rule
A clever man has said that the use of language is to conceal thought. Its primary use is certainly not to reveal thought, but to enable one person to produce an effect on the mind of another or of others, either for their or his own advantage. In the course of using speech as an instrument of command, entreaty, persuasion, menace, or fustigation, it may happen that the movements of the speaker's mind are revealed to some extent, but this is a mere incident, not the main purpose of the speech.
Grammar is the system of rules which govern the use of language in its primary and ordinary capacity.
It follows from this that language is in no sense a revelation of the reasoning processes, nor do the rulesof grammar coincide with the laws of intellect. It is just as reasonable to expect to find the metaphysic of thought revealed in any of the industrial and fine arts, as to look for it in the structure of speech. Aristotle drew his logic from the composition of the Greek sentence—he might as well have sought for logic in the constitution of the Greek buskin.16
Even when men begin to reason aloud and seek to render their logical movements as evident as possible, they are so hampered by the ordinary habits and rules of speech that their meaning is often difficult or impossible of comprehension. Whence arises the necessity, if we would reason aloud to any purpose, of redacting or translating language from the vernacular into a dialect more indicative of the logical processes that take place when we reason.
This redaction consists mainly in distinguishing clearly the four parts composing an argument, namely, the Subject of the Precedent; the Case which is brought under it for judgment; the Applicate or part of the precedent bearing on the case; the Conclusion, which is the ideal judgment concerningthe case. When these four parts are expressed and clearly understood we have a perfect argument, so far as argumentation depends on language. But probably we have spoiled the language from the grammatical and rhetorical point of view. We may have had to supply much that would be redundant and unsightly in ordinary conversation or writing, and to take away much that is appropriate to colloquial discourse. We are diverting language to a use for which it was not designed, and we need not be surprised if the result is ungraceful. This cannot be helped since there exists no other means than language by which to express our concrete reasoning.
I have already shown practically how an argument can be arranged so as to indicate the logical relations subsisting between its parts. A Greek cross is drawn, and in the four angles thus made the four parts of the argument are written, or the principal words of each. Begin with the conclusion, for that is generally the most explicitly given; then find or supply the part of the precedent that agrees or logically rhymes with it; next place the subject in the first compartment, and the case under it. These relative positions should not be varied. When this has been practiced for a while it enables one to dismember the most intricate argument with ease and exactness.
The redaction or re-writing of the language can be abbreviated by regarding the horizontal line as equivalentto a declaration of resemblance between case and precedent-subject, and (by application) between the illustrative abstraction and conclusion. If there is an argument at all there must be this resemblance, and the right-hand parts must have one of the six categorical relations to the left-hand parts. The contents of the angles may be cut down to a word or two, as—
If the category be further indicated by a numeral over the upright line, we have the essential parts of the argument in a very compact form. The cross and categorical numeral may be regarded as a sufficient substitute for grammatical syntax and punctuation.
The negative word that generally occurs in stigmatic arguments requires special attention. It should always be put in the second angle, and when it may read so as to negative the subject it should be hyphened to the predicate, thus giving it the value ofnon,un,im, or other negative prefix. To say colloquially that 'all Russians are not angels' leaves room to believe that some Russians are angels, the 'not' applying to 'all' instead of to 'angels.' By linking 'not' to 'angels' we get aterm equivalent to non-angelic, which expresses the meaning intended—that no Russians are angels.
Caution should be observed with partitive words like 'some,' 'many,' 'a few,' &c. There is little danger of ambiguity when they occur in the case, for that means that we bring only a portion of a group of things to judgment, which we are manifestly entitled to do. The conclusion however applies only to the portion in question, not to the rest of the group. 'Honest men deserve respect; some Negroes are honest men;these particular Negroes deserve respect.'
In the precedent, partitive words imply that only some of the subject have the applicate. If that portion is a dialectical 'all'—that is, if there has been no exception in the course of our experience—we may, though that experience has been limited, venture to treat the applicate as universal and ground a conclusion upon it. If the subject is really partitive—if we know for certain that some subjects have the applicate and others have it not—the conclusion must follow the greater probability. If the number and character of the observed cases is known we can express the probability arithmetically; it is the number of occurrences of a given character divided by the total number.
Redaction must not be used to correct original errors of observation; its purpose is to render explicit in language what is implicit in thought, not whatmight have been thought supposing the thinker had been more intelligent or industrious than he was.
'Conversion' is a process admitted or required in the artificial methods of syllogistic dialectic. It consists mainly in transposing the subject and predicate of a proposition, as 'some Europeans are Mohammedans'—'some Mohammedans are Europeans,' This operation never takes place in real argument, or is merely the emendation of a proposition at first awkwardly expressed. Conversion can take place only when the predicate is a class, hence the categorical propositions cannot be converted.
16:Though evidently suggested by language, the form which the syllogistic logic finally assumed is so unlike anything grammatical, that it is easily convertible into symbols having no resemblance to language. It has been put into literal symbols with algebraic values, and into geometrical diagrams. A logical machine has even been invented by Professor Jevons, 'worked by keys like a pianoforte,' which returns 'infallible answers'—of the Aristotelian sort—to every kind of question. That is sufficiently unlike both reason and language.
rule
Fallacies are counterfeit or sham arguments. They may fail to be arguments—(1) because their antecedents are false; (2) because the antecedents though true are not arranged dialectically, and do not suggest the right conclusion; (3) because the language is equivocal.
To take the last first. So many things are called by the same name, and so many different names may be applied to the same thing, that if we attempt to argue from words alone, without any personal knowledge of the things or judgments that are in question, we shall certainly make mistakes. The only security against this sort of fallacy is much experience, and the self-denial necessary to relinquish argument and the criticism of arguments, when we have no sufficient knowledge of the data.
The degree of imperfection in observation which should be considered to render the theorem fallacious, is no easy matter to determine. One class of logicians (the Formal) get over the difficulty by declaring that dialectic is not concerned with concrete knowledge at all,17but only with its general properties (as conceived by Aristotle), and they have set up as a standard of logical truth the capability of being imagined. A centaur is to them as true a fact as a horse, and they would accept as valid such a theorem as this: 'All centaurs object to be shod with iron; Gryneus is a centaur; therefore we may conclude that he would resist being shod with iron.' No amount of conceivability or formal coherence can make this other than nonsense.
J. S. Mill and his followers go to the opposite extreme. They study all the sciences and endeavour to master their methods of reasoning—which is well; but they do so with the prepossession that there exists some absolute standard of knowledge to fail in attaining which involves fallacy. They thus condemn as false all theorems based on superseded notions of nature and man. Only modern thinkers can argue rationally—the ancients were all and habituallyvictims of fallacy,—and of the moderns only the few are rational who have mastered the latest theories on every subject. This is the principle of Mill's doctrine on the fallacies of observation; we can see that he regarded all beliefs as fallacious which he had himself outgrown or did not feel a need of. 'Truth' was simply the facts and judgments that happened to suit Mill's mental constitution.
From the Substantial point of view this is an untenable position.
No degree of observation is intrinsically defective if it serve the purpose of intellect, which is to protect the mind. There is no intellectual truth as a thing in itself. The thoughts of a sparrow or a child are as perfect as those of a man, if they afford the necessary defence to the individual's sentiment. As we change our inner mental character, new intellectual ideas have to be acquired and the old are discarded, perhaps completely forgotten. They appear now to be ignorances and fallacies—mal-observation and bad reasoning. The new seem to be so much truer—perhaps infallibly true. All that is illusion. We make another advance, and the thoughts that a week before were as stable as rocks are now cast aside as absurd. Perhaps the belief in the certainty of present judgments is a condition of our making the best use of them; if so they should not be shakenuntil we are ready to enter on the next stage of knowledge.
It is quite true that one man may know more than another, but the ground on which the more intellectual is generally considered to be superior to the latter is not the right one. He is not better for his intellectual acquirements, but he is better if his mind, being of a finer sort, required a superior intellect to defend it. At bottom, then, the general cause of mal-observation—there are particular causes which interfere with the general rule—is inferiority of sentimental character. We do not see what we do not need to see, and we see imperfectly what is not essential to our well-being. That we should be ignorant or reason badly about what does not concern us is not in itself a defect.
It is inconsistent with these views on the function of intellect to admit that any sort of non-observation or mal-observation can be always and for all alike fallacious. If there are things which we habitually ignore, the presumption is that they do not concern us—that the knowledge they would confer is not essential to our welfare and would be intellectual lumber.
I should therefore abstain from condemning as fallacies theorems drawn in good faith from facts believed to be true, and which serve as motives of conduct. They are sophisms only when the reasonershave not taken ordinary pains to verify their data, or, knowing the antecedents to be false, pretend to believe them for some immoral purpose.
There is no fault of perversion, mutilation, or entanglement in the statement of an argument that we do not meet with in actual reasoning. Even in the writings of educated and honest thinkers it is rare to meet with an argument the parts of which are clearly distinguished by the author himself, and expressed so as to show the precise degree of force they ought to carry. Reasoning is still only a semi-conscious process directed by rule-of-thumb. We make certain statements and find they have a power of moving others, so we continue to make them. But whether the result is due to the rationality of the discourse or merely to the docility of the hearers, we do not know, and—so long as the desired result follows—we do not care to inquire.
For this state of things logicians are to a great extent responsible. They are uncritical imitators of the Greek philosophers, whose notions on dialecticwere quite wrong. The Greeks and their medieval and modern followers have squandered attention on a mental process which is not reason, mistaking it for reason, so that practically there has never been a science of dialectic. However much reasoners may have wished to present their thoughts coherently, they have not been provided with a method or notation adapted to the purpose. With an instinctive sense of the futility of the Syllogism, they have ignored it completely. I cannot call to mind a single controversial work that has been presented in syllogistic form, nor do even trained logicians use it overtly in argument.18Yet if it were what it professes to be, it would be as natural and convenient to express our arguments in syllogism as it is to put down on paper a sum in arithmetic. We are, as regardsthe expression of reasoning, in the position of numerical thinkers before the invention of figures and the elaboration of arithmetical rules. We have to do all our arguments 'in our head,' and so we do them badly. We can seldom be sure of the correctness of our own reasonings, and we are constantly being misled by sophistry. Nothing indeed will enable us to reason well or to detect false reasoning on a subject of which we are entirely ignorant, but a large measure of protection would be afforded by the adoption of a uniform system of presenting arguments, by which all the assumptions they involve are rendered explicit.
One of the commonest omissions in argumentation is to take the precedent for granted. This is allowable when it is a fact universally known or believed. 'If you let the glass fall it will be broken,'—the omitted precedent is the known consequences of letting brittle things fall to the ground. 'Caius is a liar, therefore he is a coward'—presupposes that every liar is a coward.
This liberty of suppression is sometimes used sophistically. The tacit precedent is not universally known or accepted, but if it is questioned the sophist is ready with an exclamation of surprise or contempt at our supposed ignorance. Persons who are afraid of appearing singular in their beliefs are liable to be deceived by this trick.
'It frequently happens,' says Whately, 'in the caseof a fallacy [of omitted precedent] that the hearers are left to the alternative of supplying either a premiss which is not true, or else one which does not prove the conclusion:e.g.if a man expatiates on the distress of the country, and thence argues that the government is tyrannical, we must suppose him to assume either that "every distressed country is under a tyranny," which is a manifest falsehood, or merely that "every country under a tyranny is distressed," which, however true, proves nothing, the Middle Term being undistributed.... Which are we to suppose the speaker meant us to understand? Surely just whichever each of his hearers might happen to prefer: some might assent to the false premiss; others allow the unsound syllogism; to the sophist himself it is indifferent, as long as they can be brought to admit the conclusion.'
'It frequently happens,' says Whately, 'in the caseof a fallacy [of omitted precedent] that the hearers are left to the alternative of supplying either a premiss which is not true, or else one which does not prove the conclusion:e.g.if a man expatiates on the distress of the country, and thence argues that the government is tyrannical, we must suppose him to assume either that "every distressed country is under a tyranny," which is a manifest falsehood, or merely that "every country under a tyranny is distressed," which, however true, proves nothing, the Middle Term being undistributed.... Which are we to suppose the speaker meant us to understand? Surely just whichever each of his hearers might happen to prefer: some might assent to the false premiss; others allow the unsound syllogism; to the sophist himself it is indifferent, as long as they can be brought to admit the conclusion.'
We sometimes attempt to reason fromContrastinstead of resemblance, with a confused notion that things which differ in some respects must differ also in others. 'Who spareth the rod hateth the child; the parent who loveth his child mustthereforespare not the rod.' The fallacy of this becomes apparent when we complete the theorem in the parallel form.
The following has often been presented as a valid argument—'What is universally believed must be true; the belief in God's existence is not universal; it is therefore not true.'
To establish the conclusion aimed at, it would be necessary to lay down as precedent—'What is not universally believed is not true.'
These theorems from contrast are on a par with the following—
This is the fallacy called in the quaint language of the syllogists 'Illicit Process of the Major Term.'
InFalse Analogythe resemblance is so slight that the application is untrustworthy, or a conclusion is drawn in excess of the resemblance. If from the habit of calling a deep bay or salt-water loch an 'arm' of the sea from its analogy to a human arm, we conclude that the sea has elbows and wrists, we commit this fallacy. The earth is like an orange, but we must not think that it is pulpy inside.
Akin to this is the fallacy ofFalse GeneralityorDoubtful Precedent. It consists in carelessly or perversely using bad antecedents when better are available. This applies to such current prejudices as that all Frenchmen are frivolous, all Germans mystical, all Jews dishonest, all Carthaginians faithless, all rich people purseproud, all nobles haughty, and so on. Even if all the Carthaginians we personally knew had proved faithless, our general knowledge of mankind should keep us from inferring that a whole nation should be faithless. The most we should conclude is thatsomeCarthaginians are faithless, but we are free to exercise caution in future dealings with members of that race. All these generalities are grounded on this prior argument: 'when a known portion of a class exhibits certain qualities, we are justified in inferring that the whole class possess these qualities'—which is only occasionally true.
The fallacy ofAccidentoccurs when the precedent is so defined as not to exclude exceptions, and the case happens to be one of the exceptions. 'What gives pain should be abstained from; surgical operations give pain; they should therefore be abstained from.' The painful things that should be universally abstained from are those which give needless or useless pain, not the sort that give less pain than they remove. Falstaff committed this fallacy when he supposed that the King would be a boon companionlike the Prince. So did the colonists who introduced rabbits and water-cress into Australia, on the supposition that they would there have the same function or value as in Great Britain. In consequence of the Accidental change the rabbits have developed into a pest, and the water-cress obstructs navigation.
If the applicate is a property of the subject only when the latter is taken collectively, it will not yield a true conclusion when the parts or individuals of the subject are taken separately. All the angles of a triangle are equal to two right angles, but it does not follow that one of them—though it resembles the triangle to some extent—is equal to two right angles. In this instance we should render the meaning clear by saying 'collectively equal,' when no argument follows and no mistake is made. This is called the fallacy ofDivision.
The fallacy ofCompositionis the converse of this. What is true of several singulars may not be true of all of them taken together. Because each of the witnesses in a law case is liable to error, it does not follow that the concurrent testimony of many is not to be credited. (Jevons.)
CircularorTautologicaltheorems (Petitio PrincipiiBegging the Question) are a breach of rule 2, sectionxviii. This fallacy often consists in proposing as a precedent the case, or information drawn from thecase and stated in other words. 'To allow every man an unbounded freedom of speech must always be, on the whole, advantageous to the State; for it is highly conducive to the interests of the Community that each individual should enjoy a liberty perfectly unlimited of expressing his sentiments.' (Whately.)
There may be tautology in a single word—the 'question-begging epithet.' We undertake to prove something, but get no further than the use of metaphors implying the point in dispute. For example, some scientific writers are anxious to promote the belief that animal life is a combination of natural forces—that there is no individual life distinct from cosmic life,—but all their proof consists in calling a man or beast a 'machine,' and calling machines 'creatures.' This might be mistaken for the Substantialist doctrine on the same subject, but the two are radically different. Substantialism asserts that man and nature havesimilarlives—materialism teaches that they have only one life in common, and that the coarse, mindless life of the cosmos as conceived realistically.
Conclusions may be used as precedents before verification, but it is not lawful to assume a hypothetical precedent on the understanding that it is to beproved in the course of the argument, and then use the conclusion so obtained to prove its own precedent. This is also dialectical tautology, but the circle includes two or more theorems. When naturalists tell us that in the struggle for life the fittest only survive, and when asked how we know which are the fittest they reply that the fittest are known by the fact of their surviving, we have a tautological argument.
Survival under competitive conditions is first assumed, and from it is deduced the superiority of the existing type of animal; then this inferential superiority is offered to justify the previously imagined competitive survival. The two hypotheses waltz round each other without making any rational advance.
When a book is quoted to prove its own authenticity we have this fallacy; or when the precedent is as unknown as the conclusion,—'Paradise was in Armenia, therefore Gihon is an Asiatic river.'
The academical syllogism as defined—not always as presented—contains two fallacies, one of which is tautology. 'AllEuropeans are white; Caius is a European; therefore he is white.' If, as logicianssay, the 'all' is absolute and includes Caius even before he is mentioned, then it is clear that the theorem amounts to saying, 'All Europeans are white, and one of them is Caius.' 'Both the twins are fair-haired; Caius is one of the twins; therefore he is fair-haired':—the pretended conclusion is merely a naming of a part of the precedent. The first of these theorems may be interpreted so as to give a valid conclusion. We are informed that an unknown person called Caius is a European; we are not told, and we do not know, what is the colour of his skin; but because all the Europeans we have known have been white, we infer—pending actual knowledge—that Caius is white. Logicians interpret the syllogism otherwise, for they have a notion that reason should give infallible certainty.
After the precedent has been divided into subject and applicate, the former is sometimes used as applicate and so generates a wrong conclusion. This may be calledCross ReasoningorDiagonal Reasoning—the fallacy termed by logicians 'Undistributed Middle.'
De Morgan has this example—'His imbecility of character might have been inferred from his pronenessto favourites; for all weak princes have this failing.'
Statements are sometimes put forward as reasoning which contain no case, either expressed or understood. This will seem hardly credible seeing that the illustration of a case is the purpose of argumentation. Not only does it occur, but a certain form of it is regarded by some logicians as valid reasoning. It is the 'particular' syllogism of the Third Figure.
Socrates was poor;Socrates was wise.
Socrates was poor;Socrates was wise.
From these premises no conclusion can be extracted, unless it be the verbal summary—'Socrates was both poor and wise.' But logicians draw from it the dialectic conclusion—
Therefore some men have been poor and wise,orTherefore one man has been poor and wise.
Therefore some men have been poor and wise,orTherefore one man has been poor and wise.
Both these conclusions are inadmissible. It is because they are empirically true that we are apt to think their truth depends on the antecedent information. If we wish to extend the qualities of Socrates to 'some men' we must make them a case with 'Socrates is poor and wise' for a precedent, but I failto see how it is to be done. If we add to the premises, 'One man was Socrates, therefore one man was poor and wise,' we have a tautological fallacy.
J. S. Mill notices a fallacy which amounts to anInversionof the Parallel: the conclusion is known or believed and the truth of the antecedents is inferred backwards.
'People continually think and express themselves as if they believed that the premises cannot be false if the conclusion is true. The truth, or supposed truth, of the inferences which follow from a doctrine, often enables it to find acceptance in spite of gross absurdities in it. How many philosophical systems which had scarcely any intrinsic recommendation have been received by thoughtful men because they were supposed to lend additional support to religion, morality, some favourite view of politics, or some other cherished persuasion; not merely because their wishes were thereby enlisted on its side, but because its leading to what they deemed sound conclusions appeared to them a strong presumption in favour of its truth, though the presumption, when viewed in its true light, amounted only to the absence of that particular evidence of falsehood which would have resulted from its leading by correct inference to something already known to be false.'19
'People continually think and express themselves as if they believed that the premises cannot be false if the conclusion is true. The truth, or supposed truth, of the inferences which follow from a doctrine, often enables it to find acceptance in spite of gross absurdities in it. How many philosophical systems which had scarcely any intrinsic recommendation have been received by thoughtful men because they were supposed to lend additional support to religion, morality, some favourite view of politics, or some other cherished persuasion; not merely because their wishes were thereby enlisted on its side, but because its leading to what they deemed sound conclusions appeared to them a strong presumption in favour of its truth, though the presumption, when viewed in its true light, amounted only to the absence of that particular evidence of falsehood which would have resulted from its leading by correct inference to something already known to be false.'19
The conclusion of an argument may sometimes be left unexpressed. If the antecedents are strong andthe conclusion obvious it weakens the argument to state the conclusion in full, besides reflecting on the capacity of the reader or hearer to draw the conclusion for himself. Hence we find at the end of controversial and indignant writings such expressions as—'Comment is superfluous'—'We leave the reader to draw his own conclusions,'—or simply a point of exclamation is appended.
Sophistical insinuations are suggested in this manner. A train of ideas is laid that generates a conclusion which the speaker is afraid or ashamed to put into words.
The second fault of the syllogism as defined may be called the fallacy ofNo Application. It consists in arranging propositions so as to end in a classification, but no applicate is detached and no rational conclusion is drawn. 'Jones is a Welshman; all Welshmen are Britons; therefore Jones is a Briton.' If in actual thinking it were ever desired to establish by argument that Jones is a Briton, it would be with the object of applying to him some quality connoted by Briton, but the presence of which in Jones is a matter of doubt. This would be a conclusion—but not the mere classification.
Irrelevant Conclusion—the fallacy called by AristoteliansIgnoratio elenchi—is an attempt to substitute a better argument for the one proposed, but which proves something which has not been denied, or stigmatisessomething that has not been asserted. It frequently arises from honest ignorance of the question at issue, as in the objections usually made to the Berkeleyan Substantialism. It can also be used as a weapon of sophistry, by confusing the matter in dispute or diverting attention to side issues. It is irrelevant to the truth of a conclusion to point out that he who now supports it formerly opposed it, or that his conduct is inconsistent with a belief in it. Appeals to passion—to reverence for authority—to popular belief—are instances of this fallacy.
The best protection against Fallacy—next to a thorough knowledge of the matter—is a clear notion of the properties of a valid argument; it is useful however to be able to distinguish and name the faulty theorems one constantly meets in controversial speeches and writings.
17:One fault of observation is noticed by formal logicians; it is that of assigning an improper cause,Non causa pro causâorPost hoc ergo propter hoc. It is evident that defects in every other category have an equal light to be noticed.
18:Whately complains of the disinclination shown by logicians to put their rules into practice. 'Whenever they have to treat of anything that is beyond the mere elements of Logic, they totally lay aside all reference to the principles they have been occupied in establishing and explaining, and have recourse to a loose, vague, and popular kind of language; such as would be best suited indeed to an exoterical discourse but seems strangely incongruous in a professed logical treatise.... Surely it affords but too much plausibility to the cavils of those who scoff at Logic altogether, that the very writers who profess to teach it should never themselves make any application of, or reference to, its principles, when, andwhen only, such application and reference are to be expected.'Logic, Book III. Introd. The fact here admitted proves that even logicians do not find their method of any practical use. But what is the meaning of the emphatic 'when only'? Why should a logical method be unsuitable for every sort of subject except those matters of logic that are beyond the mere elements?
19:Logic, 'Fallacies,' c. 6.
rule
Logicians of Greek inspiration apply the term reasoning or argument to at least eight different intellectual operations, some of them important indeed but only one of them argument. This is Analogy—which receives but little notice from logicians because it does not give certain conclusions. The operations mistaken for argument are:
Some logicians maintain that it is possible to draw a kind of conclusions from one judgment alone. These pretended conclusions are of two species.
The first is a restatement in different words of the whole or part of the single idea, and it is preceded by 'therefore' to give it the appearance of an argument. 'All men suffer, therefore some men suffer.' 'John is a man, therefore he is a living creature.' 'This weighs that down, therefore it is heavier.' These are all obvious tautologisms. It is not an inference to deny the opposite of what we have asserted, as 'The weather is warm, therefore it is not cold.' The conditional and dilemmatic examples of logicians abound in such 'inferences.' We cannot entirely avoid these locutions, as they give point and clearness to speech, but they are not argument, even when introduced by 'therefore.'
The other species of spurious conclusions arises out of what is technically called Conversion. This is a process permitted in Syllogistic in order to render propositions more explicit. The subject may change places with the predicate, a 'some' may be inserted, an 'all' suppressed, or a 'not' may be made toqualify one word instead of another. In all this there must be no change in the meaning of the proposition, and therefore there can be no inference. If the second proposition means something more or different from the first, another premise is unconsciously taken for granted, or the supposed interpretation amounts to interpolation. The reasoner may have inadvertently or sophistically added something to the original datum. Here is an example of inference by conversion—'All cabbages are plants, thereforesomeplants are cabbages.' If it is not understood from the terms of the first proposition that plants are limited to such as are cabbages, the 'some' of the converted proposition is an interpolation supplied from the reasoner's knowledge of the matter. In this case the 'quantification' of plants is not a valid inference from the original information.
Arithmetic is first a manipulation of symbols called 'figures.' There are ten of these, and they are capable of many species of combination, and an indefinite number of individual operations under each species.Certain rules govern each sort of operation, and when the rules are properly understood and recollected the operations can be performed with absolute certainty. Although the figures have names relating to number, and the problems given for exercise make mention of acres, pounds, tons, miles, and all sorts of concrete objects, the symbolic calculations of books have no necessary relation to real things, numbers, or quantities. They are a purely conventional treatment of arbitrary marks that may mean anything or nothing. That is the arithmetic of the 'schools.' There is no trace of reasoning or argument in it—it is mere rule and recollection.
There is however real Number and there is real Quantity. Number is that quality in which a group of three things (for instance) is seen to differ from a group of four or seven, even when the things are otherwise quite similar. We begin by distinguishing ten primary degrees of this difference, and then consider other degrees as multiples or parts of these primary degrees.
Quantity is degree in size, and is a property quite different from number. But, for convenience, we assume that quantities are all units or fractions of certain standard quantities, and we are thus enabled to use the same terms for both number and quantity.
The names which written language provides for the numerical degrees and their combinations are inconvenientto use, and so a set of symbols was devised exclusively for numerical designation. These are the figures of arithmetic. They are the technical vocabulary of number, and of quantity considered as number.
Number and quantity admit of but two kinds of variation—increase and diminution. These variations can be denoted so correctly by figures, that any combination we first make in figures according to rule can be reproduced in real objects, provided the objects are in other respects possible. The result of this perfection of technical nomenclature is that our study of number and quantity has been transferred from real objects to figures. It has become symbolic and indirect, and most of us never go beyond the symbols; that is, what we call arithmetic is an affair of figures, not of true quantities and numbers. We talk of miles, tons, and pounds sterling, but we do notthinkof miles, tons, and pounds sterling—we think offigures. A thousand shillings is to us, when arithmetically stated, '1000s.,' just as it is here represented on paper; we do not think of silver coins, and we could not if we tried imagine a thousand things of any sort. There is in reality an enormous difference between '0001s.' and '1000s.,' but to the arithmetician the only objective difference is one of arrangement in figures.
From these considerations it follows that there are two sciences of number. There is the true sciencewhich deals with quantities really seen in objects and imagined in the mind, and an artificial science dealing with figures which have only a historical connection with real quantity. Of the latter, unfortunately, our arithmetical education chiefly consists. We are never taught to distinguish number and size in things by the 'eye,' that is, by reason. The symbolism that was originally intended to assist real arithmetical thought has ended by supplanting it. An ignorant shepherd, bricklayer, or carpenter, who is accustomed to make a rapid estimate of the number of things in a mass, or the area of planking in a log, has a better training in real arithmetical science than some mathematicians. If we are obliged to practise genuine arithmetical thought in engineering, astronomy, and other professions, our scholastic symbolism gets realised to some extent, and is a great assistance in arithmetical estimation. But without this it has no more reference to number and quantity than a musical education, based entirely on the printed or written notation, has to the appreciation of musical sounds. A book arithmetician is in the position of a person thoroughly acquainted with theoretical music, and who can even compose musicaccording to rule, but who is unable to distinguish a high note from a low one or harmony from discord in actual sound.
It will thus be seen that it is only in the real arithmetic that reasoning can enter. The judgment infree arithmetical observation is the counting of actual groups and the measurement of actual surfaces, and the argument consists in estimating the number of individuals in other groups, and the size of other surfaces, without counting or measurement. But this exercise never enters into symbolic arithmetic. All the apparent conclusions of book arithmetic are tautological; they consist in repeating in one combination of symbols the whole or part of what has been already given in another combination. It is an exercise in expression—nothing more.
Arithmetical ratio has a resemblance to the rational parallel. 3 : 5 : : 9 : 15 might be arranged thus—
This is not argument, for two reasons. (1) The apparent conclusion is not an effort of rational imagination; it is a figure that can be obtained with infallible certainty by treating the other figures according to a rule, which has only to be recollected and applied. (2) The relation between the left-hand figures and the right-hand figures is not a categorical judgment; it is a form of resemblance, and so it cannot yield a valid conclusion.
This exercise is regarded by logicians as one of the purest forms of argument. It is nothing more than an aid to a certain kind of perception.
Take, for instance, the fifth proposition of the first book of Euclid—'The angles at the base of an isosceles triangle are equal, and if the equal sides be produced the angles on the other side shall also be equal.' The proposition is accompanied by a diagram of an isosceles triangle with the equal sides already produced, so that the conditional phrasing of the proposition does not mean that the production of the sides, and what results therefrom, are future or possible events which neither Euclid nor anybody else has yet experienced, and the probability of which is an argumentative conclusion.
What the proposition means is this: an isosceles triangle of which the equal sides have been produced, has equal angles on the same side of the base both within and without the triangle. It is an affirmation of what is, not of what we must believe to be for reasons to be given.
The truth of the proposition is seen at once from simple inspection of the diagram. It is an association of properties related in a certain manner. Ithas many relations which the geometer does not mention in this proposition, but those which he mentions are seen to be correctly described as soon as we direct attention to them. If we have any doubt on the subject we remove it by measuring the angles.
Euclid however does not appeal to the powers of inspection we can exercise in this case, and he ignores our facilities for measurement. He appeals to simpler and easier kinds of perception expressed in his axioms, which he began by assuming we were capable of exercising without demonstration. They constitute what he considers the minimum power of relational perception, which if a man have not he cannot be taught geometry. Euclid also in this proposition refers to the result of a prior demonstration, the relation in which he supposes we have seized. By means of these antecedents hepromptsour perceptive faculty up to the point of seeing the relations expressed in this proposition. If we saw them without the prompting, the latter is superfluous; if the relations do not stand the test of measurement, the prompting goes for nothing.
All Euclid's demonstrations are of this sort. They are pointings-out of what can be seen by inspection and sufficient attention. He is not bringing a case under a precedent—he is describing relations in things, that may serve as precedents in concrete or applied geometry. The service he performs is thatof a connoisseur who points out the beauties of a picture or landscape to a careless or uninterested spectator. Relations are sometimes difficult to see—much more difficult than colours or masses—and there is a legitimate sphere of usefulness for people who point out what others are apt to overlook. There is no prediction in this. We are not asked to conceive anything that is not before us. Geometrical demonstration thus assists perception, but does not imply reasoning. Euclid does not argue—he prompts.
Those who maintain that Euclid is syllogistic do so on the ground that the axioms are generalisations, and that as often as one is cited there occurs the subsumption of an object under a class-notion. That would not be argument; but let us suppose it means bringing a case under a precedent. Then if the axioms be precedents and the demonstration an application of them to new cases, the theorem is a fallacy—a useless argument written to prove a foregone certainty, for the conclusion can be and generally is perfectly known without reference to the demonstration.
It appears to me more true to regard the axioms as the simplest relations, which everybody may be supposed capable of perceiving, and that geometrical demonstration consists in showing that other relations not so apparent are really varieties or combinations of the simpler relations. By using in concert withthe axioms the relations already demonstrated, we are enabled to grasp relations that would not have been at all obvious on first beginning the geometrical praxis. Euclid's geometry is thus a series of graduated lessons in a special sort of observation, not a system of deductive arguments.
The educational theory that geometry is exceptionally good training for the reason—apart from its practical utility in mechanics—is thus evidently a mistake. Abstract geometry may induce habits of minute observation and exact definition, but reason nowhere enters into the study. As a rational gymnastic there is nothing better than the game of chess.
Those who contend that there is a kind of argument called Inductive different from the Deductive, illustrate their view by some such example as the following:—'This, that, and the other magnet' [that is, all the magnets we know] 'attract iron; therefore all possible magnets attract iron.' They say there is an irresistible compulsion in the mind to draw such a conclusion from information of the kind exemplified,and they contrast that type of thought with a deductive argument like—'All magnets attract iron; this object is a magnet; therefore it attracts iron.' They figure the former as a progress upwards, the latter as a regress downwards.
That is Induction as understood by J. S. Mill and Sir William Hamilton; on this point these philosophers happen to agree.
The first of those arguments is a deduction with the precedent omitted. Expressed in full it amounts to this—'Any relation observed several times to subsist between two classes of objects, and concerning which no exception has ever been observed, may be taken as universal; there is such a relation between known magnets and known iron; therefore it may be regarded as universal.' The precedent is not a mental compulsion, but a result of experience. Induction as above defined is therefore only a species of deductive conclusions.
Most logicians take the word Induction in its etymological sense, as meaning systematic observation carried on with a view to obtaining a general idea of some class of objects; or of establishing a categorical relation between one object or class and another, by eliminating all the alternative correlatives. In neither operation would Induction be argument.
In science a 'perfect induction' is one in which all existing objects of a class, or all objects related in acertain manner, have been perceived, so that there is no other object concerning which a conclusion can be drawn. In such cases, says Mill, there is no induction—only a summary of experience. He evidently regarded the conclusion with respect to unknown cases as the essence of induction, whereas in the scientific sense the induction is the positive content of the idea, or the abstract relation—the unknown cases are ignored, or there may be none. In scientific writings induction sometimes means themethodof observation rather than the result—the method of correcting inferences by perception, wherever possible.
This is usually put into English thus—'Whatever is affirmed or denied of a class, may be affirmed or denied of any part of that class,' and such an affirmation or denial is supposed to be an act of reason. Archbishop Whately expounds the Dictum in analysing the following theorem—Whatever exhibits marks of design had an intelligent author; the world exhibits marks of design; therefore the world had an intelligent author.
'In the first of these premises,' he says, 'we find it assumed universally of theclassof "things which exhibit marks of design," that they had an intelligent author; and in the other premise, "the world" is referred to that class as comprehended in it: now it is evident that whatever is said of the whole of a class, may be said of anything comprehended in that class: so that we are thus authorised to say of the world, that "it had an intelligent author." Again, if we examine a syllogism with a negative conclusion, as,e.g."nothing which exhibits marks of design could have been produced by chance; the world exhibits, &c.; therefore the world could not have been produced by chance:" the process of Reasoning will be found to be the same; since it is evident, that whatever isdenieduniversally of any class may be denied of anything that is comprehended in that class. On further examination it will be found, that all valid arguments whatever may be easily reduced to such a form as that of the foregoing syllogisms; and that consequently the principle on which they are constructed is theUniversal Principleof Reasoning.'20
'In the first of these premises,' he says, 'we find it assumed universally of theclassof "things which exhibit marks of design," that they had an intelligent author; and in the other premise, "the world" is referred to that class as comprehended in it: now it is evident that whatever is said of the whole of a class, may be said of anything comprehended in that class: so that we are thus authorised to say of the world, that "it had an intelligent author." Again, if we examine a syllogism with a negative conclusion, as,e.g."nothing which exhibits marks of design could have been produced by chance; the world exhibits, &c.; therefore the world could not have been produced by chance:" the process of Reasoning will be found to be the same; since it is evident, that whatever isdenieduniversally of any class may be denied of anything that is comprehended in that class. On further examination it will be found, that all valid arguments whatever may be easily reduced to such a form as that of the foregoing syllogisms; and that consequently the principle on which they are constructed is theUniversal Principleof Reasoning.'20
The examples given by Whately are perfectly valid; the first is a constructive argument in the Sixth Category, the second a stigmatic in the Fifth. I have in several places admitted that the arguments adducedby syllogists are sometimes correct, the fault complained of being in the mode in which such correct arguments are interpreted. They are interpreted wrongly, and then other theorems are found or made agreeing with theinterpretation, and the admitted soundness of the first theorems is used to procure acceptance for the second. Things brought under the same definition ought to be essentially alike, but they are not so when the utmost latitude is taken to 'assume' that predicates have properties which they obviously have not.
The objections we make to the Dictum as above interpreted are—(1) that in reasoning the precedent (major premise) need not be a class; (2) if it is a class, it consists of allknownthings of a similar kind, not of allpossiblethings of a similar kind. When interpreted in the latter sense the Dictum becomes dialectically tautological, as has been often observed.
A few pages further on Whately gives a totally different account of reasoning, without being aware of his inconsistency.
'Every syllogism has three, and only three terms: viz. the middle term and the two terms (or extremes, as they are commonly called) of the Conclusion or Question. Of these, first, the subject of the conclusion is called theminorterm; second, its predicate, themajorterm; and third, themiddleterm, (called by the older logicians "Argumentum") is that with which each of them is separately compared, in order to judge of their agreement or disagreement with each other. If therefore there were two middle terms, the extremes or terms of conclusion not being both compared to the same, could not be conclusively compared to each other.'21
'Every syllogism has three, and only three terms: viz. the middle term and the two terms (or extremes, as they are commonly called) of the Conclusion or Question. Of these, first, the subject of the conclusion is called theminorterm; second, its predicate, themajorterm; and third, themiddleterm, (called by the older logicians "Argumentum") is that with which each of them is separately compared, in order to judge of their agreement or disagreement with each other. If therefore there were two middle terms, the extremes or terms of conclusion not being both compared to the same, could not be conclusively compared to each other.'21
Here reasoning is made to consist in comparing two things by reference to a third which both resemble. There is not a word about classification, which is declared just before—in loud capitals—to be the universal principle of reasoning!
On this definition we remark—
(1) Comparison by mediation is untrustworthy, unless the qualities compared be rigidly defined or restricted, as in geometry and the use of standards (xxii). In geometry the only two qualities recognised are figure and magnitude. The axiom of mediate comparison means that things having the same magnitude as a third thing are to be considered equal, though they may have different outlines. But theaxiom is liable to be untrue in things of three or more qualities. Add colour. Then a white sphere may resemble a white cube on the one side, and a black sphere on the other, but the white cube does not at all resemble the black sphere. This axiom is therefore inadmissible or at least extremely risky in logic, which treats of things having many qualities.
(2) Comparison, however correctly performed, is never the end, but only a means, of reasoning.
We have already had two distinct definitions of syllogism. According to the first it is the application of class-attributes to individuals known to belong to the class; according to the second it is the comparison of two things or terms by reference to a third which both resemble. When we arrive at the chapters in logic books devoted to the exposition of the syllogism in detail, we find that the theorems there discussed do not conform to either of those definitions. The only sort of syllogism that can be 'converted' is one consisting of two classifications, and a conclusion which predicates a classification, as thus—