IMMEDIATE INFERENCES
Sec. 1. Under the general title of Immediate Inference Logicians discuss
three subjects, namely, Opposition, Conversion, and Obversion; to which
some writers add other forms, such as Whole and Part in Connotation,
Contraposition, Inversion, etc. Of Opposition, again, all recognise
four modes: Subalternation, Contradiction, Contrariety and
Sub-contrariety. The only peculiarities of the exposition upon which we
are now entering are, that it follows the lead of the three Laws of
Thought, taking first those modes of Immediate Inference in which
Identity is most important, then those which plainly involve
Contradiction and Excluded Middle; and that this method results in
separating the modes of Opposition, connecting Subalternation with
Conversion, and the other modes with Obversion. To make up for this
departure from usage, the four modes of Opposition will be brought
together again in Sec. 9.
Sec. 2. Subalternation.--Opposition being the relation of propositions that
have the same matter and differ only in form (as A., E., I., O.),
propositions of the forms A. and I. are said to be Subalterns in
relation to one another, and so are E. and O.; the universal of each
quality being distinguished as 'subalternans,' and the particular as
'subalternate.'
It follows from the principle of Identity that, the matter of the
propositions being the same, if A. is true I. is true, and that if E. is
true O. is true; for A. and E. predicate something of _All S_ or _All
men_; and since I. and O. make the same predication of _Some S_ or
_Some men_, the sense of these particular propositions has already been
predicated in A. or E. If _All S is P, Some S is P_; if _No S is P, Some
S is not P_; or, if _All men are fond of laughing, Some men are_; if _No
men are exempt from ridicule, Some men are not_.
Similarly, if I. is false A. is false; if O. is false E. is false. If we
deny any predication about _Some S_, we must deny it of _All S_; since
in denying it of _Some_, we have denied it of at least part of _All_;
and whatever is false in one form of words is false in any other.
On the other hand, if I. is true, we do not know that A. is; nor if O.
is true, that E. is; for to infer from _Some_ to _All_ would be going
beyond the evidence. We shall see in discussing Induction that the great
problem of that part of Logic is, to determine the conditions under
which we may in reality transcend this rule and infer from _Some_ to
_All_; though even there it will appear that, formally, the rule is
observed. For the present it is enough that I. is an immediate inference
from A., and O. from E.; but that A. is not an immediate inference from
I., nor E. from O.
Sec. 3. Connotative Subalternation.--We have seen (chap. iv. Sec. 6) that if
the connotation of one term is only part of another's its denotation is
greater and includes that other's. Hence genus and species stand in
subaltern relation, and whatever is true of the genus is true of the
species: If _All animal life is dependent on vegetation, All human life
is dependent on vegetation_. On the other hand, whatever is not true of
the species or narrower term, cannot be true of the whole genus: If it
is false that '_All human life is happy_,' it is false that '_All animal
life is happy_.'
Similar inferences may be drawn from the subaltern relation of
predicates; affirming the species we affirm the genus. To take Mill's
example, if _Socrates is a man, Socrates is a living creature_. On the
other hand, denying the genus we deny the species: if _Socrates is not
vicious, Socrates is not drunken_.
Such cases as these are recognised by Mill and Bain as immediate
inferences under the principle of Identity. But some Logicians might
treat them as imperfect syllogisms, requiring another premise to
legitimate the conclusion, thus:
_All animal life is dependent on vegetation;
All human life is animal life;
.'. All human life is dependent on vegetation._
Or again:
_All men are living creatures;
Socrates is a man;
.'. Socrates is a living creature._
The decision of this issue turns upon the question (_cf._ chap. vi. Sec. 3)
how far a Logician is entitled to assume that the terms he uses are
understood, and that the identities involved in their meanings will be
recognised. And to this question, for the sake of consistency, one of
two answers is required; failing which, there remains the rule of thumb.
First, it may be held that no terms are understood except those that are
defined in expounding the science, such as 'genus' and 'species,'
'connotation' and 'denotation.' But very few Logicians observe this
limitation; few would hesitate to substitute 'not wise' for 'foolish.'
Yet by what right? Malvolio being foolish, to prove that he is not-wise,
we may construct the following syllogism:
_Foolish is not-wise;
Malvolio is foolish;
.'. Malvolio is not-wise._
Is this necessary? Why not?
Secondly, it may be held that all terms may be assumed as understood
unless a definition is challenged. This principle will justify the
substitution of 'not-wise' for 'foolish'; but it will also legitimate
the above cases (concerning 'human life' and 'Socrates') as immediate
inferences, with innumerable others that might be based upon the
doctrine of relative terms: for example, _The hunter missed his aim_:
therefore, _The prey escaped_. And from this principle it will further
follow that all apparent syllogisms, having one premise a verbal
proposition, are immediate inferences (_cf._ chap. ix. Sec. 4).
Closely connected with such cases as the above are those mentioned by
Archbishop Thomson as "Immediate Inferences by added Determinants"
(_Laws of Thought_, Sec. 87). He takes the case: '_A negro is a
fellow-creature_: therefore, _A negro in suffering is a fellow-creature
in suffering_.' This rests upon the principle that to increase the
connotations of two terms by the same attribute or determinant does not
affect the relationship of their denotations, since it must equally
diminish (if at all) the denotations of both classes, by excluding the
same individuals, if any want the given attribute. But this principle is
true only when the added attribute is not merely the same verbally, but
has the same significance in qualifying both terms. We cannot argue _A
mouse is an animal_; therefore, _A large mouse is a large animal_; for
'large' is an attribute relative to the normal magnitude of the thing
described.
Sec. 4. Conversion is Immediate Inference by transposing the terms of a
given proposition without altering its quality. If the quantity is also
unaltered, the inference is called 'Simple Conversion'; but if the
quantity is changed from universal to particular, it is called
'Conversion by limitation' or '_per accidens._' The given proposition is
called the 'convertend'; that which is derived from it, the 'converse.'
Departing from the usual order of exposition, I have taken up Conversion
next to Subalternation, because it is generally thought to rest upon the
principle of Identity, and because it seems to be a good method to
exhaust the forms that come only under Identity before going on to those
that involve Contradiction and Excluded Middle. Some, indeed, dispute
the claims of Conversion to illustrate the principle of Identity; and
if the sufficient statement of that principle be 'A is A,' it may be a
question how Conversion or any other mode of inference can be referred
to it. But if we state it as above (chap. vi. Sec. 3), that whatever is
true in one form of words is true in any other, there is no difficulty
in applying it to Conversion.
Thus, to take the simple conversion of I.,
_Some S is P; .'. Some P is S._
_Some poets are business-like; .'. Some business-like men are poets._
Here the convertend and the converse say the same thing, and this is
true if that is.
We have, then, two cases of simple conversion: of I. (as above) and of
E. For E.:
_No S is P; .'. No P is S._
_No ruminants are carnivores; .'. No carnivores are ruminants._
In converting I., the predicate (P) when taken as the new subject, being
preindesignate, is treated as particular; and in converting E., the
predicate (P), when taken as the new subject, is treated as universal,
according to the rule in chap. v. Sec. 1.
A. is the one case of conversion by limitation:
All S is P;
.'. Some P is S.
All cats are grey in the dark;
.'. Some things grey in the dark are cats.
The predicate is treated as particular, when taking it for the new
subject, according to the rule not to go beyond the evidence. To infer
that _All things grey in the dark are cats_ would be palpably absurd;
yet no error of reasoning is commoner than the simple conversion of A.
The validity of conversion by limitation may be shown thus: if, _All S
is P_, then, by subalternation, _Some S is P_, and therefore, by simple
conversion, _Some P is S_.
O. cannot be truly converted. If we take the proposition: _Some S is
not P_, to convert this into _No P is S_, or _Some P is not S_, would
break the rule in chap. vi. Sec. 6; since _S,_ undistributed in the
convertend, would be distributed in the converse. If we are told that
_Some men are not cooks_, we cannot infer that _Some cooks are not men_.
This would be to assume that '_Some men_' are identical with '_All
men_.'
By quantifying the predicate, indeed, we may convert O. simply, thus:
_Some men are not cooks_ .'. _No cooks are some men._
And the same plan has some advantage in converting A.; for by the usual
method _per accidens_, the converse of A. being I., if we convert this
again it is still I., and therefore means less than our original
convertend. Thus:
_All S is P .'. Some P is S .'. Some S is P._
Such knowledge, as that _All S_ (the whole of it) _is P_, is too
precious a thing to be squandered in pure Logic; and it may be preserved
by quantifying the predicate; for if we convert A. to Y., thus--
_All S is P .'. Some P is all S--_
we may reconvert Y. to A. without any loss of meaning. It is the chief
use of quantifying the predicate that, thereby, every proposition is
capable of simple conversion.
The conversion of propositions in which the relation of terms is
inadequately expressed (see chap. ii., Sec. 2) by the ordinary copula (_is_
or _is not_) needs a special rule. To argue thus--
_A is followed by B_ .'. _Something followed by B is A_--
would be clumsy formalism. We usually say, and we ought to say--
_A is followed by B_ .'. _B follows A_ (or _is preceded by A_).
Now, any relation between two terms may be viewed from either side--_A:
B_ or _B: A_. It is in both cases the same fact; but, with the altered
point of view, it may present a different character. For example, in the
Immediate Inference--_A > B_ .'. _B < A_--a diminishing turns into an
increasing ratio, whilst the fact predicated remains the same. Given,
then, a relation between two terms as viewed from one to the other, the
same relation viewed from the other to the one may be called the
Reciprocal. In the cases of Equality, Co-existence and Simultaneity, the
given relation and its reciprocal are not only the same fact, but they
also have the same character: in the cases of Greater and Less and
Sequence, the character alters.
We may, then, state the following rule for the conversion of
propositions in which the whole relation explicitly stated is taken as
the copula: Transpose the terms, and for the given relation substitute
its reciprocal. Thus--
_A is the cause of B .'. B is the effect of A._
The rule assumes that the reciprocal of a given relation is definitely
known; and so far as this is true it may be extended to more concrete
relations--
_A is a genus of B .'. B is a species of A
A is the father of B .'. B is a child of A._
But not every relational expression has only one definite reciprocal. If
we are told that _A is the brother of B_, we can only infer that _B is
either the brother or the sister of A_. A list of all reciprocal
relations is a desideratum of Logic.
Sec. 5. Obversion (otherwise called Permutation or AEquipollence) is
Immediate Inference by changing the quality of the given proposition and
substituting for its predicate the contradictory term. The given
proposition is called the 'obvertend,' and the inference from it the
'obverse.' Thus the obvertend being--_Some philosophers are consistent
reasoners_, the obverse will be--_Some philosophers are not inconsistent
reasoners_.
The legitimacy of this mode of reasoning follows, in the case of
affirmative propositions, from the principle of Contradiction, that if
any term be affirmed of a subject, the contradictory term may be denied
(chap. vi. Sec. 3). To obvert affirmative propositions, then, the rule
is--Insert the negative sign, and for the predicate substitute its
contradictory term.
A. _All S is P .'. No S is not-P
All men are fallible .'. No men are infallible._
I. _Some S is P .'. some S is not-P
Some philosophers are consistent .'. Some philosophers are not
inconsistent._
In agreement with this mode of inference, we have the rule of modern
English grammar, that 'two negatives make an affirmative.'
Again, by the principle of Excluded Middle, if any term be denied of a
subject, its contradictory may be affirmed: to obvert negative
propositions, then, the rule is--Remove the negative sign, and for the
predicate substitute its contradictory term.
E. _No S is P .'. All S is not-P
No matter is destructible .'. All matter is indestructible._
O. _Some S is not P .'. Some S is not-P
Some ideals are not attainable .'. Some ideals are unattainable._
Thus, by obversion, each of the four propositions retains its quantity
but changes its quality: A. to E., I. to O., E. to A., O. to I. And all
the obverses are infinite propositions, the affirmative infinites having
the sense of negatives, and the negative infinites having the sense of
affirmatives.
Again, having obtained the obverse of a given proposition, it may be
desirable to recover the obvertend; or it may at any time be requisite
to change a given infinite proposition into the corresponding direct
affirmative or negative; and in such cases the process is still
obversion. Thus, if _No S is not-P_ be given us to recover the obvertend
or to find the corresponding affirmative; the proposition being formally
negative, we apply the rule for obverting negatives: 'Remove the
negative sign, and for the predicate substitute its contradictory.' This
yields the affirmative _All S is P_. Similarly, to obtain the obvertend
of _All S is not-P_, apply the rule for obverting Affirmatives; and this
yields _No S is P_.
Sec. 6. Contrariety.--We have seen in chap. iv. Sec. 8, that contrary terms
are such that no two of them are predicable in the same way of the same
subject, whilst perhaps neither may be predicable of it. Similarly,
Contrary Propositions may be defined as those of which no two are ever
both true together, whilst perhaps neither may be true; or, in other
words, both may be false. This is the relation between A. and E. when
concerned with the same matter: as A.--_All men are wise_; E.--_No men
are wise_. Such propositions cannot both be true; but they may both be
false, for some men may be wise and some not. They cannot both be true;
for, by the principle of Contradiction, if _wise_ may be affirmed of
_All men, not-wise_ must be denied; but _All men are not-wise_ is the
obverse of _No men are wise_, which therefore may also be denied.
At the same time we cannot apply to A. and E. the principle of Excluded
Middle, so as to show that one of them must be true of the same matter.
For if we deny that _All men are wise_, we do not necessarily deny the
attribute 'wise' of each and every man: to say that _Not all are wise_
may mean no more than that _Some are not_. This gives a proposition in
the form of O.; which, as we have seen, does not imply its subalternans,
E.
If, however, two Singular Propositions, having the same matter, but
differing in quality, are to be treated as universals, and therefore as
A. and E., they are, nevertheless, contradictory and not merely
contrary; for one of them must be false and the other true.
Sec. 7. Contradiction is a relation between two propositions analogous to
that between contradictory terms (one of which being affirmed of a
subject the other is denied)--such, namely, that one of them is false
and the other true. This is the case with the forms A. and O., and E.
and I., in the same matter. If it be true that _All men are wise_, it is
false that _Some men are not wise_ (equivalent by obversion to _Some
men are not-wise_); or else, since the 'Some men' are included in the
'All men,' we should be predicating of the same men that they are both
'wise' and 'not-wise'; which would violate the principle of
Contradiction. Similarly, _No men are wise_, being by obversion
equivalent to _All men are not-wise_, is incompatible with _Some men are
wise_, by the same principle of Contradiction.
But, again, if it be false that _All men are wise_, it is always true
that _Some are not wise_; for though in denying that 'wise' is a
predicate of 'All men' we do not deny it of each and every man, yet we
deny it of 'Some men.' Of 'Some men,' therefore, by the principle of
Excluded Middle, 'not-wise' is to be affirmed; and _Some men are
not-wise_, is by obversion equivalent to _Some men are not wise_.
Similarly, if it be false that _No men are wise_, which by obversion is
equivalent to _All men are not-wise_, then it is true at least that
_Some men are wise_.
By extending and enforcing the doctrine of relative terms, certain other
inferences are implied in the contrary and contradictory relations of
propositions. We have seen in chap. iv. that the contradictory of a
given term includes all its contraries: 'not-blue,' for example,
includes red and yellow. Hence, since _The sky is blue_ becomes by
obversion, _The sky is not not-blue_, we may also infer _The sky is not
red_, etc. From the truth, then, of any proposition predicating a given
term, we may infer the falsity of all propositions predicating the
contrary terms in the same relation. But, on the other hand, from the
falsity of a proposition predicating a given term, we cannot infer the
truth of the predication of any particular contrary term. If it be false
that _The sky is red_, we cannot formally infer, that _The sky is blue_
(_cf._ chap. iv. Sec. 8).
Sec. 8. Sub-contrariety is the relation of two propositions, concerning the
same matter that may both be true but are never both false. This is the
case with I. and O. If it be true that _Some men are wise_, it may also
be true that _Some (other) men are not wise_. This follows from the
maxim in chap. vi. Sec. 6, not to go beyond the evidence.
For if it be true that _Some men are wise_, it may indeed be true that
_All are_ (this being the subalternans): and if _All are_, it is (by
contradiction) false that _Some are not_; but as we are only told that
_Some men are_, it is illicit to infer the falsity of _Some are not_,
which could only be justified by evidence concerning _All men_.
But if it be false that _Some men are wise_, it is true that _Some men
are not wise_; for, by contradiction, if _Some men are wise_ is false,
_No men are wise_ is true; and, therefore, by subalternation, _Some men
are not wise_ is true.
Sec. 9. The Square of Opposition.--By their relations of Subalternation,
Contrariety, Contradiction, and Sub-contrariety, the forms A. I. E. O.
(having the same matter) are said to stand in Opposition: and Logicians
represent these relations by a square having A. I. E. O. at its corners:
A. Contraries E.
S Co s S
u nt e u
b ra i b
a di r a
l ct o l
t ct o t
e di r e
r ra i r
n nt e n
s Co s s
I. Sub-contraries O.
As an aid to the memory, this diagram is useful; but as an attempt to
represent the logical relations of propositions, it is misleading. For,
standing at corners of the same square, A. and E., A. and I., E. and O.,
and I. and O., seem to be couples bearing the same relation to one
another; whereas we have seen that their relations are entirely
different. The following traditional summary of their relations in
respect of truth and falsity is much more to the purpose:
(1) If A. is true, I. is true, E. is false, O. is false.
(2) If A. is false, I. is unknown, E. is unknown, O. is true.
(3) If I. is true, A. is unknown, E. is false, O. is unknown.
(4) If I. is false, A. is false, E. is true, O. is true.
(5) If E. is true, A. is false, I. is false, O. is true.
(6) If E. is false, A. is unknown, I. is true, O. is unknown.
(7) If O. is true, A. is false, I. is unknown, E. is unknown.
(8) If O. is false, A. is true, I. is true, E. is false.
Where, however, as in cases 2, 3, 6, 7, alleging either the
falsity of universals or the truth of particulars, it follows that two
of the three Opposites are unknown, we may conclude further that one of
them must be true and the other false, because the two unknown are
always Contradictories.
Sec. 10. Secondary modes of Immediate Inference are obtained by applying
the process of Conversion or Obversion to the results already obtained
by the other process. The best known secondary form of Immediate
Inference is the Contrapositive, and this is the converse of the obverse
of a given proposition. Thus:
DATUM. OBVERSE. CONTRAPOSITIVE.
A. _All S is P_ .'. _No S is not-P_ .'. _No not-P is S_
I. _Some S is P_ .'. _Some S is not not-P_ .'. (none)
E. _No S is P_ .'. _All S is not-P_ .'. _Some not-P is S_
O. _Some S is not P_ .'. _Some S is not-P_ .'. _Some not-P is S_
There is no contrapositive of I., because the obverse of I. is in the
form of O., and we have seen that O. cannot be converted. O., however,
has a contrapositive (_Some not-P is S_); and this is sometimes given
instead of the converse, and called the 'converse by negation.'
Contraposition needs no justification by the Laws of Thought, as it is
nothing but a compounding of conversion with obversion, both of which
processes have already been justified. I give a table opposite of the
other ways of compounding these primary modes of Immediate Inference.
A I E O
--------------------------------------------------------------------------------
1 All A is B Some A is B No A is B Some A is not B
--------------------------------------------------------------------------------
Obverse 2 No A is b Some A is not b All A is b Some A is b
--------------------------------------------------------------------------------
Converse 3 Some B is A Some B is A No B is A
--------------------------------------------------------------------------------
Obverse
of 4 Some B is not a Some B is not a All B is a
Converse
--------------------------------------------------------------------------------
Contra-
positive 5 No b is A Some b is A Some b is A
--------------------------------------------------------------------------------
Obverse
of 6 All b is a Some b is not a Some b is not a
Contrapos
--------------------------------------------------------------------------------
Converse
of
Obverse 7 Some a is B
of
Converse
--------------------------------------------------------------------------------
Obverse
of
Converse
of 8 Some a is not b
Obverse
of
Converse
--------------------------------------------------------------------------------
Converse
of
Obverse 9 Some a is b
of
Contrapos
--------------------------------------------------------------------------------
Obverse
of
Converse
of 10 Some a is not B
Obverse
of
Contrapos
--------------------------------------------------------------------------------
In this table _a_ and _b_ stand for _not-A_ and _not-B_ and had better
be read thus: for _No A is b, No A is not-B_; for _All b is a_ (col. 6),
_All not-B is not-A_; and so on.
It may not, at first, be obvious why the process of alternately
obverting and converting any proposition should ever come to an end;
though it will, no doubt, be considered a very fortunate circumstance
that it always does end. On examining the results, it will be found that
the cause of its ending is the inconvertibility of O. For E., when
obverted, becomes A.; every A, when converted, degenerates into I.;
every I., when obverted, becomes O.; O cannot be converted, and to
obvert it again is merely to restore the former proposition: so that the
whole process moves on to inevitable dissolution. I. and O. are
exhausted by three transformations, whilst A. and E. will each endure
seven.
Except Obversion, Conversion and Contraposition, it has not been usual
to bestow special names on these processes or their results. But the
form in columns 7 and 10 (_Some a is B--Some a is not B_), where the
original predicate is affirmed or denied of the contradictory of the
original subject, has been thought by Dr. Keynes to deserve a
distinctive title, and he has called it the 'Inverse.' Whilst the
Inverse is one form, however, Inversion is not one process, but is
obtained by different processes from E. and A. respectively. In this it
differs from Obversion, Conversion, and Contraposition, each of which
stands for one process.
The Inverse form has been objected to on the ground that the inference
_All A is B .'. Some not-A is not B_, distributes _B_ (as predicate of a
negative proposition), though it was given as undistributed (as
predicate of an affirmative proposition). But Dr. Keynes defends it on
the ground that (1) it is obtained by obversions and conversions which
are all legitimate and (2) that although _All A is B_ does not
distribute _B_ in relation to _A_, it does distribute _B_ in relation to
some _not-A_ (namely, in relation to whatever _not-A_ is _not-B_). This
is one reason why, in stating the rule in chap. vi. Sec. 6, I have
written: "an immediate inference ought to contain nothing that is not
contained, _or formally implied_, in the proposition from which it is
inferred"; and have maintained that every term formally implies its
contradictory within the _suppositio_.
Sec. 11. Immediate Inferences from Conditionals are those which
consist--(1) in changing a Disjunctive into a Hypothetical, or a
Hypothetical into a Disjunctive, or either into a Categorical; and (2)
in the relations of Opposition and the equivalences of Obversion,
Conversion, and secondary or compound processes, which we have already
examined in respect of Categoricals. As no new principles are involved,
it may suffice to exhibit some of the results.
We have already seen (chap. v. Sec. 4) how Disjunctives may be read as
Hypotheticals and Hypotheticals as Categoricals. And, as to Opposition,
if we recognise four forms of Hypothetical A. I. E. O., these plainly
stand to one another in a Square of Opposition, just as Categoricals do.
Thus A. and E. (_If A is B, C is D_, and _If A is B, C is not D_) are
contraries, but not contradictories; since both may be false (_C_ may
sometimes be _D_, and sometimes not), though they cannot both be true.
And if they are both false, their subalternates are both true, being
respectively the contradictories of the universals of opposite quality,
namely, I. of E., and O. of A. But in the case of Disjunctives, we
cannot set out a satisfactory Square of Opposition; because, as we saw
(chap. v. Sec. 4), the forms required for E. and O. are not true
Disjunctives, but Exponibles.
The Obverse, Converse, and Contrapositive, of Hypotheticals (admitting
the distinction of quality) may be exhibited thus:
DATUM. OBVERSE.
A. _If A is B, C is D_ _If A is B, C is not d_
I. Sometimes _when A is B, C is D_ Sometimes _when A is B, C is not d_
E. _If A is B, C is not D_ _If A is B, C is d_
O. Sometimes _when A is B, C is not D_ Sometimes _when A is B, C is d_
CONVERSE. CONTRAPOSITIVE.
Sometimes _when C is D, A is B_ _If C is d, A is not B_
Sometimes _when C is D, A is B_ (none)
_If C is D, A is not B_ Sometimes _when C is d, A is B_
(none) Sometimes _when C is d, A is B_
As to Disjunctives, the attempt to put them through these different
forms immediately destroys their disjunctive character. Still, given any
proposition in the form _A is either B or C_, we can state the
propositions that give the sense of obversion, conversion, etc., thus:
DATUM.--_A is either B or C;_
OBVERSE.--_A is not both b and c;_
CONVERSE.--_Something, either B or C, is A;_
CONTRAPOSITIVE.--_Nothing that is both b and c is A_.
For a Disjunctive in I., of course, there is no Contrapositive. Given a
Disjunctive in the form _Either A is B or C is D_, we may write for its
Obverse--_In no case is A b, and C at the same time d_. But no Converse
or Contrapositive of such a Disjunctive can be obtained, except by first
casting it into the hypothetical or categorical form.
The reader who wishes to pursue this subject further, will find it
elaborately treated in Dr. Keynes' _Formal Logic_, Part II.; to which
work the above chapter is indebted.