Deductive Logic

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431. Hence induction is a real process from the known to the
unknown, whereas deduction is no more than the application of
previously existing knowledge; or, to put the same thing more
technically, in an inductive inference the consequent is not contained
in the antecedent, in a deductive inference it is.

432. When, after observing that gold, silver, lead, and other
metals, are capable of being reduced to a liquid state by the
application of heat, the mind leaps to the conclusion that the same
will hold true of some other metal, as platinum, or of all metals, we
have then an inductive inference, in which the conclusion, or
consequent, is a new proposition, which was not contained in those
that went before. We are led to this conclusion, not by reason, but by
an instinct which teaches us to expect like results, under like
circumstances. Experience can tell us only of the past: but we allow
it to affect our notions of the future through a blind belief that
'the thing that hath been, it is that which shall be; and that which
is done is that which shall be done; and there is no new thing under
the sun.' Take away this conviction, and the bridge is cut which
connects the known with the unknown, the past with the future. The
commonest acts of daily life would fail to be performed, were it not
for this assumption, which is itself no product of the reason. Thus
man's intellect, like his faculties generally, rests upon a basis of
instinct. He walks by faith, not by sight.

433. It is a mistake to talk of inductive reasoning, as though it
were a distinct species from deductive. The fact is that inductive
inferences are either wholly instinctive, and so unsusceptible of
logical vindication, or else they may be exhibited under the form of
deductive inferences. We cannot be justified in inferring that
platinum will be melted by heat, except where we have equal reason for
asserting the same thing of copper or any other metal. In fact we are
justified in drawing an individual inference only when we can lay down
the universal proposition, 'Every metal can be melted by heat.' But
the moment this universal proposition is stated, the truth of the
proposition in the individual instance flows from it by way of
deductive inference. Take away the universal, and we have no logical
warrant for arguing from one individual case to another. We do so, as
was said before, only in virtue of that vague instinct which leads us
to anticipate like results from like appearances.

434. Inductive inferences are wholly extraneous to the science of
formal logic, which deals only with formal, or necessary, inferences,
that is to say with deductive inferences, whether immediate or
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 versa.

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 versa.

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 versa.

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.

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