stands as we said above). But it turns out in these also that we use
for the demonstration the very thing that is being proved: for C is
proved of B, and B of by assuming that C is said of and C is proved of
A through these premisses, so that we use the conclusion for the
demonstration.
In negative syllogisms reciprocal proof is as follows. Let B
belong to all C, and A to none of the Bs: we conclude that A belongs
to none of the Cs. If again it is necessary to prove that A belongs to
none of the Bs (which was previously assumed) A must belong to no C,
and C to all B: thus the previous premiss is reversed. If it is
necessary to prove that B belongs to C, the proposition AB must no
longer be converted as before: for the premiss 'B belongs to no A'
is identical with the premiss 'A belongs to no B'. But we must
assume that B belongs to all of that to none of which longs. Let A
belong to none of the Cs (which was the previous conclusion) and
assume that B belongs to all of that to none of which A belongs. It is
necessary then that B should belong to all C. Consequently each of the
three propositions has been made a conclusion, and this is circular
demonstration, to assume the conclusion and the converse of one of the
premisses, and deduce the remaining premiss.
In particular syllogisms it is not possible to demonstrate the
universal premiss through the other propositions, but the particular
premiss can be demonstrated. Clearly it is impossible to demonstrate
the universal premiss: for what is universal is proved through
propositions which are universal, but the conclusion is not universal,
and the proof must start from the conclusion and the other premiss.
Further a syllogism cannot be made at all if the other premiss is
converted: for the result is that both premisses are particular. But
the particular premiss may be proved. Suppose that A has been proved
of some C through B. If then it is assumed that B belongs to all A and
the conclusion is retained, B will belong to some C: for we obtain the
first figure and A is middle. But if the syllogism is negative, it
is not possible to prove the universal premiss, for the reason given
above. But it is possible to prove the particular premiss, if the
proposition AB is converted as in the universal syllogism, i.e 'B
belongs to some of that to some of which A does not belong': otherwise
no syllogism results because the particular premiss is negative.
6
In the second figure it is not possible to prove an affirmative
proposition in this way, but a negative proposition may be proved.
An affirmative proposition is not proved because both premisses of the
new syllogism are not affirmative (for the conclusion is negative) but
an affirmative proposition is (as we saw) proved from premisses
which are both affirmative. The negative is proved as follows. Let A
belong to all B, and to no C: we conclude that B belongs to no C. If
then it is assumed that B belongs to all A, it is necessary that A
should belong to no C: for we get the second figure, with B as middle.
But if the premiss AB was negative, and the other affirmative, we
shall have the first figure. For C belongs to all A and B to no C,
consequently B belongs to no A: neither then does A belong to B.
Through the conclusion, therefore, and one premiss, we get no
syllogism, but if another premiss is assumed in addition, a
syllogism will be possible. But if the syllogism not universal, the
universal premiss cannot be proved, for the same reason as we gave
above, but the particular premiss can be proved whenever the universal
statement is affirmative. Let A belong to all B, and not to all C: the
conclusion is BC. If then it is assumed that B belongs to all A, but
not to all C, A will not belong to some C, B being middle. But if
the universal premiss is negative, the premiss AC will not be
demonstrated by the conversion of AB: for it turns out that either
both or one of the premisses is negative; consequently a syllogism
will not be possible. But the proof will proceed as in the universal
syllogisms, if it is assumed that A belongs to some of that to some of
which B does not belong.
7
In the third figure, when both premisses are taken universally, it
is not possible to prove them reciprocally: for that which is
universal is proved through statements which are universal, but the
conclusion in this figure is always particular, so that it is clear
that it is not possible at all to prove through this figure the
universal premiss. But if one premiss is universal, the other
particular, proof of the latter will sometimes be possible,
sometimes not. When both the premisses assumed are affirmative, and
the universal concerns the minor extreme, proof will be possible,
but when it concerns the other extreme, impossible. Let A belong to
all C and B to some C: the conclusion is the statement AB. If then
it is assumed that C belongs to all A, it has been proved that C
belongs to some B, but that B belongs to some C has not been proved.
And yet it is necessary, if C belongs to some B, that B should
belong to some C. But it is not the same that this should belong to
that, and that to this: but we must assume besides that if this
belongs to some of that, that belongs to some of this. But if this
is assumed the syllogism no longer results from the conclusion and the
other premiss. But if B belongs to all C, and A to some C, it will
be possible to prove the proposition AC, when it is assumed that C
belongs to all B, and A to some B. For if C belongs to all B and A
to some B, it is necessary that A should belong to some C, B being
middle. And whenever one premiss is affirmative the other negative,
and the affirmative is universal, the other premiss can be proved. Let
B belong to all C, and A not to some C: the conclusion is that A
does not belong to some B. If then it is assumed further that C
belongs to all B, it is necessary that A should not belong to some
C, B being middle. But when the negative premiss is universal, the
other premiss is not except as before, viz. if it is assumed that that
belongs to some of that, to some of which this does not belong, e.g.
if A belongs to no C, and B to some C: the conclusion is that A does
not belong to some B. If then it is assumed that C belongs to some
of that to some of which does not belong, it is necessary that C
should belong to some of the Bs. In no other way is it possible by
converting the universal premiss to prove the other: for in no other
way can a syllogism be formed.
It is clear then that in the first figure reciprocal proof is made
both through the third and through the first figure-if the
conclusion is affirmative through the first; if the conclusion is
negative through the last. For it is assumed that that belongs to
all of that to none of which this belongs. In the middle figure,
when the syllogism is universal, proof is possible through the
second figure and through the first, but when particular
through the
second and the last. In the third figure all proofs are made through
itself. It is clear also that in the third figure and in the middle
figure those syllogisms which are not made through those figures
themselves either are not of the nature of circular proof or are
imperfect.
8
To convert a syllogism means to alter the conclusion and make
another syllogism to prove that either the extreme cannot belong to
the middle or the middle to the last term. For it is necessary, if the
conclusion has been changed into its opposite and one of the premisses
stands, that the other premiss should be destroyed. For if it should
stand, the conclusion also must stand. It makes a difference whether
the conclusion is converted into its contradictory or into its
contrary. For the same syllogism does not result whichever form the
conversion takes. This will be made clear by the sequel. By
contradictory opposition I mean the opposition of 'to all' to 'not
to all', and of 'to some' to 'to none'; by contrary opposition I
mean the opposition of 'to all' to 'to none', and of 'to some' to 'not
to some'. Suppose that A been proved of C, through B as middle term.
If then it should be assumed that A belongs to no C, but to all B, B
will belong to no C. And if A belongs to no C, and B to all C, A
will belong, not to no B at all, but not to all B. For (as we saw) the
universal is not proved through the last figure. In a word it is not
possible to refute universally by conversion the premiss which
concerns the major extreme: for the refutation always proceeds through
the third since it is necessary to take both premisses in reference to
the minor extreme. Similarly if the syllogism is negative. Suppose
it has been proved that A belongs to no C through B. Then if it is
assumed that A belongs to all C, and to no B, B will belong to none of
the Cs. And if A and B belong to all C, A will belong to some B: but
in the original premiss it belonged to no B.
If the conclusion is converted into its contradictory, the
syllogisms will be contradictory and not universal. For one premiss is
particular, so that the conclusion also will be particular. Let the
syllogism be affirmative, and let it be converted as stated. Then if A
belongs not to all C, but to all B, B will belong not to all C. And if
A belongs not to all C, but B belongs to all C, A will belong not to
all B. Similarly if the syllogism is negative. For if A belongs to
some C, and to no B, B will belong, not to no C at all, but-not to
some C. And if A belongs to some C, and B to all C, as was
originally assumed, A will belong to some B.
In particular syllogisms when the conclusion is converted into its
contradictory, both premisses may be refuted, but when it is converted
into its contrary, neither. For the result is no longer, as in the
universal syllogisms, refutation in which the conclusion reached by O,
conversion lacks universality, but no refutation at all. Suppose
that A has been proved of some C. If then it is assumed that A belongs
to no C, and B to some C, A will not belong to some B: and if A
belongs to no C, but to all B, B will belong to no C. Thus both
premisses are refuted. But neither can be refuted if the conclusion is
converted into its contrary. For if A does not belong to some C, but
to all B, then B will not belong to some C. But the original premiss
is not yet refuted: for it is possible that B should belong to some C,
and should not belong to some C. The universal premiss AB cannot be
affected by a syllogism at all: for if A does not belong to some of
the Cs, but B belongs to some of the Cs, neither of the premisses is
universal. Similarly if the syllogism is negative: for if it should be
assumed that A belongs to all C, both premisses are refuted: but if
the assumption is that A belongs to some C, neither premiss is
refuted. The proof is the same as before.
9
In the second figure it is not possible to refute the premiss
which concerns the major extreme by establishing something contrary to
it, whichever form the conversion of the conclusion may take. For
the conclusion of the refutation will always be in the third figure,
and in this figure (as we saw) there is no universal syllogism. The
other premiss can be refuted in a manner similar to the conversion:
I mean, if the conclusion of the first syllogism is converted into its
contrary, the conclusion of the refutation will be the contrary of the
minor premiss of the first, if into its contradictory, the
contradictory. Let A belong to all B and to no C: conclusion BC. If
then it is assumed that B belongs to all C, and the proposition AB
stands, A will belong to all C, since the first figure is produced. If
B belongs to all C, and A to no C, then A belongs not to all B: the
figure is the last. But if the conclusion BC is converted into its
contradictory, the premiss AB will be refuted as before, the
premiss, AC by its contradictory. For if B belongs to some C, and A to
no C, then A will not belong to some B. Again if B belongs to some
C, and A to all B, A will belong to some C, so that the syllogism
results in the contradictory of the minor premiss. A similar proof can
be given if the premisses are transposed in respect of their quality.
If the syllogism is particular, when the conclusion is converted
into its contrary neither premiss can be refuted, as also happened
in the first figure,' if the conclusion is converted into its
contradictory, both premisses can be refuted. Suppose that A belongs
to no B, and to some C: the conclusion is BC. If then it is assumed
that B belongs to some C, and the statement AB stands, the
conclusion will be that A does not belong to some C. But the
original statement has not been refuted: for it is possible that A
should belong to some C and also not to some C. Again if B belongs
to some C and A to some C, no syllogism will be possible: for
neither of the premisses taken is universal. Consequently the
proposition AB is not refuted. But if the conclusion is converted into
its contradictory, both premisses can be refuted. For if B belongs
to all C, and A to no B, A will belong to no C: but it was assumed
to belong to some C. Again if B belongs to all C and A to some C, A
will belong to some B. The same proof can be given if the universal
statement is affirmative.
10
In the third figure when the conclusion is converted into its
contrary, neither of the premisses can be refuted in any of the
syllogisms, but when the conclusion is converted into its
contradictory, both premisses may be refuted and in all the moods.
Suppose it has been proved that A belongs to some B, C being taken
as middle, and the premisses being universal. If then it is assumed
that A does not belong to some B, but B belongs to all C, no syllogism
is formed about A and C. Nor if A does not belong to some B, but
belongs to all C, will a syllogism be possible about B and C. A
similar proof can be given if
the premisses are not universal. For
either both premisses arrived at by the conversion must be particular,
or the universal premiss must refer to the minor extreme. But we found
that no syllogism is possible thus either in the first or in the
middle figure. But if the conclusion is converted into its
contradictory, both the premisses can be refuted. For if A belongs
to no B, and B to all C, then A belongs to no C: again if A belongs to
no B, and to all C, B belongs to no C. And similarly if one of the
premisses is not universal. For if A belongs to no B, and B to some C,
A will not belong to some C: if A belongs to no B, and to C, B will
belong to no C.
Similarly if the original syllogism is negative. Suppose it has been
proved that A does not belong to some B, BC being affirmative, AC
being negative: for it was thus that, as we saw, a syllogism could
be made. Whenever then the contrary of the conclusion is assumed a
syllogism will not be possible. For if A belongs to some B, and B to
all C, no syllogism is possible (as we saw) about A and C. Nor, if A
belongs to some B, and to no C, was a syllogism possible concerning
B and C. Therefore the premisses are not refuted. But when the
contradictory of the conclusion is assumed, they are refuted. For if A
belongs to all B, and B to C, A belongs to all C: but A was supposed
originally to belong to no C. Again if A belongs to all B, and to no
C, then B belongs to no C: but it was supposed to belong to all C. A
similar proof is possible if the premisses are not universal. For AC
becomes universal and negative, the other premiss particular and
affirmative. If then A belongs to all B, and B to some C, it results
that A belongs to some C: but it was supposed to belong to no C. Again
if A belongs to all B, and to no C, then B belongs to no C: but it was
assumed to belong to some C. If A belongs to some B and B to some C,
no syllogism results: nor yet if A belongs to some B, and to no C.
Thus in one way the premisses are refuted, in the other way they are
not.
From what has been said it is clear how a syllogism results in
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