism is more immediately recognisable. The major premiss in this shape is vaguer than in the former. For each antecedent has now a disjunctive choice of consequents, instead of being limited to one. This vagueness, however, does not affect the conclusion. For, so long as the conclusion is established, it does not matter from which members of the major its own members flow.

§ 786. It must be carefully noticed that we cannot treat the complex destructive dilemma in the same way.

If either A is B or E is F, either C is D or G is H. Either C is not D or G is not H.

Since the consequents are no longer connected individually with the antecedents, a disjunctive denial of them leaves it still possible for the antecedent as a whole to be true. For ‘C is not D’ makes it true that G is H, and ‘G is not H’ makes it true that C is D. In either case then one is true, which is all that was demanded by the consequent of the major. Hence the consequent has not really been denied.

§ 787. For the sake of simplicity we have limited the examples to the case of two antecedents or consequents. But we may have as many of either as we please, so as to have a Trilemma, a Tetralemma, and so on.


If A is B, C is D; and if E is F, G is H; and if K is L, M is N. Either A is B or E is F or K is L.
.’. Either C is D or G is H or K is L.

§ 788. Having seen what the true dilemma is, we shall now examine some forms of reasoning which resemble dilemmas without being so.

§ 789. This, for instance, is not a dilemma–

If A is B or if E is F, C is D.
But A is B and E is F.
.’. C is D.

If he observes the sabbath or if he refuses to eat pork, he is a Jew.
But he both observes the sabbath and refuses to eat pork. .’. He is a Jew.

What we have here is a combination of two partly conjunctive syllogisms with the same conclusion, which would have been established by either of them singly. The proof is redundant.

§ 790. Neither is the following a dilemma–

If A is B, C is D and E is F.
Neither C is D nor E is F.
.’. A is not B.

If this triangle is equilateral, its sides and its angles will be equal.
But neither its sides nor its angles are equal. .’. It is not equilateral.

This is another combination of two conjunctive syllogisms, both pointing to the same conclusion. The proof is again redundant. In this case we have the consequent denied in both, whereas in the former we had the antecedent affirmed. It is only for convenience that such arguments as these are thrown into the form of a single syllogism. Their real distinctness may be seen from the fact that we here deny each proposition separately, thus making two independent statements–C is not D and E is not F. But in the true instance of the simple destructive dilemma, what we deny is not the truth of the two propositions contained in the consequent, but their compatibility; in other words we make a disjunctive denial.

§ 791. Nor yet is the following a dilemma–

If A is B, either C is D or E is F.
Neither C is D nor E is F.
.’. A is not B.

If the barometer falls there will be either wind or rain. There is neither wind nor rain.
.’. The barometer has not fallen.

What we have here is simply a conjunctive major with the consequent denied in the minor. In the consequent of the major it is asserted that the two propositions, ‘C is D’ and ‘E is F’ cannot both be false; and in the minor this is denied by the assertion that they are both false.

§ 792. A dilemma is said to be rebutted or retorted, when another dilemma is made out proving an opposite conclusion. If the dilemma be a sound one, and its premisses true, this is of course impossible, and any appearance of contradiction that may present itself on first sight must vanish on inspection. The most usual mode of rebutting a dilemma is by transposing and denying the consequents in the major–

If A is B, C is D; and if E is F, G is H. Either A is B or E is F.
.’. Either C is D or G is H.

The same rebutted–

If A is B, G is not H; and if E is F, C is not D. Either A is B or E is F.
.’. Either G is not H or C is not D. = Either C is not D or G is not H.

§ 793. Under this form comes the dilemma addressed by the Athenian mother to her son–‘Do not enter public life: for, if you say what is just, men will hate you; and, if you say what is unjust, the gods will hate you’ to which the following retort was made–‘I ought to enter public life: for, if 1 say what is just, the gods will love me; and, if 1 say what is unjust, men will love me.’ But the two conclusions here are quite compatible. A man must, on the given premisses, be both hated and loved, whatever course he takes. So far indeed are two propositions of the form

Either C is D or G is H,
and Either C is not D or G is not H,

from being incompatible, that they express precisely the same thing when contradictory alternatives have been selected, e.g.–

Either a triangle is equilateral or non-equilateral. Either a triangle is non-equilateral or equilateral.

§ 794. Equally illusory is the famous instance of rebutting a dilemma contained in the story of Protagoras and Euathlus (Aul. Gell. Noct. Alt. v. 10), Euathlus was a pupil of Protagoras in rhetoric. He paid half the fee demanded by his preceptor before receiving lessons, and agreed to pay the remainder when he won his first case. But as he never proceeded to practise at the bar, it became evident that he meant to bilk his tutor. Accordingly Protagoras himself instituted a law-suit against him, and in the preliminary proceedings before the jurors propounded to him the following dilemma–‘Most foolish young man, whatever be the issue of this suit, you must pay me what I claim: for, if the verdict be given in your favour, you are bound by our bargain; and if it be given against you, you are bound by the decision of the jurors.’ The pupil, however, was equal to the occasion, and rebutted the dilemma as follows. ‘Most sapient master, whatever be the issue of this suit, I shall not pay you what you claim: for, if the verdict be given in my favour, I am absolved by the decision of the jurors; and, if it be given against me, I am absolved by our bargain.’ The jurors are said to have been so puzzled by the conflicting plausibility of the arguments that they adjourned the case till the Greek Kalends. It is evident, however, that a grave injustice was thus done to Protagoras. His dilemma was really invincible. In the counter-dilemma of Euathlus we are meant to infer that Protagoras would actually lose his fee, instead of merely getting it in one way rather than another. In either case he would both get and lose his fee, in the sense of getting it on one plea, and not getting it on another: but in neither case would he actually lose it.

§ 795. If a dilemma is correct in form, the conclusion of course rigorously follows: but a material fallacy often underlies this form of argument in the tacit assumption that the alternatives offered in the minor constitute an exhaustive division. Thus the dilemma ‘If pain is severe, it will be brief; and if it last long it will be slight,’ &c., leaves out of sight the unfortunate fact that pain may both be severe and of long continuance. Again the following dilemma–

If students are idle, examinations are unavailing; and, if they are industrious, examinations are superfluous, Students are either idle or industrious, .’. Examinations are either unavailing or superfluous,

is valid enough, so far as the form is concerned. But the person who used it would doubtless mean to imply that students could be exhaustively divided into the idle and the industrious. No deductive conclusion can go further than its premisses; so that all that the above conclusion can in strictness be taken to mean is that examinations are unavailing, when students are idle, and superfluous, when they are industrious–which is simply a reassertion as a matter of fact of what was previously given as a pure hypothesis.


_Of the Reduction of the Dilemma._

§ 796. As the dilemma is only a peculiar variety of the partly conjunctive syllogism, we should naturally expect to find it reducible in the same way to the form of a simple syllogism. And such is in fact the case. The constructive dilemma conforms to the first figure and the destructive to the second.

1) _Simple Constructive Dilemma_.

If A is B or if E is F, C is D. All cases of either A being B or E being F are cases of C being D. Either A is B or E is F. All actual cases are cases of either A being B OP E being F. .’. C is D. .’. All actual cases are cases of C being D.

(2) _Simple Destructive_.

If A is B, C is D and E is F. All cases of A being B are cases of C being D and E being F. Either C is not D or E is not F. No actual cases are cases of C being D and E being F.
.’. A is not B. .’. No actual cases are cases of A being B.

(3) _Complex Constructive_.
If A is B, C is D; and if E is F, All cases of either A being B or G is H. being F are cases of either C being D or G being H.
Either A is B or E is F. All actual cases are cases of either A being B or E being F. .’. Either C is D or G is H. .’. All actual cases are cases of either C being D or G being H.

(4) _Complex Destructive_.

If A is B, C is D; and if E is F, All cases of A being B and E being F G is H. are cases of C being D and G being H.
Either C is not D Or G is No actual cases are cases of C being not H D and G being H. Either A is not B or E is No actual cases are cases of A being not F. B and E being F.

§ 797. There is nothing to prevent our having Darii, instead of Barbara, in the constructive form, and Baroko, instead of Camestres, in the destructive. As in the case of the partly conjunctive syllogism the remaining moods of the first and second figure are obtained by taking a negative proposition as the consequent of the major premiss, e.g.–

_Simple Constructive_. Celarent or Ferio. If A is B or if E is F, C is not D No cases of either A being B or E being F are cases of C being D. Either A is B or E is F. All (or some) actual cases are cases of either A being B or E being F .’. C is not D. .’. All (or some) actual cases are not cases of C being D.


_Of the Dilemma regarded as an Immediate Inference._

§ 798. Like the partly conjunctive syllogism, the dilemma can be expressed under the forms of immediate inference. As before, the conclusion in the constructive type resolves itself into the subalternate of the major itself, and in the destructive type into the subalternate of its contrapositive. The simple constructive dilemma, for instance, may be read as follows–

If either A is B or E is F, C is D,
.’. Either A being B or E being F, C is D,

which is equivalent to

Every case of either A being B or E being F is a case of C being D. .’. Some case of either A being B or E being F is a case of C being D.

The descent here from ‘every’ to ‘some’ takes the place of the transition from hypothesis to fact.

§ 799. Again the complex destructive may be read thus–

If A is B, C is D; and if E is F, G is H, .’. It not being true that C is D and G is H, it is not true that A is B and E is F,

which may be resolved into two steps of immediate inference, namely, conversion by contraposition followed by subalternation–

All cases of A being B and E being F are cases of C being D and G being H.
.’. Whatever is not a case of C being D and G being H is not a case of A being B and E being F.
.’. Some case which is not one of C being D and G being H is not a case of A being B and E being F.


_Of Trains of Reasoning._

§ 800. The formal logician is only concerned to examine whether the conclusion duly follows from the premisses: he need not concern himself with the truth or falsity of his data. But the premisses of one syllogism may themselves be conclusions deduced from other syllogisms, the premisses of which may in their turn have been established by yet earlier syllogisms. When syllogisms are thus linked together we have what is called a Train of Reasoning.

§ 801. It is plain that all truths cannot be established by reasoning. For the attempt to do so would involve us in an infinite regress, wherein the number of syllogisms required would increase at each step in a geometrical ratio. To establish the premisses of a given syllogism we should require two preceding syllogisms; to establish their premisses, four; at the next step backwards, eight; at the next, sixteen; and so on ad infinitum. Thus the very possibility of reasoning implies truths that are known to us prior to all reasoning; and, however long a train of reasoning may be, we must ultimately come to truths which are either self-evident or are taken for granted.

§ 802. Any syllogism which establishes one of the premisses of another is called in reference to that other a Pro-syllogism, while a syllogism which has for one of its premisses the conclusion of another syllogism is called in reference to that other an Epi-syllogism.

_The Epicheirema_.

§ 803. The name Epicheirema is given to a syllogism with one or both of its premisses supported by a reason. Thus the following is a double epicheirema–

All B is A, for it is E.
All C is B, for it is F.
.’. All C is A.

All virtue is praiseworthy, for it promotes the general welfare. Generosity is a virtue, for it prompts men to postpone self to others. .’. Generosity is praiseworthy.

§ 804. An epicheirema is said to be of the first or second order according as the major or minor premiss is thus supported. The double epicheirema is a combination of the two orders.

§ 805. An epicheirema, it will be seen, consists of one syllogism fully expressed together with one, or, it may be, two enthymemes (§ 557). In the above instance, if the reasoning which supports the premisses were set forth at full length, we should have, in place of the enthymemes, the two following pro-syllogisms–

(i) All E is A.
All B is E.
.’. All B is A.

Whatever promotes the general welfare is praiseworthy. Every virtue promotes the general welfare. .’. Every virtue is praiseworthy.

(2) All F is B.
All C is F.
.’. All C is B.

Whatever prompts men to postpone self to others is a virtue. Generosity prompts men to postpone self to others. .’. Generosity is a virtue.

§ 806. The enthymemes in the instance above given are both of the first order, having the major premiss suppressed. But there is nothing to prevent one or both of them from being of the second order–

All B is A, because all F is.
All C is B, because all F is.
.’. All C is A.

All Mahometans are fanatics, because all Monotheists are. These men are Mahometans, because all Persians are. .’. These men are fanatics.

Here it is the minor premiss in each syllogism that is suppressed, namely,

(1) All Mahometans are Monotheists.

(2) These men are Persians.

_The Sorites_.

§ 807. The Sorites is the neatest and most compendious form that can be assumed by a train of reasoning.

§ 808. It is sometimes more appropriately called the chain-argument, and map be defined as–

A train of reasoning, in which one premiss of each epi-syllogism is supported by a pro-syllogism, the other being taken for granted.

This is its inner essence.

§ 809. In its outward form it may be described as–A series of propositions, each of which has one term in common with that which preceded it, while in the conclusion one of the terms in the last proposition becomes either subject or predicate to one of the terms in the first.

§ 810. A sorites may be either–

(1) Progressive,

or (2) Regressive.

_Progressive Sorites_.

All A is B.
All B is C.
All C is D.
All D is E.
.’. All A is E.

_Regressive Sorites_.

All D is E.
All C is D.
All B is C.
All A is B.
.’. All A is E.

§ 811. The usual form is the progressive; so that the sorites is commonly described as a series of propositions in which the predicate of each becomes the subject of the next, while in the conclusion the last predicate is affirmed or denied of the first subject. The regressive form, however, exactly reverses these attributes; and would require to be described as a series of propositions, in which the subject of each becomes the predicate of the next, while in the conclusion the first predicate is affirmed or denied of the last subject.

§ 812. The regressive sorites, it will be observed, consists of the same propositions as the progressive one, only written in reverse order. Why then, it may be asked, do we give a special name to it, though we do not consider a syllogism different, if the minor premiss happens to precede the major? It is because the sorites is not a mere series of propositions, but a compressed train of reasoning; and the two trains of reasoning may be resolved into their component syllogisms in such a manner as to exhibit a real difference between them.

§ 813. The Progressive Sorites is a train of reasoning in which the minor premiss of each epi-syllogism is supported by a pro-syllogism, while the major is taken for granted.

§ 814. The Regressive Sorites is a train of reasoning in which the major premiss of each epi-syllogism is supported by a pro-syllogism, while the minor is taken for granted.

_Progressive Sorites_.
(i) All B is C.
All A is B.
.’. All A is C.

(2) All C is D.
All A is C.
.’. All A is D.

(3) All D is E.
All A is D.
.’. All A is E.

_Regressive Sorites_.
(1) All D is E.
All C is D.
.’. All C is E.

(2) All C is E.
All B is C.
.’. All B is E.

(3) All B is E.
All A is B.
.’. All A is E.

§ 815. Here is a concrete example of the two kinds of sorites, resolved each into its component syllogisms–

_Progressive Sorites_.

All Bideford men are Devonshire men. All Devonshire men are Englishmen.
All Englishmen are Teutons.
All Teutons are Aryans.
.’. All Bideford men are Aryans.

(1) All Devonshire men are Englishmen. All Bideford men are Devonshire men.
.’. All Bideford men are Englishmen.

(2) All Englishmen are Teutons.
All Bideford men are Englishmen. .’. All Bideford men are Teutons.

(3) All Teutons are Aryans.
All Bideford men are Teutons.
.’. All Bideford men are Aryans.

_Regressive Sorites._

All Teutons are Aryans.
All Englishmen are Teutons.
All Devonshiremen are Englishmen.
All Bideford men are Devonshiremen. .’. All Bideford men are Aryans.

(1) All Teutons are Aryans.
All Englishmen are Teutons.
.’. All Englishmen are Aryans.

(2) All Englishmen are Aryans.
All Devonshiremen are Englishmen. .’. All Devonshiremen are Aryans.

(3) All Devonshiremen are Aryans.
All Bideford men are Devonshiremen. .’. All Bideford men are Aryans.

§ 816. When expanded, the sorites is found to contain as many syllogisms as there are propositions intermediate between the first and the last. This is evident also on inspection by counting the number of middle terms.

§ 817. In expanding the progressive form we have to commence with the second proposition of the sorites as the major premiss of the first syllogism. In the progressive form the subject of the conclusion is the same in all the syllogisms; in the regressive form the predicate is the same. In both the same series of means, or middle terms, is employed, the difference lying in the extremes that are compared with one another through them.


§ 818. It is apparent from the figure that in the progressive form we work from within outwards, in the regressive form from without inwards. In the former we first employ the term ‘Devonshiremen’ as a mean to connect ‘Bideford men’ with ‘Englishmen’; next we employ ‘Englishmen’ as a mean to connect the same subject ‘Bideford men’ with the wider term ‘Teutons’; and, lastly, we employ ‘Teutons’ as a mean to connect the original subject ‘Bideford men’ with the ultimate predicate ‘Ayrans.’

§ 819. Reversely, in the regressive form we first use ‘Teutons’ as a mean whereby to bring ‘Englishmen’ under ‘Aryans’; next we use ‘Englishmen’ as a mean whereby to bring ‘Devonshiremen’ under the dame predicate ‘Aryans’; and, lastly, we use ‘Devonshiremen’ as a mean whereby to bring the ultimate subject ‘Bideford men’ under the original predicate ‘Aryans.’

§ 820. A sorites may be either Regular or Irregular.

§ 821. In the regular form the terms which connect each proposition in the series with its predecessor, that is to say, the middle terms, maintain a fixed relative position; so that, if the middle term be subject in one, it will always be predicate in the other, and vice versâ. In the irregular form this symmetrical arrangement is violated.

§ 822. The syllogisms which compose a regular sorites, whether progressive or regressive, will always be in the first figure.

In the irregular sorites the syllogisms may fall into different figures.

§ 823. For the regular sorites the following rules may be laid down.

(1) Only one premiss can be particular, namely, the first, if the sorites be progressive, the last, if it be regressive.

(2) Only one premiss can be negative, namely, the last, if the sorites be progressive, the first, if it be regressive.

§ 824. _Proof of the Rules for the Regular Sorites_.

(1) In the progressive sorites the proposition which stands first is the only one which appears as a minor premiss in the expanded form. Each of the others is used in its turn as a major. If any proposition, therefore, but the first were particular, there would be a particular major, which involves undistributed middle, if the minor be affirmative, as it must be in the first figure.

In the regressive sorites, if any proposition except the last were particular, we should have a particular conclusion in the syllogism in which it occurred as a premiss, and so a particular major in the next syllogism, which again is inadmissible, as involving undistributed middle.

(2) In the progressive sorites, if any premiss before the last were negative, we should have a negative conclusion in the syllogism in which it occurs. This would necessitate a negative minor in the next syllogism, which is inadmissible in the first figure, as involving illicit process of the major.

In the regressive sorites the proposition which stands first is the only one which appears as a major premiss in the expanded form. Each of the others is used in its turn as a minor. If any premiss, therefore, but the first were negative, we should have a negative minor in the first figure, which involves illicit process of the major.

§ 825. The rules above given do not apply to the irregular sorites, except so far as that only one premiss can be particular and only one negative, which follows from the general rules of syllogism. But there is nothing to prevent any one premiss from being particular or any one premiss from being negative, as the subjoined examples will show. Both the instances chosen belong to the progressive order of sorites.

(1) Barbara.
All B is A.
All C is B.
All C is A.

All B is A.
All C is B.
Some C is D.
All D is E
.’. Some A is E


(2) Disamis.
Some C is D.
All C is A.
Some A is D.

(3) Darii.
All D is E
Some A is D.
Some A is E.

(1) Barbara.
All B is C.
All A is B.
All A is C.

All A is B.
All B is C.
No D is C.
All E is D.
.’. No A is E.


(2) Cesare.
No D is C.
All A is C.
.’. No A is D.

(3) Camestres.
All E is D.
No A is D.
.’. No A is E.

§ 826. A chain argument may be composed consisting of conjunctive instead of simple propositions. This is subject to the same laws as the simple sorites, to which it is immediately reducible.

_Progressive._ _Regressive._
If A is B, C is D. If E is F, G is H. If C is D, E is F. If C is D, E is F. If E is F, G is H. If A is B, C is D. .’. If A is B, G is H. .’. If A is B, G is H.


_Of Fallacies_.

§ 827. After examining the conditions on which correct thoughts depend, it is expedient to classify some of the most familiar forms of error. It is by the treatment of the Fallacies that logic chiefly vindicates its claim to be considered a practical rather than a speculative science. To explain and give a name to fallacies is like setting up so many sign-posts on the various turns which it is possible to take off the road of truth.

§ 828. By a fallacy is meant a piece of reasoning which appears to establish a conclusion without really doing so. The term applies both to the legitimate deduction of a conclusion from false premisses and to the illegitimate deduction of a conclusion from any premisses. There are errors incidental to conception and judgement, which might well be brought under the name; but the fallacies with which we shall concern ourselves are confined to errors connected with inference.

§ 829. When any inference leads to a false conclusion, the error may have arisen either in the thought itself or in the signs by which the thought is conveyed. The main sources of fallacy then are confined to two–

(1) thought,

(2) language.

§ 830. This is the basis of Aristotle’s division of fallacies, which has not yet been superseded. Fallacies, according to him, are either in the language or outside of it. Outside of language there is no source of error but thought. For things themselves do not deceive us, but error arises owing to a misinterpretation of things by the mind. Thought, however, may err either in its form or in its matter. The former is the case where there is some violation of the laws of thought; the latter whenever thought disagrees with its object. Hence we arrive at the important distinction between Formal and Material fallacies, both of which, however, fall under the same negative head of fallacies other than those of language.

| In the language
| (in the signs of thought)
Fallacy -| |–In the Form. |–Outside the language -|
| (in the thought itself) |
|–in the Matter.

§ 831. There are then three heads to which fallacies may be referred-namely, Formal Fallacies, Fallacies of Language, which are commonly known as Fallacies of Ambiguity, and, lastly, Material Fallacies.

§ 832. Aristotle himself only goes so far as the first step in the division of fallacies, being content to class them according as they are in the language or outside of it. After that he proceeds at once to enumerate the infimæ species under each of the two main heads. We shall presently imitate this procedure for reasons of expediency. For the whole phraseology of the subject is derived from Aristotle’s treatise on Sophistical Refutations, and we must either keep to his method or break away from tradition altogether. Sufficient confusion has already arisen from retaining Aristotle’s language while neglecting his meaning.

§ 833. Modern writers on logic do not approach fallacies from the same point of view as Aristotle. Their object is to discover the most fertile sources of error in solitary reasoning; his was to enumerate the various tricks of refutation which could be employed by a sophist in controversy. Aristotle’s classification is an appendix to the Art of Dialectic.

§ 834. Another cause of confusion in this part of logic is the identification of Aristotle’s two-fold division of fallacies, commonly known under the titles of In dictione and Extra diotionem, with the division into Logical and Material, which is based on quite a different principle.

§ 835. Aristotle’s division perhaps allows an undue importance to language, in making that the principle of division, and so throwing formal and material fallacies under a common head. Accordingly another classification has been adopted, which concentrates attention from the first upon the process of thought, which ought certainly to be of primary importance in the eyes of the logician. This classification is as follows.

§ 836. Whenever in the course of our reasoning we are involved in error, either the conclusion follows from the premisses or it does not. If it does not, the fault must lie in the process of reasoning, and we have then what is called a Logical Fallacy. If, on the other hand, the conclusion does follow from the premisses, the fault must lie in the premisses themselves, and we then have what is called a Material Fallacy. Sometimes, however, the conclusion will appear to follow from the premisses until the meaning of the terms is examined, when it will be found that the appearance is deceptive owing to some ambiguity in the language. Such fallacies as these are, strictly speaking, non-logical, since the meaning of words is extraneous to the science which deals with thought. But they are called Semi-logical. Thus we arrive by a different road at the same three heads as before, namely, (1) Formal or Purely Logical Fallacies, (2) Semi-logical Fallacies or Fallacies of Ambiguity, (3) Material Fallacies.

§ 837. For the sake of distinctness we will place the two divisions side by side, before we proceed to enumerate the infimae species.

|–In the language
| (Fallacy of Ambiguity)
| |–In the Form.
|–Outside the language -|
|–In the Matter.

|–Formal or purely logical. |–Logical -|
Fallacy-| |–Semi-logical
| (Fallacy of Ambiguity). |–Material

838. Of one of these three heads, namely, formal fallacies, it is not necessary to say much, as they have been amply treated of in the preceding pages. A formal fallacy arises from the breach of any of the general rules of syllogism. Consequently it would be a formal fallacy to present as a syllogism anything which had more or less than two premisses. Under the latter variety comes what is called ‘a woman’s reason,’ which asserts upon its own evidence something which requires to be proved. Schoolboys also have been known to resort to this form of argument–‘You’re a fool.’ ‘Why?’ ‘Because you are.’ When the conclusion thus merely reasserts one of the premisses, the other must be either absent or irrelevant. If, on the other hand, there are more than two premisses, either there is more than one syllogism or the superfluous premiss is no premiss at all, but a proposition irrelevant to the conclusion.

839. The remaining rules of the syllogism are more able to be broken than the first; so that the following scheme presents the varieties of formal fallacy which are commonly enumerated–

|–Four Terms.
Formal Fallacy-|–Undistributed Middle. |–Illicit Process.
|–Negative Premisses and Conclusion.

§ 840. The Fallacy of Four Terms is a violation of the second of the general rules of syllogism (§ 582). Here is a palpable instance of it–

All men who write books are authors. All educated men could write books.
.’. All educated men are authors.

Here the middle term is altered in the minor premiss to the destruction of the argument. The difference between the actual writing of books and the power to write them is precisely the difference between one who is an author and one who is not.

§ 841. Since a syllogism consists of three terms, each of which is used twice over, it would be possible to have an apparent syllogism with as many as six terms in it. The true name for the fallacy therefore is the Fallacy of More than Three Terms. But it is rare to find an attempted syllogism which has more than four terms in it, just as we are seldom tendered a line as an hexameter, which has more than seven feet.

§ 842. The Fallacies of Undistributed Middle and Illicit Process have been treated of under §§ 585, 586. The heading ‘Negative Premisses and Conclusion’ covers violations of the three general rules of syllogism relating to negative premisses (§§ 590-593). Here is an instance of the particular form of the fallacy which consists in the attempt to extract an affirmative conclusion out of two negative premisses–

All salmon are fish, for neither salmon nor fish belong to the class mammalia.

The accident of a conclusion being true often helps to conceal the fact that it is illegitimately arrived at. The formal fallacies which have just been enumerated find no place in Aristotle’s division. The reason is plain. His object was to enumerate the various modes in which a sophist might snatch an apparent victory, whereas by openly violating any of the laws of syllogism a disputant would be simply courting defeat.

§ 843. We now revert to Aristotle’s classification of fallacies, or rather of Modes of Refutation. We will take the species he enumerates in their order, and notice how modern usage has departed from the original meaning of the terms. Let it be borne in mind that, when the deception was not in the language, Aristotle did not trouble himself to determine whether it lay in the matter or in the form of thought.

§ 844. The following scheme presents the Aristotelian classification to the eye at a glance:–

| |–Equivocation.
| |–Amphiboly.
|–In the language -|–Composition. | |–Division.
| |–Accent.
| |–Figure of Speech. Modes of -|
Refutation. | |–Accident. | |–A dicto secundum quid. | |–Ignoratio Elenchi. |–Outside the language -|–Consequent. | |–Petitio Principii. | |–Non causa pro causa. | |–Many Questions.

[Footnote: for “In the language”: The Greek is [Greek: para ten lexin], the exact meaning of which is; ‘due to the statement.’]

§ 845. The Fallacy of Equivocation [Greek: òmonumía] consists in an ambiguous use of any of the three terms of a syllogism. If, for instance, anyone were to argue thus–

No human being is made of paper,
All pages are human beings,
.’. No pages are made of paper–

the conclusion would appear paradoxical, if the minor term were there taken in a different sense from that which it bore in its proper premiss. This therefore would be an instance of the fallacy of Equivocal Minor.

§ 846. For a glaring instance of the fallacy of Equivocal Major, we may take the following–

No courageous creature flies,
The eagle is a courageous creature, .’. The eagle does not fly–

the conclusion here becomes unsound only by the major being taken ambiguously.

§ 847. It is, however, to the middle term that an ambiguity most frequently attaches. In this case the fallacy of equivocation assumes the special name of the Fallacy of Ambiguous Middle. Take as an instance the following–

Faith is a moral virtue.
To believe in the Book of Mormon is faith. .’. To believe in the Book of Mormon is a moral virtue.

Here the premisses singly might be granted; but the conclusion would probably be felt to be unsatisfactory. Nor is the reason far to seek. It is evident that belief in a book cannot be faith in any sense in which that quality can rightly be pronounced to be a moral virtue.

§ 848. The Fallacy of Amphiboly ([Greek: ámphibolía]) is an ambiguity attaching to the construction of a proposition rather than to the terms of which it is composed. One of Aristotle’s examples is this–

[Greek: tò boúlesthai labeîn me toùs polemíous]

which may be interpreted to mean either ‘the fact of my wishing to take the enemy,’ or ‘the fact of the enemies’ wishing to take me.’ The classical languages are especially liable to this fallacy owing to the oblique construction in which the accusative becomes subject to the verb. Thus in Latin we have the oracle given to Pyrrhus (though of course, if delivered at all, it must have been in Greek)–

Aio te, AEacida, Romanos vincere posse. Pyrrhus the Romans shall, I say, subdue (Whately), [Footnote: Cicero, De Divinatione, ii. § 116; Quintilian, Inst. Orat. vii 9, § 6.]

which Pyrrhus, as the story runs, interpreted to mean that he could conquer the Romans, whereas the oracle subsequently explained to him that the real meaning was that the Romans could conquer him. Similar to this, as Shakspeare makes the Duke of York point out, is the witch’s prophecy in Henry VI (Second Part, Act i, sc. 4),

The duke yet lives that Henry shall depose.

An instance of amphiboly may be read on the walls of Windsor Castle–Hoc fecit Wykeham. The king mas incensed with the bishop for daring to record that he made the tower, but the latter adroitly replied that what he really meant to indicate was that the tower was the making of him. To the same head may be referred the famous sentence–‘I will wear no clothes to distinguish me from my Christian brethren.’

§ 849. The Fallacy of Composition [Greek: diaíresis] is likewise a case of ambiguous construction. It consists, as expounded by Aristotle, in taking words together which ought to be taken separately, e.g.

‘Is it possible for a man who is not writing to write?’ ‘Of course it is.’
‘Then it is possible for a man to write without writing.’

And again–

‘Can you carry this, that, and the other?’ ‘Yes.’ ‘Then you can carry this, that, and the other,’–

a fallacy against which horses would protest, if they could.

§ 850. It is doubtless this last example which has led to a convenient misuse of the term ‘fallacy of composition’ among modern writers, by whom it is defined to consist in arguing from the distributive to the collective use of a term.

§ 851. The Fallacy of Division ([Greek: diaíresis]), on the other hand, consists in taking words separately which ought to be taken together, e.g.

[Greek: ègó s’ êteka doûlon ônt’ èleúteron [Footnote: Evidently the original of the line in Terence’s _Andria_, 37,–feci ex servo ut esses libertus mihi.],

where the separation of [Greek: doûlon] from [Greek: ôntra] would lead to an interpretation exactly contrary to what is intended.

And again–

[Greek: pentékont’ àndrôn èkatòn lípe dîos Àchilleús],

where the separation of [Greek: àndrôn] from [Greek: èkatòn] leads to a ludicrous error.

Any reader whose youth may have been nourished on ‘The Fairchild Family’ may possibly recollect a sentence which ran somewhat on this wise–‘Henry,’ said Mr. Fairchild, ‘is this true? Are you a thief and a liar too?’ But I am afraid he will miss the keen delight which can be extracted at a certain age from turning the tables upon Mr. Fairchild thus–Henry said, ‘Mr. Fairchild, is this true? Are _you_ a thief and a liar too?’

§ 852. The fallacy of division has been accommodated by modern writers to the meaning which they have assigned to the fallacy of composition. So that by the ‘fallacy of division’ is now meant arguing from the collective to the distributive use of a term. Further, it is laid down that when the middle term is used distributively in the major premiss and collectively in the minor, we have the fallacy of composition; whereas, when the middle term is used collectively in the major premiss and distributively in the minor, we have the fallacy of division. Thus the first of the two examples appended would be composition and the second division.

(1) Two and three are odd and even.
Five is two and three.
.’. Five is odd and even.

(2) The Germans are an intellectual people. Hans and Fritz are Germans.
.’. They are intellectual people.

§ 853. As the possibility of this sort of ambiguity is not confined to the middle term, it seems desirable to add that when either the major or minor term is used distributively in the premiss and collectively in the conclusion, we have the fallacy of composition, and in the converse case the fallacy of division. Here is an instance of the latter kind in which the minor term is at fault–

Anything over a hundredweight is too heavy to lift. These sacks (collectively) are over a hundredweight. .’. These sacks (distributively) are too heavy to lift.

§ 854. The ambiguity of the word ‘all,’ which has been before commented upon (§ 119), is a great assistance in the English language to the pair of fallacies just spoken of.

§ 835. The Fallacy of Accent ([Greek: prosodía]) is neither more nor less than a mistake in Greek accentuation. As an instance Aristotle gives Iliad xxiii. 328, where the ancient copies of Homer made nonsense of the words [Greek: tò mèn oú katapútetai ómbro] by writing [Greek: oû] with the circumflex in place of [Greek: oú] with the acute accent. [Footnote: This goes to show that the ancient Greeks did not distinguish in pronunciation between the rough and smooth breathing any more than their modern representatives.] Aristotle remarks that the fallacy is one which cannot easily occur in verbal argument, but rather in writing and poetry.

§ 856. Modern writers explain the fallacy of accent to be the mistake of laying the stress upon the wrong part of a sentence. Thus when the country parson reads out, ‘Thou shall not bear false witness _against_ thy neighbour,’ with a strong emphasis upon the word ‘against,’ his ignorant audience leap [sic] to the conclusion that it is not amiss to tell lies provided they be in favour of one’s neighbour.

§ 857. The Fallacy of Figure of Speech [Greek: tò schêma tês léxeos] results from any confusion of grammatical forms, as between the different genders of nouns or the different voices of verbs, or their use as transitive or intransitive, e.g. [Greek: úgiaínein] has the same grammatical form as [Greek: témnein] or [Greek: oìkodomeîn], but the former is intransitive, while the latter are transitive. A sophism of this kind is put into the mouth of Socrates by Aristophanes in the Clouds (670-80). The philosopher is there represented as arguing that [Greek: kápdopos] must be masculine because [Greek: Kleónumos] is. On the surface this is connected with language, but it is essentially a fallacy of false analogy.

§ 858. To this head may be referred what is known as the Fallacy of Paronymous Terms. This is a species of equivocation which consists in slipping from the use of one part of speech to that of another, which is derived from the same source, but has a different meaning. Thus this fallacy would be committed if, starting from the fact that there is a certain probability that a hand at whist will consist of thirteen trumps, one were to proceed to argue that it was probable, or that he had proved it.

§ 859. We turn now to the tricks of refutation which lie outside the language, whether the deception be due to the assumption of a false premiss or to some unsoundness in the reasoning.

§ 860. The first on the list is the Fallacy of Accident ([Greek: tò sumbebekós]). This fallacy consists in confounding an essential with an accidental difference, which is not allowable, since many things are the same in essence, while they differ in accidents. Here is the sort of example that Aristotle gives–

‘Is Plato different from Socrates ?’ ‘Yes.’ ‘Is Socrates a man ?’ ‘Yes.’ ‘Then Plato is different from man.’

To this we answer–No: the difference of accidents between Plato and Socrates does not go so deep as to affect the underlying essence. To put the thing more plainly, the fallacy lies in assuming that whatever is different from a given subject must be different from it in all respects, so that it is impossible for them to have a common predicate. Here Socrates and Plato, though different from one another, are not so different but that they have the common predicate ‘man.’ The attempt to prove that they have not involves an illicit process of the major.

§ 861. The next fallacy suffers from the want of a convenient name. It is called by Aristotle [Greek: tò áplos tóde ê pê légestai kaì mè kupíos] or, more briefly, [Greek: tò áplôs ê mé], or [Greek: tò pê kaí áplôs], and by the Latin writers ‘Fallacia a dicto secundum quid ad dictum simpliciter.’ It consists in taking what is said in a particular respect as though it held true without any restriction, e.g., that because the nonexistent ([Greek: tò mè ôn]) is a matter of opinion, that therefore the non-existent is, or again that because the existent ([Greek: tò ôn]) is not a man, that therefore the existent is not. Or again, if an Indian, who as a whole is black, has white teeth, we should be committing this species of fallacy in declaring him to be both white and not-white. For he is only white in a certain respect ([Greek: pê]), but not absolutely ([Greek: àplôs]). More difficulty, says Aristotle, may arise when opposite qualities exist in a thing in about an equal degree. When, for instance, a thing is half white and half black, are we to say that it is white or black? This question the philosopher propounds, but does not answer. The force of it lies in the implied attack on the Law of Contradiction. It would seem in such a case that a thing may be both white and not-white at the same time. The fact is–so subtle are the ambiguities of language–that even such a question as ‘Is a thing white or not-white?’ straightforward, as it seems, is not really a fair one. We are entitled sometimes to take the bull by the horns, and answer with the adventurous interlocutor in one of Plato’s dialogues–‘Both and neither.’ It may be both in a certain respect, and yet neither absolutely.

§ 862. The same sort of difficulties attach to the Law of Excluded Middle, and may be met in the same way. It might, for instance, be urged that it could not be said with truth of the statue seen by Nebuchadnezzar in his dream either that it was made of gold or that it was not made of gold: but the apparent plausibility of the objection would be due merely to the ambiguity of language. It is not true, on the one hand, that it was made of gold (in the sense of being composed entirely of that metal); and it is not true, on the other, that it was not made of gold (in the sense of no gold at all entering into its composition). But let the ambiguous proposition be split up into its two meanings, and the stringency of the Law of Excluded Middle will at once appear–

(1) It must either have been composed entirely of gold or not.

(2) Either gold must have entered into its composition or not.

§ 863. By some writers this fallacy is treated as the converse of the last, the fallacy of accident being assimilated to it under the title of the ‘Fallacia a dicto simpliciter ad dictum secundum quid.’ In this sense the two fallacies may be defined thus.

The Fallacy of Accident consists in assuming that what holds true as a general rule will hold true under some special circumstances which may entirely alter the case. The Converse Fallacy of Accident consists in assuming that what holds true under some special circumstances must hold true as a general rule.

The man who, acting on the assumption that alcohol is a poison, refuses to take it when he is ordered to do so by the doctor, is guilty of the fallacy of accident; the man who, having had it prescribed for him when he was ill, continues to take it morning, noon, and night, commits the converse fallacy.

§ 864. There ought to be added a third head to cover the fallacy of arguing from one special case to another.

§ 865. The next fallacy is Ignoratio Elenchi [Greek: èlégchou âgnoia]. This fallacy arises when by reasoning valid in itself one establishes a conclusion other than what is required to upset the adversary’s assertion. It is due to an inadequate conception of the true nature of refutation. Aristotle therefore is at the pains to define refutation at full length, thus–

‘A refutation [Greek: êlegchos] is the denial of one and the same–not name, but thing, and by means, not of a synonymous term, but of the same term, as a necessary consequence from the data, without assumption of the point originally at issue, in the same respect, and in the same relation, and in the same way, and at the same time.’

The ELENCHUS then is the exact contradictory of the opponent’s assertion under the terms of the law of contradiction. To establish by a syllogism, or series of syllogisms, any other proposition, however slightly different, is to commit this fallacy. Even if the substance of the contradiction be established, it is not enough unless the identical words of the opponent are employed in the contradictory. Thus if his thesis asserts or denies something about [Greek: lópion], it is not enough for you to prove the contradictory with regard to [Greek: ìmátion]. There will be need of a further question and answer to identify the two, though they are admittedly synonymous. Such was the rigour with which the rules of the game of dialectic were enforced among the Greeks!

§ 866. Under the head of Ignoratio Elenchi it has become usual to speak of various forme of argument which have been labelled by the Latin writers under such names as ‘argumentum ad hominem,’ ‘ad populum,’ ‘ad verecundiam,’ ‘ad ignorantiam,’ ‘ad baculum’–all of them opposed to the ‘argumentum ad rem’ or ‘ad judicium.’

§ 867. By the ‘argumentum ad hominem’ was perhaps meant a piece of reasoning which availed to silence a particular person, without touching the truth of the question. Thus a quotation from Scripture is sufficient to stop the mouth of a believer in the inspiration of the Bible. Hume’s Essay on Miracles is a noteworthy instance of the ‘argumentum ad hominem’ in this sense of the term. He insists strongly on the evidence for certain miracles which he knew that the prejudices of his hearers would prevent their ever accepting, and then asks triumphantly if these miracles, which are declared to have taken place in an enlightened age in the full glare of publicity, are palpably imposture, what credence can be attached to accounts of extraordinary occurrences of remote antiquity, and connected with an obscure corner of the globe? The ‘argumentum ad judicium’ would take miracles as a whole, and endeavour to sift the amount of truth which may lie in the accounts we have of them in every age. [Footnote: On this subject see the author’s _Attempts at Truth_ (Trubner & Co.), pp. 46-59.]

§ 868. In ordinary discourse at the present day the term ‘argumentum ad hominem’ is used for the form of irrelevancy which consists in attacking the character of the opponent instead of combating his arguments, as illustrated in the well-known instructions to a barrister–‘No case: abuse the plaintiff’s attorney.’

§ 869. The ‘argumentum ad populum’ consists in an appeal to the passions of one’s audience. An appeal to passion, or to give it a less question-begging name, to feeling, is not necessarily amiss. The heart of man is the instrument upon which the rhetorician plays, and he has to answer for the harmony or the discord that comes of his performance.

§ 870. The ‘argumentum ad verecundiam’ is an appeal to the feeling of reverence or shame. It is an argument much used by the old to the young and by Conservatives to Radicals.

§ 871. The ‘argumentum ad ignorantiam’ consists simply in trading on the ignorance of the person addressed, so that it covers any kind of fallacy that is likely to prove effective with the hearer.

§ 872. The ‘argumentum ad baculum’ is unquestionably a form of irrelevancy. To knock a man down when he differs from you in opinion may prove your strength, but hardly your logic.

A sub-variety of this form of irrelevancy was exhibited lately at a socialist lecture in Oxford, at which an undergraduate, unable or unwilling to meet the arguments of the speaker, uncorked a bottle, which had the effect of instantaneously dispersing the audience. This might be set down as the ‘argumentum ad nasum.’

§ 873. We now come to the Fallacy of the Consequent, a term which has been more hopelessly abused than any. What Aristotle meant by it was simply the assertion of the consequent in a conjunctive proposition, which amounts to the same thing as the simple conversion of A (§ 489), and is a fallacy of distribution. Aristotle’s example is this–

If it has rained, the ground is wet. .’. If the ground is wet, it has rained.

This fallacy, he tells us, is often employed in rhetoric in dealing with presumptive evidence. Thus a speaker, wanting to prove that a man is an adulterer, will argue that he is a showy dresser, and has been seen about at nights. Both these things however may be the case, and yet the charge not be true.

§ 874. The Fallacy of Petitio or Assumptio Principii [Greek: tò èn àrchê aìteîstai or lambánein] to which we now come, consists in an unfair assumption of the point at issue. The word [Greek: aìteîstai], in Aristotle’s name for it points to the Greek method of dialectic by means of question and answer. This fact is rather disguised by the mysterious phrase ‘begging the question.’ The fallacy would be committed when you asked your opponent to grant, overtly or covertly, the very proposition originally propounded for discussion.

§ 875. As the question of the precise nature of this fallacy is of some importance we will take the words of Aristotle himself (Top. viii. 13. §§ 2, 3). ‘People seem to beg the question in five ways. First and most glaringly, when one takes for granted the very thing that has to be proved. This by itself does not readily escape detection, but in the case of “synonyms,” that is, where the name and the definition have the same meaning, it does so more easily. [Footnote: Some light is thrown upon this obscure passage by a comparison with Cat. I. § 3, where ‘synonym’ is defined. To take the word here in its later and modern sense affords an easy interpretation, which is countenanced by Alexander Aphrodisiensis, but it is flat against the usage of Aristotle, who elsewhere gives the name ‘synonym,’ not to two names for the same thing, but to two things going under the same name. See Trendelenberg on the passage.]

Secondly, when one assumes universally that which has to be proved in particular, as, if a man undertaking to prove that there is one science of contraries, were to assume that there is one science of opposites generally. For he seems to be taking for granted along with several other things what he ought to have proved by itself.

Thirdly, when one assumes the particulars where the universal has to be proved; for in so doing a man is taking for granted separately what he was bound to prove along with several other things. Again, when one assumes the question at issue by splitting it up, for instance, if, when the point to be proved is that the art of medicine deals with health and disease, one were to take each by itself for granted.

Lastly, if one were to take for granted one of a pair of necessary consequences, as that the side is incommensurable with the diagonal, when it is required to prove that the diagonal is incommensurable with the side.’

§ 876. To sum up briefly, we may beg the question in five ways–

(1) By simply asking the opponent to grant the point which requires to be proved;

(2) by asking him to grant some more general truth which involves it;

(3) by asking him to grant the particular truths which it involves;

(4) by asking him to grant the component parts of it in detail;

(5) by asking him to grant a necessary consequence of it.

§ 877. The first of these five ways, namely, that of begging the question straight off, lands us in the formal fallacy already spoken of (§ 838), which violates the first of the general rules of syllogism, inasmuch as a conclusion is derived from a single premiss, to wit, itself.

§ 878. The second, strange to say, gives us a sound syllogism in Barbara, a fact which countenances the blasphemers of the syllogism in the charge they bring against it of containing in itself a petitio principii. Certainly Aristotle’s expression might have been more guarded. But it is clear that his quarrel is with the matter, not with the form in such an argument. The fallacy consists in assuming a proposition which the opponent would be entitled to deny. Elsewhere Aristotle tells us that the fallacy arises when a truth not evident by its own light is taken to be so. [Footnote: [Greek: Ôtan tò mè dí aùtoû gnostòn dí aùtoû tis èpicheiraê deiknúnai, tót’ aìteîtai tò èx àrchês.]. Anal. Pr. II. 16. § I ad fin.]

§ 879. The third gives us an inductio per enumerationem simplicem, a mode of argument which would of course be unfair as against an opponent who was denying the universal.

§ 880. The fourth is a more prolix form of the first.

§ 881. The fifth rests on Immediate Inference by Relation (§ 534).

§ 882. Under the head of petitio principii comes the fallacy of Arguing in a Circle, which is incidental to a train of reasoning. In its most compressed form it may be represented thus–

(1) B is A.
C is B.
.’. C is A.

(2) C is A.
B is C.
.’. B is A.

§ 883. The Fallacy of Non causa pro causa ([Greek: tò mè aîtion] or [Greek: aîtoin]) is another, the name of which has led to a complete misinterpretation. It consists in importing a contradiction into the discussion, and then fathering it on the position controverted. Such arguments, says Aristotle, often impose upon the users of them themselves. The instance he gives is too recondite to be of general interest.

§ 884. Lastly, the Fallacy of Many Questions ([Greek: tò tà déo èrotémata ên poieîn]) is a deceptive form of interrogation, when a single answer is demanded to what is not really a single question. In dialectical discussions the respondent was limited to a simple ‘yes’ or ‘no’; and in this fallacy the question is so framed as that either answer would seem to imply the acceptance of a proposition which would be repudiated. The old stock instance will do as well as another–‘Come now, sir, answer “yes” or “no.” Have you left off beating your mother yet?’ Either answer leads to an apparent admission of impiety.

A late Senior Proctor once enraged a man at a fair with this form of fallacy. The man was exhibiting a blue horse; and the distinguished stranger asked him–‘With what did you paint your horse?’


These exercises should be supplemented by direct questions upon the text, which it is easy for the student or the teacher to supply for himself.



Classify the following words according as they are categorematic, syncategorematic or acategorematic;–

come peradventure why
through inordinately pshaw
therefore circumspect puss
grand inasmuch stop
touch sameness back
cage disconsolate candle.


Classify the following things according as they are substances, qualities or relations;–

God likeness weight
blueness grass imposition
ocean introduction thinness
man air spirit
Socrates raillery heat
mortality plum fire.


1. Give six instances each of-attribute, abstract, singular, privative, equivocal and relative terms.

2. Select from the following list of words such as are terms, and state whether they are (1) abstract or concrete, (2) singular or common, (3) univocal or equivocal:–

van table however
enter decidedly tiresome
very butt Solomon
infection bluff Czar
short although Caesarism
distance elderly Nihilist.

3. Which of the following words are abstract terms?–

quadruped event through
hate desirability thorough
fact expressly thoroughness faction wish light
inconvenient will garden
inconvenience volition grind.

4. Refer the following terms to their proper place under each of the divisions in the scheme:–

horse husband London
free lump empty
liberty rational capital
impotent reason Capitol
impetuosity irrationality grave
impulsive double calf.

5. Give six instances each of proper names and designations.

6. Give six instances each of connotative and non-connotative terms.

7. Give the extension and intension of–

sermon animal sky
clock square gold
sport fish element
bird student fluid
art river line
gas servant language


Arrange the following terms in order of extension–carnivorous, thing, matter, mammal, organism, vertebrate, cat, substance, animal.

* * * * *



Give a name to each of the following sentences:–

(1) Oh, that I had wings like a dove!

(2) The more, the merrier.

(3) Come rest in this bosom, my own stricken deer.

(4) Is there balm in Gilead?

(5) Hearts may be trumps.


Analyse the following propositions into subject, copula and predicate:–

(1) He being dead yet speaketh.

(2) There are foolish politicians.

(3) Little does he care.

(4) There is a land of pure delight.

(5) All’s well that ends well.

(6) Sweet is the breath of morn.

(7) Now it came to pass that the beggar died.

(8) Who runs may read.

(9) Great is Diana of the Ephesians.

(10) Such things are.

(11) Not more than others I deserve.

(12) The day will come when Ilium’s towers shall perish.


1. Express in logical form, affixing the proper symbol:–

(1) Some swans are not white.

(2) All things are possible to them that believe.

(3) No politicians are unprincipled.

(4) Some stones float on water.

(5) The snow has melted.

(6) Eggs are edible.

(7) All kings are not wise.

(8) Moths are not butterflies.

(9) Some men are born great.

(10) Not all who are called are chosen.

(11) It is not good for man to be alone.

(12) Men of talents have been known to fail in life.

(13) ‘Tis none but a madman would throw about fire.

(14) Every bullet does not kill.

(15) Amongst Unionists are Whigs.

(16) Not all truths are to be told.

(17) Not all your efforts can save him.

(18) The whale is a mammal.

(19) Cotton is grown in Cyprus.

(20) An honest man’s the noblest work of God.

(21) No news is good news.

(22) No friends are like old friends.

(23) Only the ignorant affect to despise knowledge.

(24) All that trust in Him shall not be ashamed.

(25) All is not gold that glitters.

(26) The sun shines upon the evil and upon the good.

(27) Not to go on is to go back.

(28) The king, minister, and general are a pretty trio.

(29) Amongst dogs are hounds.

(30) A fool is not always wrong.

(31) Alexander was magnanimous.

(32) Food is necessary to life.

(33) There are three things to be considered,

(34) By penitence the Eternal’s wrath’s appeased.

(35) Money is the miser’s end.

(36) Few men succeed in life.

(37) All is lost, save honour.

(38) It is mean to hit a man when he is down.

(39) Nothing but coolness could have saved him.

(40) Books are generally useful.

(41) He envies others’ virtue who has none himself.

(42) Thankless are all such offices.

(43) Only doctors understand this subject.

(44) All her guesses but two were correct.

(45) All the men were twelve.

(46) Gossip is seldom charitable.

2. Give six examples of indefinite propositions, and then quantify them according to their matter.

3. Compose three propositions of each of the following kinds:–

(1) with common terms for subjects;

(2) with abstract terms for subjects;

(3) with singular terms for predicates;

(4) with collective terms for predicates;

(5) with attributives in their subjects;

(6) with abstract terms for predicates.


1. Point out what terms are distributed or undistributed in the following propositions:–

(1) The Chinese are industrious.

(2) The angle in a semi-circle is a right angle.

(3) Not one of the crew survived.

(4) The weather is sometimes not propitious.

The same exercise may be performed upon any of the propositions in the preceding list.

2. Prove that in a negative proposition the predicate must be distributed.


Affix its proper symbol to each of the following propositions:–

(1) No lover he who is not always fond.

(2) There are Irishmen and Irishmen.

(3) Men only disagree,
Of creatures rational.

(4) Some wise men are poor.

(5) No Popes are some fallible beings.

(6) Some step-mothers are not unjust.

(7) The most original of the Roman poets was Lucretius.

(8) Some of the immediate inferences are all the forms of conversion.


1. Give six examples of terms standing one to another as genus to species.

2. To which of the heads of predicables would you refer the following statements? And why?

(1) A circle is the largest space that can be contained by one line.

(2) All the angles of a square are right angles.

(3) Man alone among animals possesses the faculty of laughter.

(4) Some fungi are poisonous.

(5) Most natives of Africa are negroes.

(6) All democracies are governments.

(7) Queen Anne is dead.


1. Define the following terms–

Sun inn-keeper tea-pot
hope anger virtue
bread diplomacy milk
carpet man death
sincerity telescope mountain
poverty Senate novel.

2. Define the following terms as used in Political Economy–

Commodity barter value
wealth land price
money labour rent
interest capital wages
credit demand profits.

3. Criticise the following as definitions–

(1) Noon is the time when the shadows of bodies are shortest.

(2) Grammar is the science of language.

(3) Grammar is a branch of philology.

(4) Grammar is the art of speaking and writing a language with propriety.

(5) Virtue is acting virtuously.

(6) Virtue is that line of conduct which tends to produce happiness.

(7) A dog is an animal of the canine species.

(8) Logic is the art of reasoning.

(9) Logic is the science of the investigation of truth by means of evidence.

(10) Music is an expensive noise.

(11) The sun is the centre of the solar system.

(12) The sun is the brightest of those heavenly bodies that move round the earth.

(13) Rust is the red desquamation of old iron.

(14) Caviare is a kind of food.

(15) Life is the opposite of death.

(16) Man is a featherless biped.

(17) Man is a rational biped.

(18) A gentleman is a person who has no visible means of subsistence.

(19) Fame is a fancied life in others’ breath.

(20) A fault is a quality productive of evil or inconvenience.

(21) An oligarchy is the supremacy of the rich in a state.

(22) A citizen is one who is qualified to exercise deliberative and judicial functions.

(23) Length is that dimension of a solid which would be measured by the longest line.

(24) An eccentricity is a peculiar idiosyncrasy.

(25) Deliberation is that species of investigation which is concerned with matters of action.

(26) Memory is that which helps us to forget.