Deductive Logic

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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 kata tou kategoroumenou legetai panta kai
kata tou hypokeimenou rhaetesetai]. 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 versa.

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 mediaeval 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.

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