_Of Immediate Inference as applied to Complex Propositions._
§ 701. So far we have treated of inference, or reasoning, whether
mediate or immediate, solely as applied to simple propositions. But it
will be remembered that we divided propositions into simple and
complex. I t becomes incumbent upon us therefore to consider the laws
of inference as applied to complex propositions. Inasmuch however as
every complex proposition is reducible to a simple one, it is evident
that the same laws of inference must apply to both.
§ 702. We must first make good this initial statement as to the
essential identity underlying the difference of form between simple
and complex propositions.
§ 703. Complex propositions are either Conjunctive or Disjunctive (§
214).
§ 704. Conjunctive propositions may assume any of the four forms, A,
E, I, O, as follows--
(A) If A is B, C is always D.
(E) If A is B, C is never D.
(I) If A is B, C is sometimes D.
(O) If A is B, C is sometimes not D.
§ 705. These admit of being read in the form of simple propositions,
thus--
(A) If A is B, C is always D = All cases of A being B are cases of C
being D. (Every AB is a CD.)
(E) If A is B, C is never D = No cases of A being B are cases of C
being D. (No AB is a CD.)
(I) If A is B, C is sometimes D = Some cases of A being B are cases
of C being D. (Some AB's are CD's.)
(O) If A is B, C is sometimes not D = Some cases of A being B are
not cases of C being D. (Some AB's are not CD's.)
§ 706. Or, to take concrete examples,
(A) If kings are ambitious, their subjects always suffer.
= All cases of ambitious kings are cases of subjects suffering.
(E) If the wind is in the south, the river never freezes.
= No cases of wind in the south are cases of the river freezing.
(I) If a man plays recklessly, the luck sometimes goes against him.
= Some cases of reckless playing are cases of going against one.
(O) If a novel has merit, the public sometimes do not buy it.
= Some cases of novels with merit are not cases of the public buying.
§ 707. We have seen already that the disjunctive differs from the
conjunctive proposition in this, that in the conjunctive the truth
of the antecedent involves the truth of the consequent, whereas in the
disjunctive the falsity of the antecedent involves the truth of the
consequent. The disjunctive proposition therefore
Either A is B or C is D
may be reduced to a conjunctive
If A is not B, C is D,
and so to a simple proposition with a negative term for subject.
All cases of A not being B are cases of C being D.
(Every not-AB is a CD.)
§ 708. It is true that the disjunctive proposition, more than any
other form, except U, seems to convey two statements in one
breath. Yet it ought not, any more than the E proposition, to be
regarded as conveying both with equal directness. The proposition 'No
A is B' is not considered to assert directly, but only implicitly,
that 'No B is A.' In the same way the form 'Either A is B or C is D'
ought to be interpreted as meaning directly no more than this, 'If A
is not B, C is D.' It asserts indeed by implication also that 'If C is
not D, A is B.' But this is an immediate inference, being, as we shall
presently see, the contrapositive of the original. When we say 'So and
so is either a knave or a fool,' what we are directly asserting is
that, if he be not found to be a knave, he will be found to be a
fool. By implication we make the further statement that, if he be not
cleared of folly, he will stand condemned of knavery. This inference
is so immediate that it seems indistinguishable from the former
proposition: but since the two members of a complex proposition play
the part of subject and predicate, to say that the two statements are
identical would amount to asserting that the same proposition can have
two subjects and two predicates. From this point of view it becomes
clear that there is no difference but one of expression between the
disjunctive and the conjunctive proposition. The disjunctive is
merely a peculiar way of stating a conjunctive proposition with a
negative antecedent.
§ 709. Conversion of Complex Propositions.
A / If A is B, C is always D.
\ .'. If C is D, A is sometimes B.
E / If A is B, C is never D.
\ .'. If C is D, A is never B.
I / If A is S, C is sometimes D.
\ .'. If C is D, A is sometimes B.
§ 710. Exactly the same rules of conversion apply to conjunctive as to
simple propositions.
§ 711. A can only be converted per accidens, as above.
The original proposition
'If A is B, C is always D'
is equivalent to the simple proposition
'All cases of A being B are cases of C being D.'
This, when converted, becomes
'Some cases of C being D are cases of A being B,'
which, when thrown back into the conjunctive form, becomes
'If C is D, A is sometimes B.'
§ 712. This expression must not be misunderstood as though it
contained any reference to actual existence. The meaning might be
better conveyed by the form
'If C is D, A may be B.'
But it is perhaps as well to retain the other, as it serves to
emphasize the fact that formal logic is concerned only with the
connection of ideas.
§ 713. A concrete instance will render the point under discussion
clearer. The example we took before of an A proposition in the
conjunctive form--
'If kings are ambitious, their subjects always suffer'
may be converted into
'If subjects suffer, it may be that their kings are ambitious,'
i.e. among the possible causes of suffering on the part of subjects is
to be found the ambition of their rulers, even if every actual case
should be referred to some other cause. It is in this sense only that
the inference is a necessary one. But then this is the only sense
which formal logic is competent to recognise. To judge of conformity
to fact is no part of its province. From 'Every AB is a CD' it follows
that ' Some CD's are AB's' with exactly the same necessity as that
with which 'Some B is A' follows from 'All A is B.' In the latter case
also neither proposition may at all conform to fact. From 'All
centaurs are animals' it follows necessarily that 'Some animals are
centaurs': but as a matter of fact this is not true at all.
§ 714. The E and the I proposition may be converted simply, as above.
§ 715. O cannot be converted at all. From the proposition
'If a man runs a race, he sometimes does not win it,'
it certainly does not follow that
'If a man wins a race, he sometimes does not run it.'
§ 716. There is a common but erroneous notion that all conditional
propositions are to be regarded as affirmative. Thus it has been
asserted that, even when we say that 'If the night becomes cloudy,
there will be no dew,' the proposition is not to be regarded as
negative, on the ground that what we affirm is a relation between the
cloudiness of night and the absence of dew. This is a possible, but
wholly unnecessary, mode of regarding the proposition. It is precisely
on a par with Hobbes's theory of the copula in a simple proposition
being always affirmative. It is true that it may always be so
represented at the cost of employing a negative term; and the same is
the case here.
§ 717. There is no way of converting a disjunctive proposition except
by reducing it to the conjunctive form.
§ 718. _Permutation of Complex Propositions_.
(A) If A is B, C is always D.
.'. If A is B, C is never not-D. (E)
(E) If A is B, C is never D.
.'. If A is B, C is always not-D. (A)
(I) If A is B, C is sometimes D.
.'. If A is B, C is sometimes not not-D. (O)
(O) If A is B, C is sometimes not D.
.'. If A is B, C is sometimes not-D. (I)
§ 719.
(A) If a mother loves her children, she is always kind to them.
.'. If a mother loves her children, she is never unkind to
them. (E)
(E) If a man tells lies, his friends never trust him.
.'. If a man tells lies, his friends always distrust him. (A)
(I) If strangers are confident, savage dogs are sometimes friendly.
.'. If strangers are confident, savage dogs are sometimes not
unfriendly. (O)
(O) If a measure is good, its author is sometimes not popular.
.'. If a measure is good, its author is sometimes
unpopular. (I)
§ 720. The disjunctive proposition may be permuted as it stands
without being reduced to the conjunctive form.
Either A is B or C is D.
.'. Either A is B or C is not not-D.
Either a sinner must repent or he will be damned.
.'. Either a sinner must repent or he will not be saved.
§ 721. _Conversion by Negation of Complex Propositions._
(A) If A is B, C is always D.
.'. If C is not-D, A is never B. (E)
(E) If A is B, C is never D.
.'. If C is D, A is always not-B. (A)
(I) If A is B, C is sometimes D.
.'. If C is D, A is sometimes not not-B. (O)
(O) If A is B, C is sometimes not D.
.'. If C is not-D, A is sometimes B. (I)
(E per acc.) If A is B, C is never D.
.'. If C is not-D, A is sometimes B. (I)
(A per ace.) If A is B, C is always D.
.'. If C is D, A is sometimes not not-D. (O)
§ 722.
(A) If a man is a smoker, he always drinks.
.'. If a man is a total abstainer, he never smokes. (E)
(E) If a man merely does his duty, no one ever thanks him.
.'. If people thank a man, he has always done more than his
duty. (A)
(I) If a statesman is patriotic, he sometimes adheres to a party.
.'. If a statesman adheres to a party, he is sometimes not
unpatriotic. (O)
(O) If a book has merit, it sometimes does not sell.
.'. If a book fails to sell, it sometimes has merit. (I)
(E per acc.) If the wind is high, rain never falls.
.'. If rain falls, the wind is sometimes high. (I)
(A per acc.) If a thing is common, it is always cheap.
.'. If a thing is cheap, it is sometimes not uncommon. (O)
§ 723. When applied to disjunctive propositions, the distinctive
features of conversion by negation are still discernible. In each of
the following forms of inference the converse differs in quality from
the convertend and has the contradictory of one of the original terms
(§ 515).
§ 724.
(A) Either A is B or C is always D.
.'. Either C is D or A is never not-B. (E)
(E) Either A is B or C is never D.
.'. Either C is not-D or A is always B. (A)
(I) Either A is B or C is sometimes D.
.'. Either C is not-D or A is sometimes not B. (O)
(O) Either A is B or C is sometimes not D.
.'. Either C is D or A is sometimes not-B. (I)
§ 725.
(A) Either miracles are possible or every ancient historian is
untrustworthy.
.'. Either ancient historians are untrustworthy or miracles are
not impossible. (E)
(E) Either the tide must turn or the vessel can not make the port.
.'. Either the vessel cannot make the port or the tide must
turn. (A)
(1) Either he aims too high or the cartridges are sometimes bad.
.'. Either the cartridges are not bad or he sometimes does not
aim too high. (0)
(O) Either care must be taken or telegrams will sometimes not be
correct.
.'. Either telegrams are correct or carelessness is sometimes
shown. (1)
§ 726. In the above examples the converse of E looks as if it had
undergone no change but the mere transposition of the
alternative. This appearance arises from mentally reading the E as an
A proposition: but, if it were so taken, the result would be its
contrapositive, and not its converse by negation.
§ 727. The converse of I is a little difficult to grasp. It becomes
easier if we reduce it to the equivalent conjunctive--
'If the cartridges are bad, he sometimes does not aim too high.'
Here, as elsewhere, 'sometimes' must not be taken to mean more than
'it may be that.'
§ 728. _Conversion by Contraposition of Complex Propositions._
As applied to conjunctive propositions conversion by contraposition
assumes the following forms--
(A) If A is B, C is always D.
.'. If C is not-D, A is always not-B.
(O) If A is B, C is sometimes not D.
.'. If C is not-D, A is sometimes not not-B.
(A) If a man is honest, he is always truthful.
.'. If a man is untruthful, he is always dishonest.
(O) If a man is hasty, he is sometimes not malevolent.
.'. If a man is benevolent, he is sometimes not unhasty.
§ 729. As applied to disjunctive propositions conversion by
contraposition consists simply in transposing the two alternatives.
(A) Either A is B or C is D.
.'. Either C is D or A is B.
For, when reduced to the conjunctive shape, the reasoning would run
thus--
If A is not B, C is D.
.'. If C is not D, A is B.
which is the same in form as
All not-A is B.
.'. All not-B is A.
Similarly in the case of the O proposition
(O) Either A is B or C is sometimes not D.
.'. Either C is D or A is sometimes not B.
§ 730. On comparing these results with the converse by negation of
each of the same propositions, A and 0, the reader will see that they
differ from them, as was to be expected, only in being permuted. The
validity of the inference may be tested, both here and in the case of
conversion by negation, by reducing the disjunctive proposition to the
conjunctive, and so to the simple form, then performing the process as
in simple propositions, and finally throwing the converse, when so
obtained, back into the disjunctive form. We will show in this manner
that the above is really the contrapositive of the 0 proposition.
(O) Either A is B or C is sometimes not D.
= If A is not B, C is sometimes not D.
= Some cases of A not being B are not cases of C being D. (Some A is
not B.)
= Some cases of C not being D are not cases of A being B. (Some
not-B is not not-A.)
= If C is not D, A is sometimes not B.
= Either C is D or A is sometimes not B.