and the following are contradictories,
Every A is B to Some A is not B
No A is B to Some A is B
A contrary, therefore, is a complete and total contradictory; and a
little consideration will make it appear that the decisive distinction
between contraries and contradictories lies in this, that contraries may
both be false, but of contradictories, one must be true and the other
false. We may say, ‘Either P is true, or _something_ in contradiction of
it is true;’ but we cannot say, ‘Either P is true, or _every thing_ in
contradiction of it is true.’ It is a very common mistake to imagine
that the _denial_ of a proposition gives a right to _affirm_ the
contrary; whereas it should be, that the _affirmation_ of a proposition
gives a right to _deny_ the contrary. Thus, if we deny that Every A is
B, we do not affirm that No A is B, but only that Some A is not B;
while, if we affirm that Every A is B, we deny No A is B, and also Some
A is not B.
But, as to contradictories, affirmation of one is a denial of the other,
and denial of one is affirmation of the other. Thus, either Every A is
B, or Some A is not B: affirmation of either is denial of the other, and
_vice versá_.
Let the student now endeavour to satisfy himself of the following.
Taking the four preceding propositions, A, E, I, O, let the simple
letter signify the affirmation, the same letter in parentheses the
denial, and the absence of the letter, that there is neither affirmation
nor denial.
From A follow│(E), I, (O)│From (A) follow O
From E │(A), (I), O│From (E) I
From I │(E) │From (I) (A), E, O
From O │(A) │From (O) A, (E), I
These may be thus summed up: The affirmation of a universal proposition,
and the denial of a particular one, enable us to affirm or deny all the
other three; but the denial of a universal proposition, and the
affirmation of a particular one, leave us unable to affirm or deny two
of the others.
In such propositions as ‘Every A is B,’ ‘Some A is not B,’ &c., A is
called the _subject_, and B the _predicate_, while the verb ‘is’ or ‘is
not,’ is called the _copula_. It is obvious that the words of the
proposition point out whether the subject is spoken of universally or
partially, but not so of the predicate, which it is therefore important
to examine. Logical writers generally give the name of _distributed_
subjects or predicates to those which are spoken of universally; but as
this word is rather technical, I shall say that a subject or predicate
enters wholly or partially, according as it is universally or
particularly spoken of.
1. In A, or ‘Every A is B,’ the subject enters wholly, but the predicate
only partially. For it obviously says, ‘Among the Bs are all the As,’
‘Every A is part of the collection of Bs, so that all the As make a part
of the Bs, the whole it _may_ be.’ Thus, ‘Every horse is an animal,’
does not speak of all animals, but states that all the horses make up a
portion of the animals.
2. In E, or ‘No A is B,’ both subject and predicate enter wholly. ‘No A
whatsoever is any one out of all the Bs;’ ‘search the whole collection
of Bs, and _every_ B shall be found to be something which is not A.’
3. In I, or ‘Some A is B,’ both subject and predicate enter partially.
‘Some of the As are found among the Bs, or make up a part (the whole
possibly, but not known from the preceding) of the Bs.’
4. In O, or ‘Some A is not B,’ the subject enters partially, and the
predicate wholly. ‘Some As are none of them any whatsoever of the Bs;
every B will be found to be no one out of a certain portion of the As.’
It appears then that,
In affirmatives, the predicate enters partially.
In negatives, the predicate enters wholly.
In contradictory propositions, both subject and predicate enter
differently in the two.
The _converse_ of a proposition is that which is made by interchanging
the subject and predicate, as follows:
The proposition. Its converse.
A Every A is B Every B is A
E No A is B No B is A
I Some A is B Some B is A
O Some A is not B Some B is not A
Now, it is a fundamental and self-evident proposition, that no
consequence must be allowed to assert more widely than its premises; so
that, for instance, an assertion which is only of some Bs can never lead
to a result which is true of all Bs. But if a proposition assert
agreement or disagreement, any other proposition which asserts the same,
to the same extent and no further, must be a legitimate consequence; or,
if you please, must amount to the whole, or part, of the original
assertion in another form. Thus, the converse of A is not true: for, in
‘Every A is B,’ the predicate enters partially; while in ‘Every B is A,’
the subject enters wholly. ‘All the As make up a part of the Bs, then a
part of the Bs are among the As, or some B is A.’ Hence, the only
_legitimate_ converse of ‘Every A is B’ is, ‘Some B is A.’ But in ‘No A
is B,’ both subject and predicate enter wholly, and ‘No B is A’ is, in
fact, the same proposition as ‘No A is B.’ And ‘Some A is B’ is also the
same as its converse ‘Some B is A;’ here both terms enter partially. But
‘Some A is not B’ admits of no converse whatever; it is perfectly
consistent with all assertions upon B and A in which B is the subject.
Thus neither of the four following lines is inconsistent with itself.
Some A is not B and Every B is A
Some A is not B and No B is A
Some A is not B and Some B is A
Some A is not B and Some B is not A.
We find then, including converses, which are not identical with their
direct propositions, _six_ different ways of asserting or denying, with
respect to agreement or non-agreement, total or partial, between A and,
say X: these we write down, designating the additional assertions by U
and Y.
│Identical. │ Identical. │
A Every A is X│E No A is X│I Some A is X│O Some A is not X
U Every X is A│„ No X is A│„ Some X is A│Y Some X is not A
We shall now repeat and extend the table of page 8 (A), &c., meaning, as
before, the denial of A, &c.
From A or (O) follow A, (E), I (O)
From E or (I) (A), E, (I), O, (U), Y
From I or (E) (E) I
From O or (A) (A), O
From U or (Y) (E) I, U (Y)
From Y or (U) (U) Y
Having thus discussed the principal points connected with the simple
assertion, we pass to the manner of making two assertions give a third.
Every instance of this is called a syllogism, the two assertions which
form the basis of the third are called premises, and the third itself
the conclusion.
If two things both agree with a third in any particular, they agree with
each other in the same; as, if A be of the same colour as X, and B of
the same colour as X, then A is of the same colour as B. Again, if A
differ from X in any particular in which B agrees with X, then A and B
differ in that particular. If A be not of the same colour as X, and B be
of the same colour as X, then A is not of the colour of B. But if A and
B both differ from X in any particular, nothing can be inferred; they
may either differ in the same way and to the same extent, or not. Thus,
if A and B be both of different colours from X, it neither follows that
they agree, nor differ, in their own colours.
The paragraph preceding contains the essential parts of all inference,
which consists in comparing two things with a third, and finding from
their agreement or difference with that third, their agreement or
difference with one another. Thus, Every A is X, every B is X, allows us
to infer that A and B have all those qualities in common which are
necessary to X. Again, from Every A is X, and ‘No B is X,’ we infer that
A and B differ from one another in all particulars which are essential
to X. The preceding forms, however, though they represent common
reasoning better than the ordinary syllogism, to which we are now
coming, do not constitute the ultimate forms of inference. Simple
_identity_ or _non-identity_ is the ultimate state to which every
assertion may be reduced; and we shall, therefore, first ask, from what
identities, &c., can other identities, &c., be produced? Again, since we
name objects in species, each species consisting of a number of
individuals, and since our assertion may include all or only part of a
species, it is further necessary to ask, in every instance, to what
extent the conclusion drawn is true, whether of all, or only of part?
Let us take the simple assertion, ‘Every living man respires;’ or, every
living man is one of the things (however varied they may be) which
respire. If we were to inclose all living men in a large triangle, and
all respiring objects in a large circle, the preceding assertion, if
true, would require that the whole of the triangle should be contained
in the circle. And in the same way we may reduce any assertion to the
expression of a coincidence, total or partial, between two figures.
Thus, a point in a circle may represent an individual of one species,
and a point in a triangle an individual of another species: and we may
express that the whole of one species is asserted to be contained or not
contained in the other by such forms as, ‘All the △ is in the ○’; ‘None
of the △ is in the ○’.
Any two assertions about A and B, each expressing agreement or
disagreement, total or partial, with or from X, and leading to a
conclusion with respect to A or B, is called a syllogism, of which X is
called the _middle term_. The plainest syllogism is the following:—
Every A is X│ All the △ is in the ○
Every X is B│ All the ○ is in the □
Therefore Every A is B│Therefore All the △ is in the □
In order to find all the possible forms of syllogism, we must make a
table of all the elements of which they can consist; namely—
A and X B and X
Every A is X A Every B is X
No A is X E No B is X
Some A is X I Some B is X
Some A is not X O Some B is not X
Every X is A U Every X is B
Some X is not A Y Some X is not B
Or their synonymes,
△ and ○ □ and ○
All the △ is in the ○ A All the □ is in the ○
None of the △ is in the ○ E None of the □ is in the ○
Some of the △ is in the ○ I Some of the □ is in the ○
Some of the △ is not in the ○ O Some of the □ is not in the ○
All the ○ is in the △ U All the ○ is in the □
Some of the ○ is not in the △ Y Some of the ○ is not in the □
Now, taking any one of the six relations between A and X, and combining
it with either of those between B and X, we have six pairs of premises,
and the same number repeated for every different relation of A and X. We
have then thirty-six forms to consider: but, thirty of these (namely,
all but (A, A) (E, E), &c.) are half of them repetitions of the other
half. Thus, ‘Every A is X, no B is X,’ and ‘Every B is X, no A is X,’
are of the same form, and only differ by changing A into B and B into A.
There are then only 15 + 6, or 21 distinct forms, some of which give a
necessary conclusion, while others do not. We shall select the former of
these, classifying them by their conclusions; that is, according as the
inference is of the form A, E, I, or O.
I. In what manner can a universal affirmative conclusion be drawn;
namely, that one figure is entirely contained in the other? This we can
only assert when we know that one figure is entirely contained in the
circle, which itself is entirely contained in the other figure. Thus,
Every A is X│ All the △ is in the ○ A
Every X is B│ All the ○ is in the □ A
∴ Every A is B│∴ All the △ is in the □ A
is the only way in which a universal affirmative conclusion can be
drawn.
II. In what manner can a universal negative conclusion be drawn; namely,
that one figure is entirely exterior to the other? Only when we are able
to assert that one figure is entirely within, and the other entirely
without, the circle. Thus,