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Deductive Logic by St. George Stock

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mediate. These are called formal, because the truth of the consequent
is apparent from the mere form of the antecedent, whatever be the
nature of the matter, that is, whatever be the terms employed in the
proposition or pair of propositions which constitutes the
antecedent. In deductive inference we never do more than vary the form
of the truth from which we started. When from the proposition 'Brutus
was the founder of the Roman Republic,' we elicit the consequence 'The
founder of the Roman Republic was Brutus,' we certainly have nothing
more in the consequent than was already contained in the antecedent;
yet all deductive inferences may be reduced to identities as palpable
as this, the only difference being that in more complicated cases the
consequent is contained in the antecedent along with a number of other
things, whereas in this case the consequent is absolutely all that the
antecedent contains.

§ 435. On the other hand, it is of the very essence of induction that
there should be a process from the known to the unknown. Widely
different as these two operations of the mind are, they are yet both
included under the definition which we have given of inference, as the
passage of the mind from one or more propositions to another. It is
necessary to point this out, because some logicians maintain that all
inference must be from the known to the unknown, whereas others
confine it to 'the carrying out into the last proposition of what was
virtually contained in the antecedent judgements.'

§ 436. Another point of difference that has to be noticed between
induction and deduction is that no inductive inference can ever attain
more than a high degree of probability, whereas a deductive inference
is certain, but its certainty is purely hypothetical.

§ 437. Without touching now on the metaphysical difficulty as to how
we pass at all from the known to the unknown, let us grant that there
is no fact better attested by experience than this--'That where the
circumstances are precisely alike, like results follow.' But then we
never can be absolutely sure that the circumstances in any two cases
are precisely alike. All the experience of all past ages in favour of
the daily rising of the sun is not enough to render us theoretically
certain that the sun will rise tomorrow We shall act indeed with a
perfect reliance upon the assumption of the coming day-break; but, for
all that, the time may arrive when the conditions of the universe
shall have changed, and the sun will rise no more.

§ 438. On the other hand a deductive inference has all the certainty
that can be imparted to it by the laws of thought, or, in other words,
by the structure of our mental faculties; but this certainty is purely
hypothetical. We may feel assured that if the premisses are true, the
conclusion is true also. But for the truth of our premisses we have to
fall back upon induction or upon intuition. It is not the province of
deductive logic to discuss the material truth or falsity of the
propositions upon which our reasonings are based. This task is left to
inductive logic, the aim of which is to establish, if possible, a test
of material truth and falsity.

§ 439. Thus while deduction is concerned only with the relative truth
or falsity of propositions, induction is concerned with their actual
truth or falsity. For this reason deductive logic has been termed the
logic of consistency, not of truth.

§ 440. It is not quite accurate to say that in deduction we proceed
from the more to the less general, still less to say, as is often
said, that we proceed from the universal to the particular. For it may
happen that the consequent is of precisely the same amount of
generality as the antecedent. This is so, not only in most forms of
immediate inference, but also in a syllogism which consists of
singular propositions only, e.g.

The tallest man in Oxford is under eight feet.
So and so is the tallest man in Oxford.
.'. So and so is under eight feet.

This form of inference has been named Traduction; but there is no
essential difference between its laws and those of deduction.

§ 441. Subjoined is a classification of inferences, which will serve
as a map of the country we are now about to explore.

Inference
________________________|__________
| |
Inductive Deductive
_________________|_______________
| |
Immediate Mediate
___________|__________ ______|______
| | | |
Simple Compound Simple Complex
______|________________ | ______|_____________|_
| | | | | | |
Opposition Conversion Permutation | Conjunctive Disjunctive Dilemma
|
_________|________
| |
Conversion Conversion
by by
Negation position

CHAPTER II.

_Of Deductive Inferences._

$ 442. Deductive inferences are of two kinds--Immediate and Mediate.

§ 443. An immediate inference is so called because it is effected
without the intervention of a middle term, which is required in
mediate inference.

§ 444. But the distinction between the two might be conveyed with at
least equal aptness in this way--

An immediate inference is the comparison of two propositions directly.

A mediate inference is the comparison of two propositions by means of
a third.

§ 445. In that sense of the term inference in which it is confined to
the consequent, it may be said that--

An immediate inference is one derived from a single proposition.

A mediate inference is one derived from two propositions conjointly.

§ 446. There are never more than two propositions in the antecedent of
a deductive inference. Wherever we have a conclusion following from
more than two propositions, there will be found to be more than one
inference.

§ 447. There are three simple forms of immediate inference, namely
Opposition, Conversion and Permutation.

§ 448. Besides these there are certain compound forms, in which
permutation is combined with conversion.

CHAPTER III.

_Of Opposition._

§ 449. Opposition is an immediate inference grounded on the relation
between propositions which have the same terms, but differ in quantity
or in quality or in both.

§ 450. In order that there should be any formal opposition between two
propositions, it is necessary that their terms should be the
same. There can be no opposition between two such propositions as
these--

(1) All angels have wings.

(2) No cows are carnivorous.

§ 451. If we are given a pair of terms, say A for subject and B for
predicate, and allowed to affix such quantity and quality as we
please, we can of course make up the four kinds of proposition
recognised by logic, namely,

A. All A is B.

E. No A is B.

I. Some A is B.

O. Some A is not B.

§ 452. Now the problem of opposition is this: Given the truth or
falsity of any one of the four propositions A, E, I, O, what can be
ascertained with regard to the truth or falsity of the rest, the
matter of them being supposed to be the same?

§ 453. The relations to one another of these four propositions
are usually exhibited in the following scheme--

A . . . . Contrary . . . . E
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
Subaltern Contradictory Subaltern
. . . .
. . . .
. . . .
. . . .
. . . .
. . . .
I . . . Sub-contrary . . . O

§ 454. Contrary Opposition is between two universals which differ in
quality.

§ 455. Sub-contrary Opposition is between two particulars which differ
in quality.

§ 456. Subaltern Opposition is between two propositions which differ
only in quantity.

§ 457. Contradictory Opposition is between two propositions which
differ both in quantity and in quality.

§ 458. Subaltern Opposition is also known as Subalternation, and of
the two propositions involved the universal is called the Subalternant
and the particular the Subalternate. Both together are called
Subalterns, and similarly in the other forms of opposition the two
propositions involved are known respectively as Contraries,
Sub-contraries and Contradictories.

§ 459. For the sake of convenience some relations are classed under
the head of opposition in which there is, strictly speaking, no
opposition at all between the two propositions involved.

§ 460. Between sub-contraries there is an apparent, but not a real
opposition, since what is affirmed of one part of a term may often
with truth be denied of another. Thus there is no incompatibility
between the two statements.

(1) Some islands are inhabited.

(2) Some islands are not inhabited.

§ 461. In the case of subaltern opposition the truth of the universal
not only may, but must, be compatible with that of the particular.

§ 462. Immediate Inference by Relation would be a more appropriate
name than Opposition; and Relation might then be subdivided into
Compatible and Incompatible Relation. By 'compatible' is here meant
that there is no conflict between the _truth_ of the two
propositions. Subaltern and sub-contrary opposition would thus fall
under the head of compatible relation; contrary and contradictory
relation under that of incompatible relation.

Relation
______________|_____________
| |
Compatible Incompatible
______|_____ _____|_______
| | | |
Subaltern Sub-contrary Contrary Contradictory.

§ 463. It should be noticed that the inference in the case of
opposition is from the truth or falsity of one of the opposed
propositions to the truth or falsity of the other.

§ 464. We will now lay down the accepted laws of inference with regard
to the various kinds of opposition.

§ 465. Contrary propositions may both be false, but cannot both be
true. Hence if one be true, the other is false, but not vice versâ.

§ 466. Sub-contrary propositions may both be true, but cannot both be
false. Hence if one be false, the other is true, but not vice versâ.

§ 467. In the case of subaltern propositions, if the universal be
true, the particular is true; and if the particular be false, the
universal is false; but from the truth of the particular or the
falsity of the universal no conclusion can be drawn.

§ 468. Contradictory propositions cannot be either true or false
together. Hence if one be true, the other is false, and vice versâ.

§ 469. By applying these laws of inference we obtain the following
results--

If A be true, E is false, O false, I true.

If A be false, E is unknown, O true, I unknown.

If E be true, O is true, I false, A false.

If E be false, O is unknown, I true, A unknown.

If O be true, I is unknown, A false, E unknown.

If O be false, I is true, A true, E false.

If I be true, A is unknown, E false, O unknown.

If I be false, A is false, E true, O true.

§ 470. It will be seen from the above that we derive more information
from deriving a particular than from denying a universal. Should this
seem surprising, the paradox will immediately disappear, if we reflect
that to deny a universal is merely to assert the contradictory
particular, whereas to deny a particular is to assert the
contradictory universal. It is no wonder that we should obtain more
information from asserting a universal than from asserting a
particular.

§ 471. We have laid down above the received doctrine with regard to
opposition: but it is necessary to point out a flaw in it.

When we say that of two sub-contrary propositions, if one be false,
the other is true, we are not taking the propositions I and O in their
now accepted logical meaning as indefinite (§ 254), but rather in
their popular sense as 'strict particular' propositions. For if I and
O were taken as indefinite propositions, meaning 'some, if not all,'
the truth of I would not exclude the possibility of the truth of A,
and, similarly, the truth of O would not exclude the possibility of
the truth of E. Now A and E may both be false. Therefore I and O,
being possibly equivalent to them, may both be false also. In that
case the doctrine of contradiction breaks down as well. For I and O
may, on this showing, be false, without their contradictories E and A
being thereby rendered true. This illustrates the awkwardness, which
we have previously had occasion to allude to, which ensures from
dividing propositions primarily into universal and particular, instead
of first dividing them into definite and indefinite, and particular (§
256).

§ 472. To be suddenly thrown back upon the strictly particular view of
I and O in the special case of opposition, after having been
accustomed to regard them as indefinite propositions, is a manifest
inconvenience. But the received doctrine of opposition does not even
adhere consistently to this view. For if I and O be taken as strictly
particular propositions, which exclude the possibility of the
universal of the same quality being true along with them, we ought not
merely to say that I and O may both be true, but that if one be true
the other must also be true. For I being true, A is false, and
therefore O is true; and we may argue similarly from the truth of O to
the truth of I, through the falsity of E. Or--to put the Same thing in
a less abstract form--since the strictly particular proposition means
'some, but not all,' it follows that the truth of one sub-contrary
necessarily carries with it the truth of the other, If we lay down
that some islands only are inhabited, it evidently follows, or rather
is stated simultaneously, that there are some islands also which are
not inhabited. For the strictly particular form of proposition 'Some A
only is B' is of the nature of an exclusive proposition, and is really
equivalent to two propositions, one affirmative and one negative.

§ 473. It is evident from the above considerations that the doctrine
of opposition requires to be amended in one or other of two
ways. Either we must face the consequences which follow from regarding
I and O as indefinite, and lay down that sub-contraries may both be
false, accepting the awkward corollary of the collapse of the doctrine
of contradiction; or we must be consistent with ourselves in regarding
I and O, for the particular purposes of opposition, as being strictly
particular, and lay down that it is always possible to argue from the
truth of one sub-contrary to the truth of the other. The latter is
undoubtedly the better course, as the admission of I and O as
indefinite in this connection confuses the theory of opposition
altogether.

§ 474. Of the several forms of opposition contradictory opposition is
logically the strongest. For this three reasons may be given--

(1) Contradictory opposites differ both in quantity and in quality,
whereas others differ only in one or the other.

(2) Contradictory opposites are incompatible both as to truth and
falsity, whereas in other cases it is only the truth _or_
falsity of the two that is incompatible.

(3) Contradictory opposition is the safest form to adopt in
argument. For the contradictory opposite refutes the adversary's
proposition as effectually as the contrary, and is not so hable to a
counter-refutation.

§ 475. At first sight indeed contrary opposition appears stronger,
because it gives a more sweeping denial to the adversary's
assertion. If, for instance, some person with whom we were arguing
were to lay down that 'All poets are bad logicians,' we might be
tempted in the heat of controversy to maintain against him the
contrary proposition 'No poets are bad logicians.' This would
certainly be a more emphatic contradiction, but, logically considered,
it would not be as sound a one as the less obtrusive contradictory,
'Some poets are not bad logicians,' which it would be very difficult
to refute.

§ 476. The phrase 'diametrically opposed to one another' seems to be
one of the many expressions which have crept into common language from
the technical usage of logic. The propositions A and O and E and I
respectively are diametrically opposed to one another in the sense
that the straight lines connecting them constitute the diagonals of
the parallelogram in the scheme of opposition.

§ 477. It must be noticed that in the case of a singular proposition
there is only one mode of contradiction possible. Since the quantity
of such a proposition is at the minimum, the contrary and
contradictory are necessarily merged into one. There is no way of
denying the proposition 'This house is haunted,' save by maintaining
the proposition which differs from it only in quality, namely, 'This
house is not haunted.'

478. A kind of generality might indeed he imparted even to a singular
proposition by expressing it in the form 'A is always B.' Thus we may
say, 'This man is always idle'--a proposition which admits of being
contradicted under the form 'This man is sometimes not idle.'

CHAPTER IV.

_Of Conversion._

§ 479. Conversion is an immediate inference grounded On the
transposition of the subject and predicate of a proposition.

§ 480. In this form of inference the antecedent is technically known
as the Convertend, i.e. the proposition to be converted, and the
consequent as the Converse, i.e. the proposition which has been
converted.

§ 481. In a loose sense of the term we may be said to have converted a
proposition when we have merely transposed the subject and predicate,
when, for instance, we turn the proposition 'All A is B' into 'All B
is A' or 'Some A is not B' into 'Some B is not A.' But these
propositions plainly do not follow from the former ones, and it is
only with conversion as a form of inference--with Illative Conversion
as it is called--that Logic is concerned.

§ 482. For conversion as a form of inference two rules have been laid
down--

(1) No term must be distributed in the converse which was not
distributed in the convertend.

(2) The quality of the converse must be the same as that of the
convertend.

§ 483. The first of these rules is founded on the nature of things. A
violation of it involves the fallacy of arguing from part of a term to
the whole.

§ 484. The second rule is merely a conventional one. We may make a
valid inference in defiance of it: but such an inference will be seen
presently to involve something more than mere conversion.

§ 485. There are two kinds of conversion--

(1) Simple.

(2) Per Accidens or by Limitation.

§ 486. We are said to have simply converted a proposition when the
quantity remains the same as before.

§ 487. We are said to have converted a proposition per accidens, or by
limitation, when the rules for the distribution of terms necessitate a
reduction in the original quantity of the proposition.

§ 488.

A can only be converted per accidens.

E and I can be converted simply.

O cannot be converted at all.

§ 489. The reason why A can only be converted per accidens is that,
being affirmative, its predicate is undistributed (§ 293). Since 'All
A is B' does not mean more than 'All A is some B,' its proper converse
is 'Some B is A.' For, if we endeavoured to elicit the inference, 'All
B is A,' we should be distributing the term B in the converse, which
was not distributed in the convertend. Hence we should be involved in
the fallacy of arguing from the part to the whole. Because 'All
doctors are men' it by no means follows that 'All men are doctors.'

§ 499. E and I admit of simple conversion, because the quantity of the
subject and predicate is alike in each, both subject and predicate
being distributed in E and undistributed in I.

/ No A is B.
E <
\ .'. No B is A.

/ Some A is B.
I <
\ .'. Some B is A.

§ 491. The reason why O cannot be converted at all is that its subject
is undistributed and that the proposition is negative. Now, when the
proposition is converted, what was the subject becomes the predicate,
and, as the proposition must still be negative, the former subject
would now be distributed, since every negative proposition distributes
its predicate. Hence we should necessarily have a term distributed in
the converse which was not distributed in the convertend. From 'Some
men are not doctors,' it plainly does not follow that 'Some doctors
are not men'; and, generally from 'Some A is not B' it cannot be
inferred that 'Some B is not A,' since the proposition 'Some A is not
B' admits of the interpretation that B is wholly contained in A.

[Illustration]

§ 492. It may often happen as a matter of fact that in some given
matter a proposition of the form 'All B is A' is true simultaneously
with 'All A is B.' Thus it is as true to say that 'All equiangular
triangles are equilateral' as that 'All equilateral triangles are
equiangular.' Nevertheless we are not logically warranted in inferring
the one from the other. Each has to be established on its separate
evidence.

§ 493. On the theory of the quantified predicate the difference
between simple conversion and conversion by limitation disappears. For
the quantity of a proposition is then no longer determined solely by
reference to the quantity of its subject. 'All A is some B' is of no
greater quantity than 'Some B is all A,' if both subject and predicate
have an equal claim to be considered.

§ 494. Some propositions occur in ordinary language in which the
quantity of the predicate is determined. This is especially the case
when the subject is a singular term. Such propositions admit of
conversion by a mere transposition of their subject and predicate,
even though they fall under the form of the A proposition, e.g.

Virtue is the condition of happiness.
.'. The condition of happiness is virtue.

And again,

Virtue is a condition of happiness.
.'. A condition of happiness is virtue.

In the one case the quantity of the predicate is determined by the
form of the expression as distributed, in the other as undistributed.

§ 495. Conversion offers a good illustration of the principle on which
we have before insisted, namely, that in the ordinary form of
proposition the subject is used in extension and the predicate in
intension. For when by conversion we change the predicate into the
subject, we are often obliged to attach a noun substantive to the
predicate, in order that it may be taken in extension, instead of, as
before, in intension, e.g.

Some mothers are unkind.
.'. Some unkind persons are mothers.

Again,

Virtue is conducive to happiness.
.'. One of the things which are conducive to happiness is virtue.

CHAPTER V.

_Of Permutation._

§ 496. Permutation [Footnote: Called by some writers Obversion.] is an
immediate inference grounded on a change of quality in a proposition
and a change of the predicate into its contradictory-term.

§ 497. In less technical language we may say that permutation is
expressing negatively what was expressed affirmatively and vice versâ.

§ 498. Permutation is equally applicable to all the four
forms of proposition.

(A) All A is B.
.'. No A is not-B (E).

(E) No A is B.
.'. All A is not-B (A).

(I) Some A is B.
.'. Some A is not not-B (O).

(O) Some A is not B.
.'. Some A is not-B (I).

§ 499, Or, to take concrete examples--

(A) All men are fallible.
.'. No men are not-fallible (E).

(E) No men are perfect.
.'. All men are not-perfect (A).

(I) Some poets are logical.
.'. Some poets are not not-logical (O).

(O) Some islands are not inhabited.
.'. Some islands are not-inhabited (I).

§ 500. The validity of permutation rests on the principle of excluded
middle, namely--That one or other of a pair of contradictory terms
must be applicable to a given subject, so that, when one may be
predicated affirmatively, the other may be predicated negatively, and
vice versâ (§ 31).

§ 501. Merely to alter the quality of a proposition would of course
affect its meaning; but when the predicate is at the same time changed
into its contradictory term, the original meaning of the proposition
is retained, whilst the form alone is altered. Hence we may lay down
the following practical rule for permutation--

Change the quality of the proposition and change the predicate into
its contradictory term.

§ 502. The law of excluded middle holds only with regard to
contradictories. It is not true of a pair of positive and privative
terms, that one or other of them must be applicable to any given
subject. For the subject may happen to fall wholly outside the sphere
to which such a pair of terms is limited. But since the fact of a term
being applied is a sufficient indication of its applicability, and
since within a given sphere positive and privative terms are as
mutually destructive as contradictories, we may in all cases
substitute the privative for the negative term in immediate inference
by permutation, which will bring the inferred proposition more into
conformity with the ordinary usage of language. Thus the concrete
instances given above will appear as follows--

(A) All men are fallible.
.'. No men are infallible (E).

(E) No men are perfect.
.'. All men are imperfect (A).

(I) Some poets are logical.
.'. Some poets are not illogical (O).

(O) Some islands are not inhabited.
.'. Some islands are uninhabited (I).

CHAPTER VI.

_Of Compound Forms of Immediate Inference._

§ 503. Having now treated of the three simple forms of immediate
inference, we go on to speak of the compound forms, and first of

_Conversion by Negation._

§ 504. When A and O have been permuted, they become respectively E and
I, and, in this form, admit of simple conversion. We have here two
steps of inference: but the process may be performed at a single
stroke, and is then known as Conversion by Negation. Thus from 'All A
is B' we may infer 'No not-B is A,' and again from 'Some A is not B'
we may infer 'Some not-B is A.' The nature of these inferences will be
seen better in concrete examples.

§ 505.

(A) All poets are imaginative.
.'. No unimaginative persons are poets (E).

(O) Some parsons are not clerical.
.'. Some unclerical persons are parsons (I).

§ 506. The above inferences, when analysed, will be found to resolve
themselves into two steps, namely,

(1) Permutation.

(2) Simple Conversion.

(A) All A is B.
.'. No A is not-B (by permutation).
.'. No not-B is A (by simple conversion).

(O) Some A is not B.
.'. Some A is not-B (by permutation).
.'. Some not-B is A (by simple conversion).

§ 507. The term conversion by negation has been arbitrarily limited to
the exact inferential procedure of permutation followed by simple
conversion. Hence it necessarily applies only to A and 0 propositions,
since these when permuted become E and 1, which admit of simple
conversion; whereas E and 1 themselves are permuted into A and 0,
which do not. There seems to be no good reason, however, why the term
'conversion by negation' should be thus restricted in its meaning;
instead of being extended to the combination of permutation with
conversion, no matter in what order the two processes may be
performed. If this is not done, inferences quite as legitimate as
those which pass under the title of conversion by negation are left
without a name.

§ 508. From E and 1 inferences may be elicited as follows--

(E) No A is B.
.'. All B is not-A (A).

(I) Some A is B.
.'. Some B is not not-A (O).

(E) No good actions are unbecoming.
.'. All unbecoming actions are not-good (A).

(I) Some poetical persons are logicians.
.'. Some logicians are not unpoetical (O).

Or, taking a privative term for our subject,

Some unpractical persons are statesmen.
.'. Some statesmen are not practical.

§ 509. When the inferences just given are analysed, it will be found
that the process of simple conversion precedes that of permutation.

§ 510. In the case of the E proposition a compound inference can be
drawn even in the original order of the processes,

No A is B.
.'. Some not-B is A.

No one who employs bribery is honest.
.'. Some dishonest men employ bribery.

The inference here, it must be remembered, does not refer to matter of
fact, but means that one of the possible forms of dishonesty among men
is that of employing bribery.

§ 511. If we analyse the preceding, we find that the second step is
conversion by limitation.

No A is B.
.'. All A is not-B (by permutation).
.'. Some not-B is A (by conversion per accidens).

§ 512. From A again an inference can be drawn in the reverse order of
conversion per accidens followed by permutation--

All A is B.
.'. Some B is not not-A.

All ingenuous persons are agreeable.
.'. Some agreeable persons are not disingenuous.

§ 513. The intermediate link between the above two propositions is the
converse per accidens of the first--'Some B is A.' This inference,
however, coincides with that from 1 (§ 508), as the similar inference
from E (§ 510) coincides with that from 0 (§ 506).

§ 514. All these inferences agree in the essential feature of
combining permutation with conversion, and should therefore be classed
under a common name.

§ 515. Adopting then this slight extension of the term, we define
conversion by negation as--A form of conversion in which the converse
differs in quality from the convertend, and has the contradictory of
one of the original terms.

§ 516. A still more complex form of immediate inference is known as

_Conversion by Contraposition._

This mode of inference assumes the following form--

All A is B.
.'. All not-B is not-A.

All human beings are fallible.
.'. All infallible beings are not-human.

§ 517. This will be found to resolve itself on analysis into three
steps of inference in the following order--

(1) Permutation.

(2) Simple Conversion.

(3) Permutation.

§ 518. Let us verify this statement by performing the three steps.

All A is B.
.'. No A is not-B (by permutation).
.'. No not-B is A (by simple conversion).
.'. All not-B is not-A (by permutation).

All Englishmen are Aryans.
.'. No Englishmen are non-Aryans.
.'. No non-Aryans are Englishmen.
.'. All non-Aryans are non-Englishmen.

§ 519. Conversion by contraposition may be complicated in appearance
by the occurrence of a negative term in the subject or predicate or
both, e.g.

All not-A is B.
.'. All not-B is A.

Again,

All A is not-B.
.'. All B is not-A.

Lastly,

All not-A is not-B.
.'. All B is A.

§ 520. The following practical rule will be found of use for the right
performing of the process--

Transpose the subject and predicate, and substitute for each its
contradictory term.

§ 521. As concrete illustrations of the above forms of inference we
may take the following--

All the men on this board that are not white are red.
.'. All the men On this board that are not red are white.

Again,

All compulsory labour is inefficient.
.'. All efficient labour is free (=non-compulsory).

Lastly,

All inexpedient acts are unjust.
.'. All just acts are expedient.

§ 522. Conversion by contraposition may be said to
rest on the following principle--

If one class be wholly contained in another, whatever is external to
the containing class is external also to the class contained.

[Illustration]

§ 523. The same principle may be expressed intensively as follows:--

If an attribute belongs to the whole of a subject, whatever fails to
exhibit that attribute does not come under the subject.

§ 524. This statement contemplates conversion by contraposition only
in reference to the A proposition, to which the process has hitherto
been confined. Logicians seem to have overlooked the fact that
conversion by contraposition is as applicable to the O as to the A
proposition, though, when expressed in symbols, it presents a more
clumsy appearance.

Some A is not B.
.'. Some not-B is not not-A.

Some wholesome things are not pleasant.
.'. Some unpleasant things are not unwholesome.

§ 525. The above admits of analysis in exactly the same way as the
same process when applied to the A proposition.

Some A is not B.
.'. Some A is not-B (by permutation).
.'. Some not-B is A (by simple conversion).
.'. Some not-B is not not-A (by permutation).

The result, as in the case of the A proposition, is the converse by
negation of the original proposition permuted.

§ 526. Contraposition may also be applied to the E proposition by the
use of conversion per accidens in the place of simple conversion. But,
owing to the limitation of quantity thus effected, the result arrived
at is the same as in the case of the O proposition. Thus from 'No
wholesome things are pleasant' we could draw the same inference as
before. Here is the process in symbols, when expanded.

No A is B.
.'. All A is not-B (by permutation).
.'. Some not-B is A (by conversion per accidens).
.'. Some not-B is not not-A (by permutation).

§ 527. In its unanalysed form conversion by contraposition may be
defined generally as--A form of conversion in which both subject and
predicate are replaced by their contradictories.

§ 528. Conversion by contraposition differs in several respects from
conversion by negation.

(1) In conversion by negation the converse differs in quality from
the convertend: whereas in conversion by contraposition the quality
of the two is the same.

(2) In conversion by negation we employ the contradictory either of
the subject or predicate, but in conversion by contraposition we
employ the contradictory of both.

(3) Conversion by negation involves only two steps of immediate
inference: conversion by contraposition three.

§ 529. Conversion by contraposition cannot be applied to the ordinary
E proposition except by limitation (§ 526).

From 'No A is B' we cannot infer 'No not-B is not-A.' For, if we
could, the contradictory of the latter, namely, 'Some not-B is not-A'
would be false. But it is manifest that this is not necessarily
false. For when one term is excluded from another, there must be
numerous individuals which fall under neither of them, unless it
should so happen that one of the terms is the direct contradictory of
the other, which is clearly not conveyed by the form of the expression
'No A is B. 'No A is not-A' stands alone among E propositions in
admitting of full conversion by contraposition, and the form of that
is the same after it as before.

§ 530. Nor can conversion by contraposition be applied at all to I.

[Illustration]

From 'Some A is B' we cannot infer that 'Some not-B is not-A.' For
though the proposition holds true as a matter of fact, when A and B
are in part mutually exclusive, yet this is not conveyed by the form
of the expression. It may so happen that B is wholly contained under
A, while A itself contains everything. In this case it will be true
that 'No not-B is not-A,' which contradicts the attempted
inference. Thus from the proposition 'Some things are substances' it
cannot be inferred that 'Some not-substances are not-things,' for in
this case the contradictory is true that 'No not-substances are
not-things'; and unless an inference is valid in every case, it is not
formally valid at all.

§ 531. It should be noticed that in the case of the [nu] proposition
immediate inferences are possible by mere contraposition without
conversion.

All A is all B.
.'. All not-A is not-B.

For example, if all the equilateral triangles are all the equiangular,
we know at once that all non-equilateral triangles are also
non-equiangular.

§ 532. The principle upon which this last kind of inference rests is
that when two terms are co-extensive, whatever is excluded from the
one is excluded also from the other.

CHAPTER VII.

_Of other Forms of Immediate Inference._

§ 533. Having treated of the main forms of immediate inference,
whether simple or compound, we will now close this subject with a
brief allusion to some other forms which have been recognised by
logicians.

§ 534. Every statement of a relation may furnish us with ail immediate
inference in which the same fact is presented from the opposite
side. Thus from 'John hit James' we infer 'James was hit by John';
from 'Dick is the grandson of Tom' we infer 'Tom is the grandfather of
Dick'; from 'Bicester is north-east of Oxford' we infer 'Oxford is
south-west of Bicester'; from 'So and so visited the Academy the day
after he arrived in London' we infer 'So and so arrived in London the
day before he visited the Academy'; from 'A is greater than B' we
infer 'B is less than A'; and so on without limit. Such inferences as
these are material, not formal. No law can be laid down for them
except the universal postulate, that

'Whatever is true in one form of words is true in every other form
of words which conveys the same meaning.'

§ 535. There is a sort of inference which goes under the title of
Immediate Inference by Added Determinants, in which from some
proposition already made another is inferred, in which the same
attribute is attached both to the subject and the predicate, e.g.,

A horse is a quadruped.
.'. A white horse is a white quadruped.

§ 536. Such inferences are very deceptive. The attributes added must
be definite qualities, like whiteness, and must in no way involve a
comparison. From 'A horse is a quadruped' it may seem at first sight
to follow that 'A swift horse is a swift quadruped.' But we need not
go far to discover how little formal validity there is about such an
inference. From 'A horse is a quadruped' it by no means follows that
'A slow horse is a slow quadruped'; for even a slow horse is swift
compared with most quadrupeds. All that really follows here is that
'A slow horse is a quadruped which is slow for a horse.' Similarly,
from 'A Bushman is a man' it does not follow that 'A tall Bushman is a
tall man,' but only that 'A tall Bushman is a man who is tall for a
Bushman'; and so on generally.

§ 537. Very similar to the preceding is the process known as Immediate
Inference by Complex Conception, e.g.

A horse is a quadruped.
.'. The head of a horse is the head of a quadruped.

§ 538. This inference, like that by added determinants, from which it
differs in name rather than in nature, may be explained on the
principle of Substitution. Starting from the identical proposition,
'The head of a quadruped is the head of a quadruped,' and being given
that 'A horse is a quadruped,' so that whatever is true of 'quadruped'
generally we know to be true of 'horse,' we are entitled to substitute
the narrower for the wider term, and in this manner we arrive at the
proposition,

The head of a horse is the head of a quadruped.

§ 539. Such an inference is valid enough, if the same caution be
observed as in the case of added determinants, that is, if no
difference be allowed to intervene in the relation of the fresh
conception to the generic and the specific terms.

CHAPTER VIII.

_Of Mediate Inferences or Syllogisms._

§ 540. A Mediate Inference, or Syllogism, consists of two
propositions, which are called the Premisses, and a third proposition
known as the Conclusion, which flows from the two conjointly.

§ 541. In every syllogism two terms are compared with one another by
means of a third, which is called the Middle Term. In the premisses
each of the two terms is compared separately with the middle term; and
in the conclusion they are compared with one another.

§ 542. Hence every syllogism consists of three terms, one of which
occurs twice in the premisses and does not appear at all in the
conclusion. This term is called the Middle Term. The predicate of the
conclusion is called the Major Term and its subject the Minor Term.

§ 543. The major and minor terms are called the Extremes, as opposed
to the Mean or Middle Term.

§ 544. The premiss in which the major term is compared with the middle
is called the Major Premiss.

§ 545. The other premiss, in which the minor term is compared with the
middle, is called the Minor Premiss.

§ 546. The order in which the premisses occur in a syllogism is
indifferent, but it is usual, for convenience, to place the major
premiss first.

§ 547. The following will serve as a typical instance of a syllogism--

Middle term Major term \
Major Premiss. All mammals are warm-blooded | Antecedent
> or
Minor term Middle term | Premisses
Minor Premiss. All whales are mammals /

Minor term Major term \ Consequent or
.'. All whales are warm-blooded > Conclusion.

§ 548. The reason why the names 'major, 'middle' and 'minor' terms
were originally employed is that in an affirmative syllogism such as
the above, which was regarded as the perfect type of syllogism, these
names express the relative quantity in extension of the three terms.

[Illustration]

§ 549. It must be noticed however that, though the middle term cannot
be of larger extent than the major nor of smaller extent than the
minor, if the latter be distributed, there is nothing to prevent all
three, or any two of them, from being coextensive.

§ 550. Further, when the minor term is undistributed, we either have a
case of the intersection of two classes, from which it cannot be told
which of them is the larger, or the minor term is actually larger than
the middle, when it stands to it in the relation of genus to species,
as in the following syllogism--

All Negroes have woolly hair.
Some Africans are Negroes.
.'. Some Africans have woolly hair.

[Illustration]

§ 551. Hence the names are not applied with strict accuracy even in
the case of the affirmative syllogism; and when the syllogism is
negative, they are not applicable at all: since in negative
propositions we have no means of comparing the relative extension of
the terms employed. Had we said in the major premiss of our typical
syllogism, 'No mammals are cold-blooded,' and drawn the conclusion 'No
whales are cold-blooded,' we could not have compared the relative
extent of the terms 'mammal' and 'cold-blooded,' since one has been
simply excluded from the other.

[Illustration]

§ 552. So far we have rather described than defined the syllogism. All
the products of thought, it will be remembered, are the results of
comparison. The syllogism, which is one of them, may be so regarded in
two ways--

(1) As the comparison of two propositions by means of a third.

(2) As the comparison of two terms by means of a third or middle
term.

§ 553. The two propositions which are compared with one another are
the major premiss and the conclusion, which are brought into
connection by means of the minor premiss. Thus in the syllogism above
given we compare the conclusion 'All whales are warm-blooded' with the
major premiss 'All mammals are warm-blooded,' and find that the former
is contained under the latter, as soon as we become acquainted with
the intermediate proposition 'All whales are mammals.'

§ 554. The two terms which are compared with one another are of course
the major and minor.

§ 555. The syllogism is merely a form into which our deductive
inferences may be thrown for the sake of exhibiting their
conclusiveness. It is not the form which they naturally assume in
speech or writing. Practically the conclusion is generally stated
first and the premisses introduced by some causative particle as
'because,' 'since,' 'for,' &c. We start with our conclusion, and then
give the reason for it by supplying the premisses.

§ 556. The conclusion, as thus stated first, was called by logicians
the Problema or Quaestio, being regarded as a problem or question, to
which a solution or answer was to be found by supplying the premisses.

§ 557. In common discourse and writing the syllogism is usually stated
defectively, one of the premisses or, in some cases, the conclusion
itself being omitted. Thus instead of arguing at full length

All men are fallible,
The Pope is a man,
.'. The Pope is fallible,

we content ourselves with saying 'The Pope is fallible, for he is a
man,' or 'The Pope is fallible, because all men are so'; or perhaps we
should merely say 'All men are fallible, and the Pope is a man,'
leaving it to the sagacity of our hearers to supply the desired
conclusion. A syllogism, as thus elliptically stated, is commonly,
though incorrectly, called an Enthymeme. When the major premiss is
omitted, it is called an Enthymeme of the First Order; when the minor
is omitted, an Enthymeme of the Second Order; and when the conclusion
is omitted an Enthymeme of the Third Order.

CHAPTER IX.

_Of Mood and Figure._

§ 558. Syllogisms may differ in two ways--

(1) in Mood;

(2) in Figure.

§ 559. Mood depends upon the kind of propositions employed. Thus a
syllogism consisting of three universal affirmatives, AAA, would be
said to differ in mood from one consisting of such propositions as EIO
or any other combination that might be made. The syllogism previously
given to prove the fallibility of the Pope belongs to the mood
AAA. Had we drawn only a particular conclusion, 'Some Popes are
fallible,' it would have fallen into the mood AAI.

§ 560. Figure depends upon the arrangement of the terms in the
propositions. Thus a difference of figure is internal to a difference
of mood, that is to say, the same mood can be in any figure.

§ 561. We will now show how many possible varieties there are of mood
and figure, irrespective of their logical validity.

§ 562. And first as to mood.

Since every syllogism consists of three propositions, and each of
these propositions may be either A, E, I, or O, it is clear that there
will be as many possible moods as there can be combinations of four
things, taken three together, with no restrictions as to
repetition. It will be seen that there are just sixty-four of such
combinations. For A may be followed either by itself or by E, I, or
O. Let us suppose it to be followed by itself. Then this pair of
premisses, AA, may have for its conclusion either A, E, I, or O, thus
giving four combinations which commence with AA. In like manner there
will be four commencing with AE, four with AI, and four with AO,
giving a total of sixteen combinations which commence with
A. Similarly there will be sixteen commencing with E, sixteen with I,
sixteen with O--in all sixty-four. It is very few, however, of these
possible combinations that will be found legitimate, when tested by
the rules of syllogism.

§ 563. Next as to figure.

There are four possible varieties of figure in a syllogism, as may be
seen by considering the positions that can be occupied by the middle
term in the premisses. For as there are only two terms in each
premiss, the position occupied by the middle term necessarily
determines that of the others. It is clear that the middle term must
either occupy the same position in both premisses or not, that is, it
must either be subject in both or predicate in both, or else subject
in one and predicate in the other. Now, if we are not acquainted with
the conclusion of our syllogism, we do not know which is the major and
which the minor term, and have therefore no means of distinguishing
between one premiss and another; consequently we must Stop here, and
say that there are only three different arrangements possible. But, if
the Conclusion also be assumed as known, then we are able to
distinguish one premiss as the major and the other as the minor; and
so we can go further, and lay down that, if the middle term does not
hold the same position in both premisses, it must either be subject in
the major and predicate in the minor, or else predicate in the major
and subject in the minor.

§ 564. Hence there result

_The Four Figures._

When the middle term is subject in the major and predicate in the
minor, we are said to have the First Figure.

When the middle term is predicate in both premisses, we are said to
have the Second Figure.

When the middle term is subject in both premisses, we are said to have
the Third Figure.

When the middle term is predicate in the major premiss and subject in
the minor, we are said to have the Fourth Figure.

§ 565. Let A be the major term; B the middle. C the minor.

Figure I. Figure II. Figure III. Figure IV.
B--A A--B B--A A--B
C--B C--B B--C B--C
C--A C--A C--A C--A

All these figures are legitimate, though the fourth is comparatively
valueless.

§ 566. It will be well to explain by an instance the meaning of the
assertion previously made, that a difference of figure is internal to
a difference of mood. We will take the mood EIO, and by varying the
position of the terms, construct a syllogism in it in each of the four
figures.

I.
E No wicked man is happy.
I Some prosperous men are wicked.
O .'. Some prosperous men are not happy.

II.
E No happy man is wicked.
I Some prosperous men are wicked.
O .'. Some prosperous men are not happy.

III.
E No wicked man is happy.
I Some wicked men are prosperous.
O .'. Some prosperous men are not happy.

IV.
E No happy man is wicked.
I Some wicked men are prosperous.
O .'. Some prosperous men are not happy.

§ 567. In the mood we have selected, owing to the peculiar nature of
the premisses, both of which admit of simple conversion, it happens
that the resulting syllogisms are all valid. But in the great majority
of moods no syllogism would be valid at all, and in many moods a
syllogism would be valid in one figure and invalid in another. As yet
however we are only concerned with the conceivable combinations, apart
from the question of their legitimacy.

§ 568. Now since there are four different figures and sixty-four
different moods, we obtain in all 256 possible ways of arranging three
terms in three propositions, that is, 256 possible forms of syllogism.

CHAPTER X.

_Of the Canon of Reasoning._

& 569. The first figure was regarded by logicians as the only perfect
type of syllogism, because the validity of moods in this figure may be
tested directly by their complying, or failing to comply, with a
certain axiom, the truth of which is self-evident. This axiom is known
as the Dictum de Omni et Nullo. It may be expressed as follows--

Whatever may be affirmed or denied of a whole class may be affirmed
or denied of everything contained in that class.

§ 570. This mode of stating the axiom contemplates predication as
being made in extension, whereas it is more naturally to be regarded
as being made in intension.

§ 571. The same principle may be expressed intensively as follows--

Whatever has certain attributes has also the attributes which
invariably accompany them .[Footnote: Nota notae est nota rei
ipsius. 'Whatever has any mark has that which it is a mark of.'
Mill, vol. i, p. 201,]

§ 572. By Aristotle himself the principle was expressed in a neutral
form thus--

'Whatever is stated of the predicate will be stated also of the
subject [Footnote: [Greek: osa katà toû kategorouménou légetai pánta kaì
katà toû hypokeiménou rhaetésetai]. Cat. 3, § I].'

This way of putting it, however, is too loose.

§ 573. The principle precisely stated is as follows--

Whatever may be affirmed or denied universally of the predicate of
an affirmative proposition, may be affirmed or denied also of the
subject.

§ 574. Thus, given an affirmative proposition 'Whales are mammals,' if
we can affirm anything universally of the predicate 'mammals,' as, for
instance, that 'All mammals are warm-blooded,' we shall be able to
affirm the same of the subject 'whales'; and, if we can deny anything
universally of the predicate, as that 'No mammals are oviparous,' we
shall be able to deny the same of the subject.

§ 575. In whatever way the supposed canon of reasoning may be stated,
it has the defect of applying only to a single figure, namely, the
first. The characteristic of the reasoning in that figure is that some
general rule is maintained to hold good in a particular case. The
major premiss lays down some general principle, whether affirmative or
negative; the minor premiss asserts that a particular case falls under
this principle; and the conclusion applies the general principle to
the particular case. But though all syllogistic reasoning may be
tortured into conformity with this type, some of it finds expression
more naturally in other ways.

§ 576. Modern logicians therefore prefer to abandon the Dictum de Omni
et Nullo in any shape, and to substitute for it the following three
axioms, which apply to all figures alike.

_Three Axioms of Mediale Inference._

(1) If two terms agree with the same third term, they agree with one
another.

(2) If one term agrees, and another disagrees, with the same third
term, they disagree with one another.

(3) If two terms disagree with the same third term, they may or may
not agree with one another.

§ 577. The first of these axioms is the principle of all affirmative,
the second of all negative, syllogisms; the third points out the
conditions under which no conclusion can be drawn. If there is any
agreement at all between the two terms and the third, as in the cases
contemplated in the first and second axioms, then we have a conclusion
of some kind: if it is otherwise, we have none.

§ 578. It must be understood with regard to these axioms that, when we
speak of terms agreeing or disagreeing with the same third term, we
mean that they agree or disagree with the same part of it.

§ 579. Hence in applying these axioms it is necessary to bear in mind
the rules for the distinction of terms. Thus from

All B is A,
No C is B,

the only inference which can be drawn is that Some A is not C (which
alters the figure from the first to the fourth). For it was only part
of A which was known to agree with B. On the theory of the quantified
predicate we could draw the inference No C is some A.

§ 580. It is of course possible for terms to agree with different
parts of the same third term, and yet to have no connection with one
another. Thus

All birds fly.
All bats fly.

But we do not infer therefrom that bats are birds or vice versâ.

§ 581. On the other hand, had we said,--

All birds lay eggs,
No bats lay eggs,

we might confidently have drawn the conclusion

No bats are birds

For the term 'bats,' being excluded from the whole of the term 'lay
eggs,' is thereby necessarily excluded from that part of it which
coincides with 'birds.'

[Illustration]

CHAPTER XI.

_Of the Generad Rules of Syllogism._

§ 582. We now proceed to lay down certain general rules to which all
valid syllogisms must conform. These are divided into primary and
derivative.

I. _Primary_.

(1) A syllogism must consist of three propositions only.

(2) A syllogism must consist of three terms only.

(3) The middle term must be distributed at least once in the
premisses.

(4) No term must be distributed in the conclusion which was not
distributed in the premisses.

(5) Two negative premisses prove nothing.

(6) If one premiss be negative, the conclusion must be negative.

(7) If the conclusion be negative, one of the premisses must be
negative: but if the conclusion be affirmative, both premisses must
be affirmative.

II. _Derivative_.

(8) Two particular premisses prove nothing.

(9) If one premiss be particular, the conclusion must be particular.

§ 583. The first two of these rules are involved in the definition of
the syllogism with which we started. We said it might be regarded
either as the comparison of two propositions by means of a third or as
the comparison of two terms by means of a third. To violate either of
these rules therefore would be inconsistent with the fundamental
conception of the syllogism. The first of our two definitions indeed
(§ 552) applies directly only to the syllogisms in the first figure;
but since all syllogisms may be expressed, as we shall presently see,
in the first figure, it applies indirectly to all. When any process
of mediate inference appears to have more than two premisses, it will
always be found that there is more than one syllogism. If there are
less than three propositions, as in the fallacy of 'begging the
question,' in which the conclusion simply reiterates one of the
premisses, there is no syllogism at all.

With regard to the second rule, it is plain that any attempted
syllogism which has more than three terms cannot conform to the
conditions of any of the axioms of mediate inference.

§ 584. The next two rules guard against the two fallacies which are
fatal to most syllogisms whose constitution is unsound.

§ 585. The violation of Rule 3 is known as the Fallacy of
Undistributed Middle. The reason for this rule is not far to seek.
For if the middle term is not used in either premiss in its whole
extent, we may be referring to one part of it in one premiss and to
quite another part of it in another, so that there will be really no
middle term at all. From such premisses as these--

All pigs are omnivorous,
All men are omnivorous,

it is plain that nothing follows. Or again, take these premisses--

Some men are fallible,
All Popes are men.

Here it is possible that 'All Popes' may agree with precisely that
part of the term 'man,' of which it is not known whether it agrees
with 'fallible' or not.

§ 586. The violation of Rule 4 is known as the Fallacy of Illicit
Process. If the major term is distributed in the conclusion, not
having been distributed in the premiss, we have what is called Illicit
Process of the Major; if the same is the case with the minor term, we
have Illicit Process of the Minor.

§ 587. The reason for this rule is that if a term be used in its whole
extent in the conclusion, which was not so used in the premiss in
which it occurred, we would be arguing from the part to the whole. It
is the same sort of fallacy which we found to underlie the simple
conversion of an A proposition.

§ 588. Take for instance the following--

All learned men go mad.
John is not a learned man.
.'. John will not go mad.

In the conclusion 'John' is excluded from the whole class of persons
who go mad, whereas in the premisses, granting that all learned men go
mad, it has not been said that they are all the men who do so. We have
here an illicit process of the major term.

§ 589. Or again take the following--

All Radicals are covetous.
All Radicals are poor.
.'. All poor men are covetous.

The conclusion here is certainly not warranted by our premisses. For
in them we spoke only of some poor men, since the predicate of an
affirmative proposition is undistributed.

§ 590. Rule 5 is simply another way of stating the third axiom of
mediate inference. To know that two terms disagree with the same third
term gives us no ground for any inference as to whether they agree or
disagree with one another, e.g.

Ruminants are not oviparous.
Sheep are not oviparous.

For ought that can be inferred from the premisses, sheep may or may
not be ruminants.

§ 591. This rule may sometimes be violated in appearance, though not
in reality. For instance, the following is perfectly legitimate
reasoning.

No remedy for corruption is effectual that does not render it
useless.
Nothing but the ballot renders corruption useless.
.'. Nothing but the ballot is an effectual remedy for corruption.

But on looking into this we find that there are four terms--

No not-A is B.
No not-C is A.
.'. No not-C is B.

The violation of Rule 5 is here rendered possible by the additional
violation of Rule 2. In order to have the middle term the same in both
premisses we are obliged to make the minor affirmative, thus

No not-A is B.
All not-C is not-A.
.'. No not-C is B.

No remedy that fails to render corruption useless is effectual.
All but the ballot fails to render corruption useless.
.'. Nothing but the ballot is effectual.

§ 592. Rule 6 declares that, if one premiss be negative, the
conclusion must be negative. Now in compliance with Rule 5, if one
premiss be negative, the other must be affirmative. We have therefore
the case contemplated in the second axiom, namely, of one term
agreeing and the other disagreeing with the same third term; and we
know that this can only give ground for a judgement of disagreement
between the two terms themselves--in other words, to a negative
conclusion.

§ 593. Rule 7 declares that, if the conclusion be negative, one of the
premisses must be negative; but, if the conclusion be affirmative,
both premisses must be affirmative. It is plain from the axioms that a
judgement of disagreement can only be elicited from a judgement of
agreement combined with a judgement of disagreement, and that a
judgement of agreement can result only from two prior judgements of
agreement.

§ 594. The seven rules already treated of are evident by their own
light, being of the nature of definitions and axioms: but the two
remaining rules, which deal with particular premisses, admit of being
proved from their predecessors.

§ 595. Proof of Rule 8.--_That two particular premisses prove
nothing_.

We know by Rule 5 that both premisses cannot be negative. Hence they
must be either both affirmative, II, or one affirmative and one
negative, IO or OI.

Now II premisses do not distribute any term at all, and therefore the
middle term cannot be distributed, which would violate Rule 3.

Again in IO or OI premisses there is only one term distributed,
namely, the predicate of the O proposition. But Rule 3 requires that
this one term should be the middle term. Therefore the major term must
be undistributed in the major premiss. But since one of the premisses
is negative, the conclusion must be negative, by Rule 6. And every
negative proposition distributes its predicate. Therefore the major
term must be distributed where it occurs as predicate of the
conclusion. But it was not distributed in the major premiss. Therefore
in drawing any conclusion we violate Rule 4 by an illicit process of
the major term.

§ 596. Proof of Rule 9.--_That_, _if_ one _premiss be
particular_, _the conclusion must be particular_.

Two negative premisses being excluded by Rule 5, and two particular by
Rule 8, the only pairs of premisses we can have are--

AI, AO, EI.

Of course the particular premiss may precede the universal, but the
order of the premisses will not affect the reasoning.

AI premisses between them distribute one term only. This must be the
middle term by Rule 3. Therefore the conclusion must be particular, as
its subject cannot be distributed,

AO and EI premisses each distribute two terms, one of which must be
the middle term by Rule 3: so that there is only one term left which
may be distributed in the conclusion. But the conclusion must be
negative by Rule 4. Therefore its predicate must be distributed.
Hence its subject cannot be so. Therefore the conclusion must be
particular.

§ 597. Rules 6 and 9 are often lumped together in a single
expression--'The conclusion must follow the weaker part,' negative
being considered weaker than affirmative, and particular than
universal.

§ 598. The most important rules of syllogism are summed up in the
following mnemonic lines, which appear to have been perfected, though
not invented, by a mediæval logician known as Petrus Hispanus, who was
afterwards raised to the Papal Chair under the title of Pope John XXI,
and who died in 1277--

Distribuas medium, nec quartus terminus adsit;
Utraque nec praemissa negans, nec particularis;
Sectetur partem conclusio deteriorem,
Et non distribuat, nisi cum praemissa, negetve.

CHAPTER XII.

_Of the Determination of the Legitimate Moods of Syllogism._

§ 599. It will be remembered that there were found to be 64 possible
moods, each of which might occur in any of the four figures, giving us
altogether 256 possible varieties of syllogism. The task now before us
is to determine how many of these combinations of mood and figure are
legitimate.

§ 600. By the application of the preceding rules we are enabled to
reduce the 64 possible moods to 11 valid ones. This may be done by a
longer or a shorter method. The longer method, which is perhaps easier
of comprehension, is to write down the 64 possible moods, and then
strike out such as violate any of the rules of syllogism.

AAA -AEA- -AIA- -AOA-
-AAE- AEE -AIE- -AOE-
AAI -AEI- AII -AOI-
-AAO- AEO -AIO- AOO

-EAA- -EEA- -EIA- -EOA-
EAE -EEE- -EIE- -EOE-
-EAI- -EEI- -EII- -EOI-
EAO -EEO- EIO -EOO-

[Illustration]

§ 601. The batches which are crossed are those in which the premisses
can yield no conclusion at all, owing to their violating Rule 6 or 9;
in the rest the premises are legitimate, but a wrong conclusion is
drawn from each of them as are translineated.

§ 602. IEO stands alone, as violating Rule 4. This may require a
little explanation.

Since the conclusion is negative, the major term, which is its
predicate, must be distributed. But the major premiss, being 1, does
not distribute either subject or predicate. Hence IEO must always
involve an illicit process of the major.

§ 603. The II moods which have been left valid, after being tested by
the syllogistic rules, are as follows--

AAA. AAI. AEE. AEO. AII. AOO.
EAE. EAO. EIO.
IAI.
OAO.

§ 604. We will now arrive at the same result by a shorter and more
scientific method. This method consists in first determining what
pairs of premisses are valid in accordance with Rules 6 and g, and
then examining what conclusions may be legitimately inferred from them
in accordance with the other rules of syllogism.

§ 605. The major premiss may be either A, E, I or O. If it is A, the
minor also may be either A, E, I or O. If it is E, the minor can only
be A or I. If it is I, the minor can only be A or E. If it is O, the
minor can only be A. Hence there result 9 valid pairs of premisses.

AA. AE. AI. AO.
EA. EI.
IA. IE.
OA.

Three of these pairs, namely AA, AE, EA, yield two conclusions apiece,
one universal and one particular, which do not violate any of the
rules of syllogism; one of them, IE, yields no conclusion at all; the
remaining five have their conclusion limited to a single proposition,
on the principle that the conclusion must follow the weaker part.
Hence we arrive at the same result as before, of II legitimate moods--

AAA. AAI. AEE. AEO. EAE. EAO.
AII. AOO. EIO. IAI. OAO.

CHAPTER XIII.

_Of the Special Rules of the Four Figures_.

§ 606. Our next task must be to determine how far the 11 moods which
we arrived at in the last chapter are valid in the four figures. But
before this can be done, we must lay down the

_Special Rules of the Four Figures_.

FIGURE 1.

Rule 1, The minor premiss must be affirmative.

Rule 2. The major premiss must be universal.

FIGURE II.

Rule 1. One or other premiss must be negative.

Rule 2. The conclusion must be negative.

Rule 3. The major premiss must be universal.

FIGURE III.

Rule 1. The minor premiss must be affirmative.

Rule 2. The conclusion must be particular.

FIGURE IV.

Rule 1. When the major premiss is affirmative, the minor must be
universal.

Rule 2. When the minor premiss is particular, the major must be
negative.

Rule 3, When the minor premiss is affirmative, the conclusion must
be particular.

Rule 4. When the conclusion is negative, the major premiss must be
universal.

Rule 5. The conclusion cannot be a universal affirmative.

Rule 6. Neither of the premisses can be a particular negative.

§ 607. The special rules of the first figure are merely a reassertion
in another form of the Dictum de Omni et Nullo. For if the major
premiss were particular, we should not have anything affirmed or
denied of a whole class; and if the minor premiss were negative, we
should not have anything declared to be contained in that class.
Nevertheless these rules, like the rest, admit of being proved from
the position of the terms in the figure, combined with the rules for
the distribution of terms (§ 293).

_Proof of the Special Rules of the Four Figures._

FIGURE 1.

§ 608. Proof of Rule 1.--_The minor premiss must be affirmative_.

B--A
C--B
C--A

If possible, let the minor premiss be negative. Then the major must be
affirmative (by Rule 5), [Footnote: This refers to the General Rules
of Syllogism.] and the conclusion must be negative (by Rule 6). But
the major being affirmative, its predicate is undistributed; and the
conclusion being negative, its predicate is distributed. Now the major
term is in this figure predicate both in the major premiss and in the
conclusion. Hence there results illicit process of the major
term. Therefore the minor premiss must be affirmative.

§ 609. Proof of Rule 2.--_The major premiss must be universal._

Since the minor premiss is affirmative, the middle term, which is its
predicate, is undistributed there. Therefore it must be distributed in
the major premiss, where it is subject. Therefore the major premiss
must be universal.

FIGURE II.

§ 610. Proof of Rule 1,--_One or other premiss must be negative_.

A--B
C--B
C--A

The middle term being predicate in both premisses, one or other must
be negative; else there would be undistributed middle.

§ 611. Proof of Rule 2.--_The conclusion must be negative._

Since one of the premisses is negative, it follows that the conclusion
also must be so (by Rule 6).

§ 612. Proof of Rule 3.--_The major premiss must be universal._

The conclusion being negative, the major term will there be
distributed. But the major term is subject in the major
premiss. Therefore the major premiss must be universal (by Rule 4).

FIGURE III.

§ 613. Proof of Rule 1.--_The minor premiss must be affirmative._

B--A
B--C
C--A

The proof of this rule is the same as in the first figure, the two
figures being alike so far as the major term is concerned.

§ 614. Proof of Rule 2.--_The conclusion must be particular_.

The minor premiss being affirmative, the minor term, which is its
predicate, will be undistributed there. Hence it must be undistributed
in the conclusion (by Rule 4). Therefore the conclusion must be
particular.

FIGURE IV.

§ 615. Proof of Rule I.--_When the major premiss is affirmative,
the minor must be universal_.

If the minor were particular, there would be undistributed
middle. [Footnote: Shorter proofs are employed in this figure, as the
student is by this time familiar with the method of procedure.]

§ 616. Proof of Rule 2.--_When the minor premiss is particular, the
major must be negative._

A--B
B--C
C--A

This rule is the converse of the preceding, and depends upon the same
principle.

§ 617. Proof of Rule 3.--_When the minor premiss is affirmative, the
conclusion must be particular._

If the conclusion were universal, there would be illicit process of
the minor.

§ 618. Proof of Rule 4.--_When the conclusion is negative, the major
premiss must_ be universal.

If the major premiss were particular, there would be illicit process
of the major.

§ 619. Proof of Rule 5.--_The conclusion CANNOT be A UNIVERSAL
affirmative_.

The conclusion being affirmative, the premisses must be so too (by
Rule 7). Therefore the minor term is undistributed in the minor
premiss, where it is predicate. Hence it cannot be distributed in the
conclusion (by Rule 4). Therefore the affirmative conclusion must be
particular.

§ 620. Proof of Rule 6.--_Neither of the premisses can lie a,
PARTICULAR NEGATIVE_.

If the major premiss were a particular negative, the conclusion would
be negative. Therefore the major term would be distributed in the
conclusion. But the major premiss being particular, the major term
could not be distributed there. Therefore we should have an illicit
process of the major term.

If the minor premiss were a particular negative, then, since the major
must be affirmative (by Rule 5), we should have undistributed middle.

CHAPTER XIV

_Of the Determination of the Moods that are valid in the Four
Figures._

§ 621. By applying the special rules just given we shall be able to
determine how many of the eleven legitimate moods are valid in the
four figures.

$622. These eleven legitimate moods were found to be

AAA. AAI. AEE. AEO. AII. AOO. EAE.
EAO. EIO. IAI. OAO.

FIGURE 1.

§ 623. The rule that the major premiss must be universal excludes the
last two moods, IAI, OAO. The rule that the minor premiss must be
affirmative excludes three more, namely, AEE, AEO, AOO.

Thus we are left with six moods which are valid in the first figure,
namely,

AAA. EAE. AII. EIO. AAI. EAO.

FIGURE II.

§ 624. The rule that one premiss must be negative excludes four moods,
namely, AAA, AAI, AII, IAI. The rule that the major must be universal
excludes OAO. Thus we are left with six moods which are valid in the
second figure, namely,

EAE. AEE. EIO. AOO. EAO. AEO.

FIGURE III.

§ 625. The rule that the conclusion must be particular confines us to
eight moods, two of which, namely AEE and AOO, are excluded by the
rule that the minor premiss must be affirmative.

Thus we are left with six moods which are valid in the third figure,
namely,

AAI. IAI. AII. EAO. OAO. EIO.

FIGURE IV.

§ 626. The first of the eleven moods, AAA, is excluded by the rule
that the conclusion cannot be a universal affirmative.

Two more moods, namely AOO and OAO, are excluded by the rule that
neither of the premisses can be a particular negative.

AII violates the rule that when the major premiss is affirmative, the
minor must be universal.

EAE violates the rule that, when the minor premiss is affirmative, the
conclusion must be particular. Thus we are left with six moods which
are valid in the fourth figure, namely,

AAI. AEE. IAI. EAO. EIO. AEO.

§ 627. Thus the 256 possible forms of syllogism have been reduced to
two dozen legitimate combinations of mood and figure, six moods being
valid in each of the four figures.

FIGURE I. AAA. EAE. AII. EIO. (AAI. EAO.)

FIGURE II. EAE. AEE. EIO. AGO. (EAO. AEO.)

FIGURE III. AAI. IAI. AII. EAO. OAO. EIO.

FIGURE IV. AAI. AEE. IAI. EAO. EIO. (AEO.)

§ 628. The five moods enclosed in brackets, though valid, are
useless. For the conclusion drawn is less than is warranted by the
premisses. These are called Subaltern Moods, because their conclusions
might be inferred by subalternation from the universal conclusions
which can justly be drawn from the same premisses. Thus AAI is
subaltern to AAA, EAO to EAE, and so on with the rest.

§ 629. The remaining 19 combinations of mood and figure, which are
loosely called 'moods,' though in strictness they should be called
'figured moods,' are generally spoken of under the names supplied by
the following mnemonics--

Barbara, Celarent, Darii, Ferioque prioris;
Cesare, Camestres, Festino, Baroko secundæ;
Tertia Darapti, Disamis, Datisi, Felapton,
Bokardo, Ferison habet; Quarta insuper addit
Bramantip, Camenes, Dimaris, Fesapo, Fresison:
Quinque Subalterni, totidem Generalibus orti,

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