(5) _Ambiguity of pronunciation_ ([Greek: prosodia], _fallacia accentus_).

Analogies to words that differ only in accent, such as [Greek: ou-with accents {psili and persipomeni}] and [Greek: ou-with accents {psili and oxia}], may be found in differences of pronunciation. "Hair very thick, sir," said a barber to a customer, whose hair was bushy, but beginning to turn grey. "Yes, I daresay. But I would rather have it thick than thin." "Ah, too thick to-day, sir." "But I don't want to dye it." "Excuse me, sir, I mean the hair of the hatmosphere, t-o-d-a-y, to-day."

"He said, saddle me the ass. And they saddled _him_."

(6) _Ambiguity of inflexion_ ([Greek: schema tes lexeos], _Figura dictionis_).

This is not easy to make intelligible in English. The idea is that a termination may be ambiguously interpreted, a neuter participle, _e.g._, taken for an active. Thus: "George is ailing". "Doing what, did you say? Ailing? What is he ailing? Ginger-aleing?"

Non-Verbal Fallacies, or Fallacies in thought, are a more important division. Aristotle distinguishes seven.

Of these, three are comparatively unimportant and trifling. One of them, known to the Schoolmen as _Fallacia Plurium Interrogationum_, was peculiar to Interrogative disputation. It is the trick of putting more than one question as one, so that a simple Yes commits the respondent to something implied. "Have you left off beating your father?" If you answer Yes, that implies that you have been in the habit of beating him. "Has the practice of excessive drinking ceased in your part of the country?" Such questions were unfair when the Respondent could answer only Yes or No. The modern disputant who demands a plain answer Yes or No, is sometimes guilty of this trick.

Two others, the fallacies known as _A dicto simpliciter ad dictum secundum quid_, and _A dicto secundum quid ad dictum simpliciter_, are as common in modern dialectic as they were in ancient. The trick, conscious or unconscious, consists in getting assent to a statement with a qualification and proceeding to argue as if it had been conceded without qualification, and _vice versa_. For example, it being admitted that culture is good, a disputant goes on to argue as if the admission applied to some sort of culture in special, scientific, aesthetic, philosophical or moral. The fallacy was also known as _Fallacia Accidentis_. Proving that the Syllogism is useless for a certain purpose, and then claiming to have proved that it is useless for any purpose is another example. Getting a limited admission and then extending it indefinitely is perhaps the more common of the two forms. It is common enough to deserve a shorter name.

The _Fallacia Consequentis_, or _Non-Sequitur_, which consists specially in ignoring the possibility of a plurality of causes, has already been partly explained in connexion with the Hypothetical Syllogism, and will be explained further in the Logic of Induction.

_Post hoc ergo proper hoc_ is a purely Inductive Fallacy, and will be explained in connexion with the Experimental Methods.

There remain the two typical Deductive Fallacies, PETITIO PRINCIPII (Surreptitious Assumption) and IGNORATIO ELENCHI (Irrelevant Argument) about which we must speak more at length.

The phrase of which Petitio Principii or Begging the Question is a translation--[Greek: to en arche aiteisthai]--was applied by Aristotle to an argumentative trick in debate by Question and Answer. The trick consisted in taking for granted a proposition necessary to the refutation without having obtained the admission of it. Another expression for the same thing--[Greek: to en arche lambanein]--taking the principle for granted--is more descriptive.

Generally speaking, Aristotle says, Begging the Question consists in not demonstrating the theorem. It would be in accordance with this general description to extend the name to all cases of tacitly or covertly, unwittingly to oneself or to one's opponent, assuming any premiss necessary to the conclusion. It is the fallacy of Surreptitious Assumption, and all cases of Enthymematic or Elliptical argument, where the unexpressed links in the chain of argument are not fully understood, are examples of it. By contrast, the articulate and explicit Syllogism is an _Expositio Principii_. The only remedy for covert assumptions is to force them into the light.[1]

_Ignoratio Elenchi_, ignoring the refutation ([Greek: tou elenchou agnoia]), is simply arguing beside the point, distracting the attention by irrelevant considerations. It often succeeds by proving some other conclusion which is not the one in dispute, but has a superficial resemblance to it, or is more or less remotely connected with it.

It is easier to explain what these fallacies consist in than to illustrate them convincingly. It is chiefly in long arguments that the mischief is done. "A Fallacy," says Whately, "which when stated barely in a few sentences would not deceive a child, may deceive half the world if diluted in a quarto volume." Very rarely is a series of propositions put before us in regular form and order, all bearing on a definite point. A certain conclusion is in dispute, not very definitely formulated perhaps, and a mixed host of considerations are tumbled out before us. If we were perfectly clear-headed persons, capable of protracted concentration of attention, incapable of bewilderment, always on the alert, never in a hurry, never over-excited, absolutely without prejudice, we should keep our attention fixed upon two things while listening to an argument, the point to be proved, and the necessary premisses. We should hold the point clearly in our minds, and watch indefatigably for the corroborating propositions. But none of us being capable of this, all of us being subject to bewilderment by a rapid whirl of statements, and all of us biased more or less for or against a conclusion, the sophist has facilities for doing two things--taking for granted that he has stated the required premisses (_petitio principii_), and proving to perfect demonstration something which is not the point in dispute, but which we are willing to mistake for it (_ignoratio elenchi_).

It is chiefly in the heat of argument that either Petitio or Ignoratio succeeds. When a fallacy continues to perplex us in cold blood, it must have in its favour either some deeply-rooted prejudice or some peculiar intricacy in the language used, or some abstruseness in the matter. If we are not familiar with the matter of the argument, and have but a vague hold of the words employed, we are, of course, much more easily imposed upon.

The famous Sophisms of antiquity show the fascination exercised over us by proving something, no matter how irrelevant. If certain steps in an argument are sound, we seem to be fascinated by them so that we cannot apply our minds to the error, just as our senses are fascinated by an expert juggler. We have seen how plausibly Zeno's argument against the possibility of motion hides a Petitio: the Fatalist Dilemma is another example of the same sort.

If it is fated that you die, you will die whether you call in a doctor or not, and if it is fated that you will recover, you will recover whether you call in a doctor or not. But it must be fated either that you die or that you recover. _Therefore_, you will either die or recover whether you call in a doctor or not.

Here it is tacitly assumed in the Major Premiss that the calling in of a doctor cannot be a link in the fated chain of events. In the statement of both the alternative conditions, it is assumed that Fate does not act through doctors, and the conclusion is merely a repetition of this assumption, a verbal proposition after an imposing show of argument. "If Fate does not act through doctors, you will die whether you call in a doctor or not."

The fallacy in this case is probably aided by our veneration for the grand abstraction of Fate and the awful idea of Death, which absorbs our attention and takes it away from the artful _Petitio_.

The Sophism of Achilles and the Tortoise is the most triumphant of examples of _Ignoratio Elenchi_.

The point that the Sophism undertakes to prove is that Achilles can never overtake a Tortoise once it has a certain start: what it really proves, and proves indisputably, is that he cannot overtake the Tortoise within a certain space or time.

For simplicity of exposition, let us assume that the Tortoise has 100 yards start and that Achilles runs ten times as fast. Then, clearly, Achilles will not come up with it at the end of 100 yards, for while he has run 100, the Tortoise has run 10; nor at the end of 110, for then the Tortoise has run 1 more; nor at the end of 111, for then the Tortoise has run 1/10 more; nor at the end of 111-1/10, for then the Tortoise has gained 1/100 more. So while Achilles runs this 1/100, the Tortoise runs 1/1000; while he runs the 1/1000, it runs 1/10000. Thus it would seem that the Tortoise must always keep ahead: he can never overtake it.

But the conclusion is only a confusion of ideas: all that is really proved is that Achilles will not overtake the Tortoise while running

100 + 10 + 1 + 1/10 + 1/100 + 1/1000 + 1/10000, etc.

That is, that he will not overtake it till he has completed the sum of this series, 111-1/9 yards. To prove this is an _ignoratio elenchi_; what the Sophist undertakes to prove is that Achilles will never overtake it, and he really proves that Achilles passes it between the 111th and 112th yards.

The exposure of this sophism is an example also of the value of a technical term. All attempts to expose it without using the term _Ignoratio Elenchi_ or something equivalent to it, succeed only in bewildering the student. It is customary to say that the root of the fallacy lies in assuming that the sum of an infinite series is equal to infinity. This profound error may be implied: but if any assumption so hard to understand were really required, the fallacy would have little force with the generality.

It has often been argued that the Syllogism involves a _petitio principii_, because the Major Premiss contains the Conclusion, and would not be true unless the Conclusion were true. But this is really an _Ignoratio Elenchi_. The fact adduced, that the Major Premiss contains the Conclusion, is indisputable; but this does not prove the Syllogism guilty of Petitio. _Petitio principii_ is an argumentative trick, a conscious or unconscious act of deception, a covert assumption, and the Syllogism, so far from favouring this, is an _expositio principii_, an explicit statement of premisses such that, if they are true, the conclusion is true. The Syllogism merely shows the interdependence of premisses and conclusion; its only tacit assumption is the _Dictum de Omni_.

If, indeed, an opponent challenges the truth of the conclusion, and you adduce premisses necessarily containing it as a refutation, that is an _ignoratio elenchi_ unless your opponent admits those premisses.

If he admits them and denies the conclusion, you convict him of inconsistency, but you do not prove the truth of the conclusion.

Suppose a man to take up the position: "I am not mortal, for I have procured the _elixir vitae_". You do not disprove this by saying, "All men are mortal, and you are a man". In denying that he is mortal, he denies that all men are mortal. Whatever is sufficient evidence that he is not mortal, is sufficient evidence that all men are not mortal.

Perhaps it might be said that in arguing, "All men are mortal, and you are a man," it is not so much _ignoratio elenchi_ as _petitio principii_ that you commit. But be it always remembered that you may commit both fallacies at once. You may both argue beside the point and beg the question in the course of one and the same argument.

[Footnote 1: Cp. Mr. Sidgwick's instructive treatise on Fallacies, International Scientific Series, p. 199.]

CHAPTER IX.

FORMAL OR ARISTOTELIAN INDUCTION.--INDUCTIVE ARGUMENT.

The distinction commonly drawn between Deduction and Induction is that Deduction is reasoning from general to particular, and Induction reasoning from particular to general.

But it is really only as modes of argumentation that the two processes can be thus clearly and fixedly opposed. The word Induction is used in a much wider sense when it is the title of a treatise on the Methods of Scientific Investigation. It is then used to cover all the processes employed in man's search into the system of reality; and in this search deduction is employed as well as induction in the narrow sense.

We may call Induction in the narrow sense Formal Induction or Inductive Argument, or we may simply call it Aristotelian Induction inasmuch as it was the steps of Inductive argument that Aristotle formulated, and for which he determined the conditions of validity.

Let us contrast it with Deductive argument. In this the questioner's procedure is to procure the admission of a general proposition with a view to forcing the admission of a particular conclusion which is in dispute. In Inductive argument, on the other hand, it is a general proposition that is in dispute, and the procedure is to obtain the admission of particular cases with a view to forcing the admission of this general proposition.

Let the question be whether All horned animals ruminate. You engage to make an opponent admit this. How do you proceed? You ask him whether he admits it about the various species. Does the ox ruminate? The sheep? The goat? And so on. The bringing in of the various particulars is the induction ([Greek: epagoge]).

When is this inductive argument complete? When is the opponent bound to admit that all horned animals ruminate? Obviously, when he has admitted it about every one. He must admit that he has admitted it about every one, in other words, that the particulars enumerated constitute the whole, before he can be held bound in consistency to admit it about the whole.

The condition of the validity of this argument is ultimately the same with that of Deductive argument, the identity for purposes of predication of a generic whole with the sum of its constituent parts.

The Axiom of Inductive Argument is, _What is predicated of every one of the parts is predicable of the whole._ This is the simple converse of the Axiom of Deductive argument, the _Dictum de Omni_, "What is predicated of the whole is predicable about every one of the parts".

The Axiom is simply convertible because for purposes of predication generic whole and specific or individual parts taken all together are identical.

Practically in inductive argument an opponent is worsted when he cannot produce an instance to the contrary. Suppose he admits the predicate in question to be true of this, that and the other, but denies that this, that and the other constitute the whole class in question, he is defeated in common judgement if he cannot instance a member of the class about which the predicate does not hold. Hence this mode of induction became technically known as _Inductio per enumerationem simplicem ubi non reperitur instantia contradictoria_.

When this phrase is applied to a generalisation of fact, Nature or Experience is put figuratively in the position of a Respondent unable to contradict the inquirer.

Such in plain language is the whole doctrine of Inductive Argument.

Aristotle's Inductive Syllogism is, in effect, an expression of this simple doctrine tortuously in terms of the Deductive Syllogism. The great master was so enamoured of his prime invention that he desired to impress its form upon everything: otherwise, there was no reason for expressing the process of Induction syllogistically. Here is his description of the Inductive Syllogism:--

"Induction, then, and the Inductive Syllogism, consists in syllogising one extreme with the middle through the other extreme. For example, if B is middle to A and C, to prove through C that A belongs to B."[1]

This may be interpreted as follows: Suppose a general proposition is in dispute, and that you wish to make it good by obtaining severally the admission of all the particulars that it sums up. The type of a general proposition in Syllogistic terminology is the Major Premiss, All M is P. What is the type of the particulars that it sums up?

Obviously, the Conclusion, S is P. This particular is contained in the Major Premiss, All M is P; its truth is accepted as contained in the truth of All M is P. S is one of the parts of the generic whole M; one of the individuals or species contained in the class M. If you wish, then, to establish P of All M by Induction, you must establish P of all the parts, species, or individuals contained in M, that is, of all possible S_s:_ you must make good that this, that and the other S is P, and also that this, that and the other S constitute the whole of M.

You are then entitled to conclude that All M is P: you have syllogised one Extreme with the Middle through the other Extreme. The formal statement of these premisses and conclusion is the Inductive Syllogism.

This, that and the other S is P, _Major_.