Chapter XII: Introduction: I--Relation of Berkeley's Philosophy to Other Systems (5)

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 of _Division_.

The fallacy of _Composition_ is 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.)

_Circular_ or _Tautological_ theorems (_Petitio Principii_ Begging the Question) are a breach of rule 2, section XVIII. This fallacy often consists in proposing as a precedent the case, or information drawn from the case 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.)

It is conducive | that each individual should enjoy
-------------------+------------------------------
It is advantageous | to allow

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 have _similar_ lives--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 be proved 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.

Animals that survive | are the fittest | Fittest animals | survive
---------------------+------------------+-----------------+--------------
A particular animal | _hence it is the | This animal is | _which is the
has survived | fittest of its | the fittest of | reason it has
| species_ | its species. | survived_

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. 'ALL Europeans are white; Caius is a European; therefore he is white.' If, as logicians say, 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 called _Cross Reasoning_ or _Diagonal Reasoning_--the fallacy termed by logicians 'Undistributed Middle.'

Manx cats | have no tails
---------------------+----------------------
This cat has no tail | _it must be a Manx cat_

De Morgan has this example--'His imbecility of character might have been inferred from his proneness to favourites; for all weak princes have this failing.'

All weak princes | are prone
-----------------+------------------------
He was prone | _he must have been weak_

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._

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_,
or
_Therefore 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 fail to 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 an _Inversion_ of 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]

The conclusion of an argument may sometimes be left unexpressed. If the antecedents are strong and the 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 of _No 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 Aristotelians _Ignoratio elenchi_--is an attempt to substitute a better argument for the one proposed, but which proves something which has not been denied, or stigmatises something 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.

[Footnote 17: One fault of observation is noticed by formal
logicians; it is that of assigning an improper cause, _Non
causa pro causâ_ or _Post hoc ergo propter hoc_. It is evident
that defects in every other category have an equal light to be
noticed.]

[Footnote 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, and _when 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?]

[Footnote 19: _Logic_, 'Fallacies,' c. 6.]

ACADEMICAL DIALECTIC

XXXI--ANALOGY

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:

Immediate Inference--
Arithmetical Calculation--
Geometrical Demonstration--
Induction--
Aristotle's Dictum--
Mediate Comparison--
Syllogism.

XXXII--IMMEDIATE INFERENCE

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 to qualify 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, therefore _some_ plants 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.

XXXIII--ARITHMETICAL CALCULATION

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 inconvenient to 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 not _think_ of miles, tons, and pounds sterling--we think of _figures_. A thousand shillings is to us, when arithmetically stated, '1000_s._,' 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 '0001_s._' and '1000_s._,' 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 science which 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 music _according 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 in free 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--

5 | 15
--+---
3 | 9

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.

XXXIV--GEOMETRICAL DEMONSTRATION

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. It has 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 he _prompts_ our 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 that of 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 with the 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.

XXXV--INDUCTION

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 a certain 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 the _method_ of observation rather than the result--the method of correcting inferences by perception, wherever possible.

XXXVI--ARISTOTLE'S DICTUM

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 the _class_ of "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 is _denied_
universally 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
the UNIVERSAL PRINCIPLE of 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 adduced by 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 the _interpretation_, 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 all _known_ things of a similar kind, not of all _possible_ things of a similar kind. When interpreted in the latter sense the Dictum becomes dialectically tautological, as has been often observed.

XXXVII--MEDIATE COMPARISON

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 the _minor_
term; second, its predicate, the _major_ term; and third, the
_middle_ term, (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 the axiom 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.

XXXVIII--SYLLOGISM

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--

_All Englishmen are Europeans;
John Smith is an Englishman;
therefore John Smith is a European._

Observe the difference between this theorem and that adduced in illustration of the Dictum (XXXVI). In the latter the first premise is a categorical judgment and so therefore is the conclusion; in the theorem just given the first premise is a classification, and the conclusion is necessarily a classification.

We first remark that such an 'argument' is never met with in real spontaneous thinking--it occurs only in logic books. It is manufactured exclusively for Peripatetic consumption. The reason it is not to be found is simple--the conclusion it yields is a classification, and that is not enough for valid argument. In reasoning we may introduce a classification as the _minor premise_--that is, the proposition which brings the case under the precedent--but the applicate is never a general or class idea. It is one or more properties abstracted from the subject (whether the latter be a single object or general idea), and applied to the case. Merely to classify a case and so leave it would answer no rational purpose.

Logicians urge in recommendation of this syllogism that it gives a certain conclusion. The premises being correct, the conclusion is infallibly true.

No doubt it is, for in contemplating a thing we can mentally enter it into all the classes to which it appears to belong, whatever be their generality. Knowing the class European and the individual John Smith, we see at once that the latter is contained in the former, and we can do this without putting him first in the minor class English. It is like saying, 'The pavilion is in the garden, John Smith is in the pavilion, therefore he is in the garden.' Of course he is! The minor premise of a double classification is superfluous. The fact that such conclusions are certain, shows how nugatory they are. We are not certain of anything till it has been experienced. In legitimate reasoning the conclusion is never more than probable. The certainty of these double classifications shows that we are stating what we already know--not imagining an ideal addition to our positive knowledge.

_Doctrine of the Predicate._ So long as logicians are permitted to fabricate their own examples, all is plain sailing with the syllogism. But they are sometimes obliged to deal with genuine arguments. In this case what they do is to assume that _for logical purposes_ every predicate of the precedent--that is, the applicate--is a general or class term. Even when an argument is good they spoil it with a bad theory.

Sir William Hamilton states that up to his time logicians recognised but one type of proposition--that called by him the proposition 'in extension,' which means the classifying of the subject. He announced that he intended to introduce a proposition 'in comprehension,' meaning a judgment in the category of inherence--as for instance, 'man is responsible.' He further said that he recognised a third type of proposition, that concerning 'cause and effect.'

But in the course of working out these logical novelties he seems to have discovered that they were irreconcilable with conversion, and so he dropped them. The judgment in comprehension, he then declared, was to all intents and purposes the same as one in extension, and as to causation--why, a cause is a class, and an effect is an individual belonging to that class![22]

Let us see what is the result of treating applicates as general ideas. Take an example in each of the categories.

'The paper is white.' This means that the paper has the property or attribute of whiteness. In logic it is interpreted to mean that paper is an individual of the class _white_. This is wrong, for there is no such class. No sane person would form a class out of salt, snow, milk, china, silver, the moon, and other white things; for though they have a common property it is not the sign of a common human utility.

The confusing a single property with a class is not always owing to exigencies of syllogism. It pervades the writings of most Western metaphysicians, and may be accounted for in this manner.

General ideas and abstract properties or ideas have in common that they are _partial_ recognitions of what we perceive (XIV). The partition in each is however made in a different way, and for a different purpose. In generalisation the selection is done almost mechanically. We see many things that have some common relation, function, or utility for us, and we remember only so much of them as appears to be necessary for the recognition of that relation or utility--just so much of the Intellectual experience as has always accompanied the Sentimental experience. The process is very like that of putting a piece of wood or ivory in a turning-lathe, and whittling off all that we do not want. A general idea is the useful core of a multitude of superposed observations, each of which had something irrelevant--something which it is better to forget. We whittle this off and remember only the core.

Abstraction, on the other hand, is a conscious and deliberate operation from beginning to end. It consists in distinguishing one by one the properties of a thing, and even treating each property as if it had an independent existence. For this exercise it is not necessary to observe many things: we can analyse one alone, though an acquaintance with other cognate objects is sometimes necessary to call our attention to single properties. We need the shock of difference to be able to distinguish well a fine abstraction--the difference between shades of colours, for example. Abstraction is thus a minute attention to individuals, and need not for a moment be confounded with generalisation.

Another cause of the confusion in question can be traced to the use of the verb 'is' to represent both the relation of a thing to the general idea it has contributed to form, and the relation of a single property to the thing in which it inheres. We say 'The man _is_ a British subject'--classifying him; we say also 'The man _is_ cold'--mentioning one of his attributes. There is no class of cold men, and the two relations have nothing in common. A class does not inhere in a man as cold inheres in him. There is no _object_ corresponding to class--it is a conceptual creation.

The ambiguity of 'is' favours the syllogistic doctrine of predication, and there is a rule to the effect that in syllogising propositions, all verbs are to be converted into 'is' (or its conjugates) with a participle or noun, so that if they were not before statements of classification they now become such. 'He walks' is clearly no classification; but 'he is walking' is assimilated by false analogy to such a classification as 'he is human,' and so is treated as a classification by those who reason according to the Letter.

The substantive verb has no positive and uniform meaning. As an auxiliary it is a mere sign of tense, and in other positions it is an indefinite mark of relationship, the precise meaning of which must be determined by the subject and the context. It may sometimes be dispensed with in classification, as 'Victoria Regina'--'Phillips, Dentist.'

In the second category we have such propositions as 'the book lies on the table.' In syllogistic this is first altered to 'the book is lying on the table,' and it is feigned that 'lying on the table' is a class or general idea, and 'book' an individual of that class. To interpret 'the groom stands by the horse' a class has to be created, composed of the persons who happen to be standing by horses.

'The mountain is ten miles off' is a judgment in perspection. Syllogistically we are asked to believe that a class of things exists having the common property of being ten miles off, and that the mountain is entered in that class. The absurdity of this doctrine is self-evident.

In the remaining categories the reduction to 'is' has, if possible, a worse effect. In changing 'Canada lies west of Ireland' into 'Canada is a country lying west of Ireland,' we lose the relation in concretion, and express instead a verbal definition. Instead of affirming a position we explain a name. In such a proposition as 'the town of A lies 100 miles due north of B,' it is plain the predicate cannot be a class, for only one place has the quality expressed.

In the fifth category we have such a proposition as 'water freezes when the temperature falls to zero Centigrade.' This is turned into a substantive sentence by saying 'water is that liquid which freezes,' &c., which is a verbal or identical proposition.

'Cecrops founded Athens' is a judgment in causation. In turning it into 'Cecrops was (or is) the founder of Athens,' we emphasise the man's name, but the relation signified by 'founded' is slurred over or lost sight of. Boole converts 'Caesar conquered the Gauls' into 'Caesar is he who conquered the Gauls,'[23] and this he interpreted as classification. We need not be surprised that he should suppose a class could be formed by one individual, for he elsewhere tells us that _Nothing_ is a class.[24]

Classification is not judgment of any sort--it is a variety of recollection. Logicians imagine it is the only judgment, and so far as they can they degrade true judgments to that spurious form.

_Moods of the Syllogism._ Having persuaded themselves that classification is the beginning, middle, and end of reasoning, logicians next proceed to divide the matter of their science.

Modern logicians who have some acquaintance with real thinking as exemplified in works of physical science, can, if acting according to their natural intelligence, lay down correct rules for dividing a subject. These are simple and obvious: divide according to fundamental resemblance--let each division correspond to some definite human utility--let the more important properties take precedence of the less important, and so forth: the merest common sense.

But in the division of their own subject they follow Aristotle, and so lose their way.

It is plain that an act of reasoning is a mental thing in the first place, and only when uttered, and thus in a secondary sense, is it a material object. The classification of arguments should therefore follow mental characteristics. Logicians make it follow the material characteristics of the terms in which the arguments are uttered. Their moods of the syllogism are mere varieties of expression, not varieties of reason.

The number of these moods is accidental, depending on flexibility of language and ingenuity in inventing varieties of syntax. Mere transposition of premises constitutes a difference of mood. Logicians however pretend to base their numeration on a more general necessity. They calculate from the distinctive parts of the three propositions forming a syllogism, varied by negation, &c., that there _ought to be_ sixty-four moods. Experience proves that in spite of their free and easy method of multiplying syllogistic varieties they cannot produce anything like that number. One logician has thirty-six moods, another thirty-two, a third eleven; the more orthodox fix the number at nineteen. But they all admit that every argument can be reduced to one of four fundamental types--the moods of the First Figure. Why then have more classes than these four? Because, says Whately, it would be 'occasionally tedious' to reduce every argument to the first figure.

If the 11, 19, 32, or 36 classes were natural arguments taken down untouched from men's lips, and it was found to be useless and troublesome to reduce them to four artificial forms, the plea might be admitted. But the so-called valid syllogisms are themselves artificial, and just as tedious to make as the moods of the first figure. Not only so, but an elaborate system of mnemonic rules is provided for reducing the valid moods to the fundamental moods, thus admitting that the former are only intermediate halting places between the natural speech and the fundamental moods. It is _expected_ that the intermediates should be reduced to the first figure.

Is there anything analogous to this sort of division in any science or branch of practical thought? Would logicians themselves sanction such a classification in a natural science? If a zoologist, for example, were to determine beforehand how many classes of animals there ought to be, would they not say he was acting improperly? If, after discovering that he had five times as many classes as he could find animals to put into them, he still retained his classification and required his pupils to write out the names or symbols of all the useless classes--would not logicians be apt to call him a pedant? Yet in a modern work on logic such a task is prescribed for students:--

'Write out the sixty-four moods of the syllogism, _and strike
out the fifty-three invalid ones_.'

We might have excused the existence of a merely verbal classification in logic, if it were accompanied by and subordinated to a classification of theorems considered as mental facts. But in syllogistic the verbal is the dominant classification, and we have seen from the procedure of Sir William Hamilton--in dropping his categorical judgments--that when the two principles of division conflict, it is the mental which has to give way. The Letter is allowed to kill the Spirit.

_All the Moods reducible to One._ Syllogists appear not to know their own schematism very well. They say there are four ultimate moods, which it is impossible to reduce to any lower number. But since each of the four is, mentally, a double classification, it must be possible to reflect this common property in the mode of expression. The difference between them can only be verbal. Let us adopt another than the ordinary symbolism.

Cut a card into three triangular pieces of unequal size, and call them by the letters A, B, C, beginning with the largest. These are the terms of the syllogism.

A A
/\B /\ B
/ /\C / \ /\C
/ / /\ / \ / /\
/ / / \ / \ / / \
/ / / \ / \ / / \
/ / / \ / \ / / \
+------------+ +------------+ +----------+
_Barbara._ _Celarent._

A A
/\B /\ B
/ /\ C / \ /\ C
/ / \ : / \ / \ :
/ / \: : / \ / \: :
/ / /\ : / \ / /\ :
/ / / \ : / \ / / \ :
+------------+... +------------+ +----------+...
_Darii._ _Ferio._

The first mood _Barbara_ is formed by placing the cards on top of each other, so that B is within the margin of A, and C within the margin of B. This is the syllogism, 'All B is A, all C is B, therefore all C is A.'

Next let B and C be as above, but let A be wholly apart from both. This is _Celarent_: 'No B is A, all C is B, therefore no C is A.'

In _Darii_ the whole of B is in A, but only a part of C coincides with B. The syllogism is: 'All B is A, some C is B, therefore some C is A.'

In _Ferio_ A is again wholly separated from the others, and C is only partially in B. Argument: 'No B is A, some C is B, therefore some C is not A.'

It is to be remembered that all the other figures and moods are reducible to the above figure of four moods, so that the reduction applicable to the latter is equally applicable to the former.

To reduce _Darii_ to _Barbara_ all that is necessary is to ignore the dotted part of C. That is suggested by the use of the word 'some,' which has a correlative 'all' or 'others.' But the correlative quantity does not enter into the syllogism, and we know nothing about it. It may not even exist. We are therefore at liberty to substitute for 'some C' the name D, and consider it an integer instead of a fraction. Then we have the _Barbara_ syllogism: 'All B is A, all D (= some C) is B, therefore all D is A.' The phrase 'all of some' is quite allowable: 'I met some firemen, all of whom wore brass helmets.'

_Ferio_ in the same manner is reduced to _Celarent_. The dotted part of C is cut away, and the part really significant in the syllogism is called E. Then 'No B is A, all E is B, no E is A.'

Finally _Celarent_ can be reduced to _Barbara_. B cannot indeed be enclosed in A, but we assume the existence of a whole having all the characters which A has _not_, or having none of the characters which A has. This is the whole F = Not-A. Then _Celarent_ becomes _Barbara_ thus: 'All B is F, all C is B, therefore all C is F.'

This demonstrates that there is only one fundamental operation where syllogists suppose there are at least four. The difference is wholly a matter of language, and disappears on changing the names of the terms and ignoring irrelevant suggestions. But the syllogism, I repeat, does not represent the act of reasoning, and its moods and figures are fit only to be a game for children.

[Footnote 20: _Logic_, Book I. § 3.]

[Footnote 21: _Logic_, Book II. c. 3. § 2.]

[Footnote 22: _Lectures_, iii. pp. 287 and 356. The
impossibility of reconciling their definitions and rules to
real thinking and argument is the despair of logicians. Most
of them take to symbols, which are more accommodating than
real experience, having just such properties as their makers
choose to put in them. Sir William Hamilton had the courage to
declare that a logician might use arguments of a concrete or
real form, but that it is not necessary they should agree with
real fact. 'The logician has a right to suppose any material
impossibility, any material falsity; he takes no account of
what is objectively impossible or false, he has a right to
assume what premises he please, provided that they do not
involve a contradiction in terms.'--_Id._ 322. That means
in plain English that a logician may misrepresent matters of
fact, if he cannot otherwise establish his theory!]

[Footnote 23: _Laws of Thought_, p. 35.]

[Footnote 24: _Ibid._ p. 47.]

STUDIES IN DIALECTIC

XXXIX

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

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

***

Here are examples of arguments in inherence--

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

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

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

I.
Cloven feet | inhere with horns
-------------------------+-------------------
Fossil animal appears to | _it is probable he
have had cloven feet | had horns_

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

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

***

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