Deductive Logic

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735. Of these two kinds of conjunctive syllogisms we will first take
that which consists throughout of conjunctive propositions.

_The Wholly Conjunctive Syllogism_.

736. Wholly conjunctive syllogisms do not differ essentially from
simple ones, to which they are immediately reducible. They admit of
being constructed in every mood and figure, and the moods of the
imperfect figures may be brought into the first by following the
ordinary rules of reduction. For instance--

Cesare. Celarent.

If A is B, C is never D. \ / If C is D, A is never B.
If E is F, C is always D. | = | If E is F, C is always D.
.'. If E is F, A is never B. / \ .'. If E is F, A is never B.

If it is day, the stars never shine.\ /If the stars shine, it is never day.
If it is night, the stars always \=/ If it is night, the stars always
shine. / \ shine.
.'. If it is night, it is never day / \.'. If it is night, it is never day.

Disamis. Darii.
If C is D, A is sometimes B. \ / If C is D, E is always F.
If C is D, E is always F. | = | If A is B, C is sometimes D.
If E is F, A is sometimes B. / \ .'. If A is B, E is sometimes F.
.'. If E is F, A is sometimes B.

If she goes, I sometimes go. \ / If she goes, he always goes,
If she goes, he always goes. | = | If I go, she sometimes goes.
.'. If he goes, I sometimes go. / \ .'. If I go, he sometimes goes.
.'. If he goes, I sometimes go.

_The Partly Conjunctive Syllogism._

737. It is this kind which is usually meant when the Conjunctive or
Hypothetical Syllogism is spoken of.

738. Of the two premisses, one conjunctive and one simple, the
conjunctive is considered to be the major, and the simple premiss the
minor. For the conjunctive premiss lays down a certain relation to
hold between two propositions as a matter of theory, which is applied
in the minor to a matter of fact.

739. Taking a conjunctive proposition as a major premiss, there are
four simple minors possible. For we may either assert or deny the
antecedent or the consequent of the conjunctive.

Constructive Mood. Destructive Mood.
(1) If A is B, C is D. (2) If A is B, C is D.
A is B. C is not D.
.'. C is D. .'. A is not B.

(3) If A is B, C is D. (4) If A is B, C is D.
A is not B. C is D.
No conclusion. No conclusion.

740. When we take as a minor 'A is not B ' (3), it is clear that we
can get no conclusion. For to say that C is D whenever A is B gives us
no right to deny that C can be D in the absence of that
condition. What we have predicated has been merely inclusion of the
case AB in the case CD.

[Illustration]

741. Again, when we take as a minor, 'C is D' (4), we can get no
universal conclusion. For though A being B is declared to involve as a
consequence C being D, yet it is possible for C to be D under other
circumstances, or from other causes. Granting the truth of the
proposition 'If the sky falls, we shall catch larks,' it by no means
follows that there are no other conditions under which this result can
be attained.

742. From a consideration of the above four cases we elicit the
following

_Canon of the Conjunctive Syllogism._

To affirm the antecedent is to affirm the consequent, and to deny the
consequent is to deny the antecedent: but from denying the antecedent
or affirming the consequent no conclusion follows.

743. There is a case, however, in which we can legitimately deny the
antecedent and affirm the consequent of a conjunctive proposition,
namely, when the relation predicated between the antecedent and the
consequent is not that of inclusion but of coincidence--where in fact
the conjunctive proposition conforms to the type u.

For example--

_Denial of the Antecedent_.
If you repent, then only are you forgiven.
You do not repent.
.'. You are not forgiven.

_Affirmation of the Consequent_.
If you repent, then only are you forgiven.
You are forgiven.
.'. You repent.

CHAPTER XXI.

_Of the Reduction of the Partly Conjunctive Syllogism._

744. Such syllogisms as those just treated of, if syllogisms they
are to be called, have a major and a middle term visible to the eye,
but appear to be destitute of a minor. The missing minor term is
however supposed to be latent in the transition from the conjunctive
to the simple form of proposition. When we say 'A is B,' we are taken
to mean, 'As a matter of fact, A is B' or 'The actual state of the
case is that A is B.' The insertion therefore of some such expression
as 'The case in hand,' or 'This case,' is, on this view, all that is
wanted to complete the form of the syllogism. When reduced in this
manner to the simple type of argument, it will be found that the
constructive conjunctive conforms to the first figure and the
destructive conjunctive to the second.

_Constructive Mood_. _Barbara_.

If A is B, C is D. \ / All cases of A being B are cases of
\ = / C being D.
A is B. / \ This is a case of A being B.
.'. C is D. / \ .'. This is a case of C being D.

_Destructive Mood._ Camestres.

If A is B, C is D. \ / All cases of A being B are cases of
\ = / C being D.
C is not D. / \ This is not a case of C being D.
.'. A is not B. / \ .'. This is not a case of A being B.

745. It is apparent from the position of the middle term that the
constructive conjunctive must fall into the first figure and the
destructive conjunctive into the second. There is no reason, however,
why they should be confined to the two moods, Barbara and
Carnestres. If the inference is universal, whether as general or
singular, the mood is Barbara or Carnestres; if it is particular, the
mood is Darii or Baroko.

Barbara. Camestres.
If A is B, C is always D. \ If A is B, C is always D. \
A is always B. \ C is never D. \
.'. C is always D. \ .'. A is never B. \
| |
If A is B, C is always D. / If A is B, C is always D. /
A is in this case B. / C is not in this case D. /
.'. C is in this case D. / .'. A is not in this case B. /

Darii. Baroko.

If A is B, C is always D. If A is B, C is never D.
A is sometimes B. C is sometimes not D.
.'. C is sometimes D. .'. A is sometimes not B.

746. The remaining moods of the first and second figure are obtained
by taking a negative proposition as the consequent in the major
premiss.

Celarent. Ferio.
If A is B, C is never D. If A is B, C is never D.
A is always B. A is sometimes B.
.'. C is never D. .'. C is sometimes not D.

_Cesare_. Festino.
If A is B, C is never D. If A is B, C is never D.
C is always D. C is sometimes D.
.'. A is never B. .'. A is sometimes not B.

747. As the partly conjunctive syllogism is thus reducible to the
simple form, it follows that violations of its laws must correspond
with violations of the laws of simple syllogism. By our throwing the
illicit moods into the simple form it will become apparent what
fallacies are involved in them.

_Denial of Anteceded_.

If A is B, C is D. \ / All cases of A being B are cases of C
\ = / being D.
A is not B. / \ This is not a case of A being B.
.'. C is not D. / \ .'. This is not a case of C being D.

Here we see that the denial of the antecedent amounts to illicit
process of the major term.

7481 _Affirmation of Consequent_.

If A is B, C is D. \ / All Cases of A being B are cases of C
| = | being D.
C is D. / \ This is a case of C being D.

Here we see that the affirmation of the consequent amounts to
undistributed middle.

749. If we confine ourselves to the special rules of the four
figures, we see that denial of the antecedent involves a negative
minor in the first figure, and affirmation of the consequent two
affirmative premisses in the second. Or, if the consequent in the
major premiss were itself negative, the affirmation of it would amount
to the fallacy of two negative premisses. Thus--

If A is B, C is not D. \ / No cases of A being B are cases of C
| = | being D.
C is not D. / \ This is not a case of C being D.

750. The positive side of the canon of the conjunctive
syllogism--'To affirm the antecedent is to affirm the consequent,'
corresponds with the Dictum de Omni. For whereas something (viz. C
being D) is affirmed in the major of all conceivable cases of A being
B, the same is affirmed in the conclusion of something which is
included therein, namely, 'this case,' or 'some cases,' or even 'all
actual cases.'

751. The negative side--'to deny the consequent is to deny the
antecedent'--corresponds with the Dictum de Diverse ( 643). For
whereas in the major all conceivable cases of A being B are included
in C being D, in the minor 'this case,' or 'some cases,' or even 'all
actual cases' of C being D, are excluded from the same notion.

752. The special characteristic of the partly conjunctive syllogism
lies in the transition from hypothesis to fact. We might lay down as
the appropriate axiom of this form of argument, that 'What is true in
the abstract is true--in the concrete,' or 'What is true in theory is
also true in fact,' a proposition which is apt to be neglected or
denied. But this does not vitally distinguish it from the ordinary
syllogism. For though in the latter we think rather of the transition
from a general truth to a particular application of it, yet at bottom
a general truth is nothing but a hypothesis resting upon a slender
basis of observed fact. The proposition 'A is B' may be expressed in
the form 'If A is, B is.' To say that 'All men are mortal' may be
interpreted to mean that 'If we find in any subject the attributes of
humanity, the attributes of mortality are sure to accompany them.'

CHAPTER XXII.

_Of the Partly Conjunctive Syllogism regarded as an Immediate
Inference_.

753. It is the assertion of fact in the minor premiss, where we have
the application of an abstract principle to a concrete instance, which
alone entitles the partly conjunctive syllogism to be regarded as a
syllogism at all. Apart from this the forms of semi-conjunctive
reasoning run at once into the moulds of immediate inference.

754. The constructive mood will then be read in this way--

If A is B, C is D,
.'. A being B, C is D.

reducing itself to an instance of immediate inference by subaltern
opposition--

Every case of A being B, is a case of C being D.
.'. Some particular case of A being B is a case of C being D.

755. Again, the destructive conjunctive will read as follows--

If A is B, C is D,
.'. C not being D, A is not B.

which is equivalent to

All cases of A being B are cases of C being D.
.'. Whatever is not a case of C being D is not a case of A being B.
.'. Some particular case of C not being D is not a case of A being
B.

But what is this but an immediate inference by contraposition, coming
under the formula

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

and followed by Subalternation?

756. The fallacy of affirming the consequent becomes by this mode of
treatment an instance of the vice of immediate inference known as the
simple conversion of an A proposition. 'If A is B, C is D' is not
convertible with 'If C is D, A is B' any more than 'All A is B' is
convertible with 'All B is A.'

757. We may however argue in this way

If A is B, C is D,
C is D,
.'. A may be B,

which is equivalent to saying,

When A is B, C is always D,
.'. When C is D, A is sometimes B,

and falls under the legitimate form of conversion of A per accidens--

All cases of A being B are cases of C being D.
.'. Some cases of C being D are cases of A being B.

758. The fallacy of denying the antecedent assumes the following
form--

If A is B, C is D,
.'. If A is not B, C is not D,

equivalent to--

All cases of A being B are cases of C being D.
.'. Whatever is not a case of A being B is not a case of C being D.

This is the same as to argue--

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

an erroneous form of immediate inference for which there is no special
name, but which involves the vice of simple conversion of A, since
'All not-A is not-B' is the contrapositive, not of 'All A is B,' but
of its simple converse 'All B is A.'

759. The above-mentioned form of immediate inference, however
(namely, the employment of contraposition without conversion), is
valid in the case of the U proposition; and so also is simple
conversion. Accordingly we are able, as we have seen, in dealing with
a proposition of that form, both to deny the antecedent and to assert
the consequent with impunity--

If A is B, then only C is D,
.'. A not being B, C is not D;

and again, C being D, A must be B.

CHAPTER XXIII.

_Of the Disjunctive Syllogism_.

760. Roughly speaking, a Disjunctive Syllogism results from the
combination of a disjunctive with a simple premiss. As in the
preceding form, the complex proposition is regarded as the major
premiss, since it lays down a hypothesis, which is applied to fact in
the minor.

761. The Disjunctive Syllogism may be exactly defined as follows--

A complex syllogism, which has for its major premiss a disjunctive
proposition, either the antecedent or consequent of which is in the
minor premiss simply affirmed or denied.

762. Thus there are four types of disjunctive syllogism possible.

_Constructive Moods._

(1) Either A is B or C is D. (2) Either A is B or C is D.
A is not B. C is not D.
.'. C is D. .'. A is B.

Either death is annihilation or we are immortal.
Death is not annihilation.
.'. We are immortal.

Either the water is shallow or the boys will be drowned.
The boys are not drowned.
.'. The water is shallow.

_Destructive Moods_.

(3) Either A is B or C is D. (4) Either A is B or C is D.
A is B. C is D.
.'. C is not D. .'. A is not B.

763. Of these four, however, it is only the constructive moods that
are formally conclusive. The validity of the two destructive moods is
contingent upon the kind of alternatives selected. If these are such
as necessarily to exclude one another, the conclusion will hold, but
not otherwise. They are of course mutually exclusive whenever they
embody the result of a correct logical division, as 'Triangles are
either equilateral, isosceles or scalene.' Here, if we affirm one of
the members, we are justified in denying the rest. When the major thus
contains the dividing members of a genus, it may more fitly be
symbolized under the formula, 'A is either B or C.' But as this admits
of being read in the shape, 'Either A is B or A is C,' we retain the
wider expression which includes it. Any knowledge, however, which we
may have of the fact that the alternatives selected in the major are
incompatible must come to us from material sources; unless indeed we
have confined ourselves to a pair of contradictory terms (A is either
B or not-B). There can be nothing in the form of the expression to
indicate the incompatibility of the alternatives, since the same form
is employed when the alternatives are palpably compatible. When, for
instance, we say, 'A successful student must be either talented or
industrious,' we do not at all mean to assert the positive
incompatibility of talent and industry in a successful student, but
only the incompatibility of their negatives--in other words, that, if
both are absent, no student can be successful. Similarly, when it is
said, 'Either your play is bad or your luck is abominable,' there is
nothing in the form of the expression to preclude our conceiving that
both may be the case.

764. There is no limit to the number of members in the disjunctive
major. But if there are only two alternatives, the conclusion will be
a simple proposition; if there are more than two, the conclusion will
itself be a disjunctive. Thus--

Either A is B or C is D or E is F or G is H.
E is not F.
.'. Either A is B or C is D or G is H.

765. The Canon of the Disjunctive Syllogism may be laid down as
follows--

To deny one member is to affirm the rest, either simply or
disjunctively; but from affirming any member nothing follows.

CHAPTER XXIV.

_Of the Reduction of the Disjunctive Syllogism._

766. We have seen that in the disjunctive syllogism the two
constructive moods alone are formally valid. The first of these,
namely, the denial of the antecedent, will in all cases give a simple
syllogism in the first figure; the second of them, namely, the denial
of the consequent, will in all cases give a simple syllogism in the
second figure.

_Denial of Antecedent_ = Barbara.

Either A is B or C is D.
A is not B.
.'.C is D

is equal to

If A is not B, C is D.
A is not B.
.'. C is D.

is equal to

All cases of A not being B are cases of C being D.
This is a case of A not being B.
.'. This is a case of C being D.

_Denial of Consequent_ = Camestres.

Either A is E or C is D.
C is not D.
.'. A is B.

is equal to

If A is not B, C is D.
C is not D.
.'. A is B.

is equal to

All cases of A not being B are cases of C being D.
This is not a case of C being D.
.'. This is not a case of A being B.

767. The other moods of the first and second figures can be obtained
by varying the quality of the antecedent and consequent in the major
premiss and reducing the quantity of the minor.

768. The invalid destructive moods correspond with the two invalid
types of the partly conjunctive syllogism, and have the same fallacies
of simple syllogism underlying them. Affirmation of the antecedent of
a disjunctive is equivalent to the semi-conjunctive fallacy of denying
the antecedent, and therefore involves the ordinary syllogistic
fallacy of illicit process of the major.

Affirmation of the consequent of a disjunctive is equivalent to the
same fallacy in the semi-conjunctive form, and therefore involves the
ordinary syllogistic fallacy of undistributed middle.

_Affirmation of Antecedent_ = _Illicit Major_.

Either A is B or C is D.
A is B.
.'. C is not D.

is equal to

If A is not B, C is D.
A is B.
.'. C is not D.

is equal to

All cases of A not being B are cases of C being D.
This is not a case of A not being B.
.'. This is not a case of C not being D.

_Affirmation of Consequent_ = _Undistributed Middle_.

Either A is B or C is D.
C is D.

is equal to

If A is not B, C is D.
C is D.

is equal to

All cases of A not being B are cases of C being D.
This is a case of C being D.

769. So far as regards the consequent, the two species of complex
reasoning hitherto discussed are identical both in appearance and
reality. The apparent difference of procedure in the case of the
antecedent, namely, that it is affirmed in the partly conjunctive, but
denied in the disjunctive syllogism, is due merely to the fact that in
the disjunctive proposition the truth of the consequent is involved in
the falsity of the antecedent, so that the antecedent being
necessarily negative, to deny it in appearance is in reality to assert
it.

CHAPTER XXV.

_The Disjunctive Syllogism regarded as an Immediate Inference_.

770. If no stress be laid on the transition from disjunctive
hypothesis to fact, the disjunctive syllogism will run with the same
facility as its predecessor into the moulds of immediate inference.

771.

_Denial of Antecedent_. Subalternation.

Either A is B or C is D, Every case of A not being B
is a case of C being D.
.'. A not being B, C is D. .'. Some case of A not being B
is a case of C being D.

772.

_Denial of Consequent_. Conversion by Contraposition
+ Subalternation.

Either A is B or C is D. All cases of A not being B
are cases of C being D.
.'. C not being D, A is B .'. All cases of C not being D are
cases of A being B.
.'. Some case of C not being D is
a case of A being B.

773. Similarly the two invalid types of disjunctive syllogism will
be found to coincide with fallacies of immediate inference.

774.

_Affirmation of Antecedent_. Contraposition without
Conversion.

Either A is B or C is D. All cases of A not being B are
cases of C being D.
.'. A being B, C is not D .'. All cases of A being B are
cases of C not being D.

775. The affirmation of the antecedent thus comes under the
formula--

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

a form of inference which cannot hold except where A and B are known
to be incompatible. Who, for instance, would assent to this?--

All non-boating men play cricket.
.'. All boating men are non-cricketers.

776.

_Affirmation of Consequent_. Simple Conversion of A.

Either A is B or C is D. All cases of A not being B are
cases of C being D.
.'.C being D, A is not B. .'. All cases of C being D are
cases of A not being B.

777. We may however argue in this way--

Conversion of A per accidens.
Either A is B or C is D. All cases of A not being B
are cases of C being D.
.'. C being D, A is sometimes B. .'. Some cases of C being D are
cases of A not being B.

The men who pass this examination must have either talent or industry.
.'. Granting that they are industrious, they may be without talent.

CHAPTER XXVI.

_Of the Mixed Form of Complex Syllogism_.

778. Under this head are included all syllogisms in which a
conjunctive is combined with a disjunctive premiss. The best known
form is

_The Dilemma_.

779. The Dilemma may be defined as--

A complex syllogism, having for its major premiss a conjunctive
proposition with more than one antecedent, or more than one
consequent, or both, which (antecedent or consequent) the minor
premiss disjunctively affirms or denies.

780. It will facilitate the comprehension of the dilemma, if the
following three points are borne in mind--

(1) that the dilemma conforms to the canon of the partly conjunctive
syllogism, and therefore a valid conclusion can be obtained only by
affirming the antecedent or denying the consequent;

(2) that the minor premiss must be disjunctive;

(3) that if only the antecedent be more than one, the conclusion
will be a simple proposition; but if both antecedent and consequent
be more than one, the conclusion will itself be disjunctive.

781. The dilemma, it will be seen, differs from the partly
conjunctive syllogism chiefly in the fact of having a disjunctive
affirmation of the antecedent or denial of the consequent in the
minor, instead of a simple one. It is this which constitutes the
essence of the dilemma, and which determines its possible
varieties. For if only the antecedent or only the consequent be more
than one, we must, in order to obtain a disjunctive minor, affirm the
antecedent or deny the consequent respectively; whereas, if there be
more than one of both, it is open to us to take either course. This
gives us four types of dilemma.

782.

(1). _Simple Constructive._

If A is B or C is D, E is F.
Either A is B or C is D.
.'. E is F.

(2). _Simple Destructive._

If A is B, C is D and E is F.
Either C is not D or E is not F.
.'. A is not B.

(3). _Complex Constructive._

If A is B, C is D; and if E is F, G is H.
Either A is B or E is F.
.'. Either C is D or G is H.

(4). _Complex Destructive_.

If A is B, C is D; and if E is F, G is H.
Either C is not D or G is not H.
.'. Either A is not B or E is not F.

783.

(1). _Simple Constructive_.

If she sinks or if she swims, there will be an end of her.
She must either sink or swim.
.'. There will be an end of her.

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