Deductive Logic

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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 versa.

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 versa ( 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.

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