would have been considered as formed by a spurious and unnecessary
excess of assertion. The minimum of assertion would be contained in
either of the following,
Every X is A, Some X is B, therefore Some A is B
Some X is A, Every X is B, therefore Some A is B
In this tract, syllogisms have been divided into two classes: first,
those which prove a universal conclusion; secondly, those which prove a
partial conclusion, and which are (all but one) derived from the first
by weakening one of the premises, in such manner as to produce a
legitimate but weakened conclusion. Those of the first class are placed
in the first column, and the other in the second.
Universal. Particular.
A Every A is X Some A is X I
A Every X is B ────── Every X is B A
———————————— —————————————
A Every A is B Some A is B I
Some A is X I
No X is B E
┌ ———————————————
A Every A is X │ Some A is not B O
E No X is B ─────┼ Every A is X A
———————————— │
E No A is B │ Some B is not X O
└ ———————————————
Some B is not A O
Every X is A A
...... Some X is not B O
———————————————
Some A is not B O
In all works on logic, it is customary to write that premiss first which
contains the predicate of the conclusion. Thus,
Every X is B Every A is X
Every A is X would be written, and not Every X is B
———————————— ————————————
Every A is B Every A is B
The premises thus arranged are called major and minor; the predicate of
the conclusion being called the major term, and its subject the minor.
Again, in the preceding case we see the various subjects coming in the
order X, B; A, X; A, B: and the number of different orders which can
appear is four, namely—
XB BX XB BX
AX AX XA XA
—— —— —— ——
AB AB AB AB
which are called the four figures, and every kind of syllogism in each
figure is called a mood. I now put down the various moods of each
figure, the letters of which will be a guide to find out those of the
preceding list from which they are derived. Co means that a premiss of
the preceding list has been converted; + that it has been strengthened;
Co +, that both changes have taken place. Thus,
A Every X is B A Every X is B
I Some A is X becomes A Every X is A: (Co +)
———————————— ————————————
I Some A is B I Some A is B
And Co + abbreviates the following: If some A be X, then some X is A
(Co); and all that is true when Some X is A, is true when Every X is A
(+); therefore the second is legitimate, if the first be so.
_First Figure._
A Every X is B A Every X is B
A Every A is X I Some A is X
———————————— ——————————————
A Every A is B I Some A is B
E No X is B E No X is B
A Every A is X I Some A is X
———————————— ————————————————
E No A is B O Some A is not B
_Second Figure._
E No B is X (Co) E No B is X (Co)
A Every A is X I Some A is X
———————————— —————————————————
E No A is B O Some A is not B
A Every B is X A Every B is X
E No A is X (Co) O Some A is not X
———————————— ————————————————
E No A is B O Some A is not B
_Third Figure._
A Every X is B E No X is B
A Every X is A (Co+) A Every X is A (Co+)
———————————— ——————————————————
I Some A is B O Some A is not B
I Some X is B (Co) O Some X is not B
A Every X is A A Every X is A
———————————— ————————————————
I Some A is B O Some A is not B
A Every X is B E No X is B
I Some X is A (Co) I Some X is A (Co)
———————————— —————————————————
I Some A is B O Some A is not B
_Fourth Figure._
A Every B is X (+) I Some B is X
A Every X is A A Every X is A
———————————— ————————————
I Some A is B I Some B is A
A Every B is X E No B is X (Co)
E No X is A A Every X is A (Co+)
———————————— ——————————————————
E No A is B O Some A is not B
E No B is X (Co)
I Some X is A (Co)
————————————————
O Some A is not B
The above is the ancient method of dividing syllogisms; but, for the
present purpose, it will be sufficient to consider the six from which
the rest can be obtained. And since some of the six have A in the
predicate of the conclusion, and not B, we shall join to them the six
other syllogisms which are found by transposing B and A. The complete
list, therefore, of syllogisms with the weakest premises and the
strongest conclusions, in which a comparison of A and B is obtained by
comparison of both with X, is as follows:
Every A is X Every B is X │Some A is X Some B is X
Every X is B Every X is A │No X is B No X is A
———————————— ———————————— │———————————————— ————————————————
Every A is B Every B is A │Some A is not B Some B is not A
│
Every A is X Every B is X │Every A is X Every B is X
No X is B No X is A │Some B is not X Some A is not X
———————————— ———————————— │———————————————— ————————————————
No A is B No B is A │Some B is not A Some A is not B
│
Some A is X Some B is X │Every X is A Every X is B
Every X is B Every X is A │Some X is not B Some X is not A
———————————— ———————————— │———————————————— ————————————————
Some A is B Some B is A │Some A is not B Some B is not A
In the list of page 19, there was nothing but recapitulation of forms,
each form admitting a variation by interchanging A and B. This
interchange having been made, and the results collected as above, if we
take every case in which B is the predicate, or can be made the
predicate by allowable conversion, we have a collection of all possible
_weakest_ forms in which the result is one of the four ‘Every A is B,’
‘No A is B,’ ‘Some A is B,’ ‘Some A is not B’; as follows. The premises
are written in what appeared the most natural order, without distinction
of major or minor; and the letters prefixed are according to the forms
of the several premises, as in page 10.
A Every A is X
U Every X is B
————————————
A Every A is B
I Some A is X I Some B is X
U Every X is B U Every X is A
———————————— ————————————
I Some A is B I Some A is B
A Every A is X A Every B is X
E No B is X E No A is X
———————————— ————————————
E No A is B E No A is B
I Some A is X A Every B is X U Every X is A
E No B is X O Some A is not X Y Some X is not B
———————————————— ———————————————— ————————————————
O Some A is not B O Some A is not B O Some A is not B
Every assertion which can be made upon two things by comparison with any
third, that is, every simple inference, can be reduced to one of the
preceding forms. Generally speaking, one of the premises is omitted, as
obvious from the conclusion; that is, one premiss being named and the
conclusion, that premiss is implied which is necessary to make the
conclusion good. Thus, if I say, “That race must have possessed some of
the arts of life, for they came from Asia,” it is obviously meant to be
asserted, that all races coming from Asia must have possessed some of
the arts of life. The preceding is then a syllogism, as follows:
‘That race’ is ‘a race of Asiatic origin:’
Every ‘race of Asiatic origin’ is ‘a race which must have
possessed some of the arts of life:’
Therefore, That race _is_ a race which must have possessed some of the
arts of life.
A person who makes the preceding assertion either means to imply,
antecedently to the conclusion, that all Asiatic races must have
possessed arts, or he talks nonsense if he asserts the conclusion
positively. ‘A must be B, for it is X,’ can only be true when ‘Every X
is B.’ This latter proposition may be called the suppressed premiss; and
it is in such suppressed propositions that the greatest danger of error
lies. It is also in such propositions that men convey opinions which
they would not willingly express. Thus, the honest witness who said, ‘I
always thought him a respectable man—he kept his gig,’ would probably
not have admitted in direct terms, ‘Every man who keeps a gig must be
respectable.’