The Complete Aristotle (eng.)

hypothetical arguments are formed: but at present
this much must be clear, that it is not possible to resolve such
arguments into the figures. And we have explained the reason.
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    Whatever problems are proved in more than one figure, if they
have been established in one figure by syllogism, can be reduced to
another figure, e.g. a negative syllogism in the first figure can
be reduced to the second, and a syllogism in the middle figure to
the first, not all however but some only. The point will be clear
in the sequel. If A belongs to no B, and B to all C, then A belongs
to no C. Thus the first figure; but if the negative statement is
converted, we shall have the middle figure. For B belongs to no A,
and to all C. Similarly if the syllogism is not universal but
particular, e.g. if A belongs to no B, and B to some C. Convert the
negative statement and you will have the middle figure.
    The universal syllogisms in the second figure can be reduced to
the first, but only one of the two particular syllogisms. Let A
belong to no B and to all C. Convert the negative statement, and
you will have the first figure. For B will belong to no A and A to
all C. But if the affirmative statement concerns B, and the
negative C, C must be made first term. For C belongs to no A, and A
to all B: therefore C belongs to no B. B then belongs to no C: for
the negative statement is convertible.
    But if the syllogism is particular, whenever the negative
statement concerns the major extreme, reduction to the first figure
will be possible, e.g. if A belongs to no B and to some C: convert
the negative statement and you will have the first figure. For B
will belong to no A and A to some C. But when the affirmative
statement concerns the major extreme, no resolution will be
possible, e.g. if A belongs to all B, but not to all C: for the
statement AB does not admit of conversion, nor would there be a
syllogism if it did.
    Again syllogisms in the third figure cannot all be resolved into
the first, though all syllogisms in the first figure can be
resolved into the third. Let A belong to all B and B to some C.
Since the particular affirmative is convertible, C will belong to
some B: but A belonged to all B: so that the third figure is
formed. Similarly if the syllogism is negative: for the particular
affirmative is convertible: therefore A will belong to no B, and to
some C.
    Of the syllogisms in the last figure one only cannot be resolved
into the first, viz. when the negative statement is not universal:
all the rest can be resolved. Let A and B be affirmed of all C:
then C can be converted partially with either A or B: C then
belongs to some B. Consequently we shall get the first figure, if A
belongs to all C, and C to some of the Bs. If A belongs to all C
and B to some C, the argument is the same: for B is convertible in
reference to C. But if B belongs to all C and A to some C, the
first term must be B: for B belongs to all C, and C to some A,
therefore B belongs to some A. But since the particular statement
is convertible, A will belong to some B. If the syllogism is
negative, when the terms are universal we must take them in a
similar way. Let B belong to all C, and A to no C: then C will
belong to some B, and A to no C; and so C will be middle term.
Similarly if the negative statement is universal, the affirmative
particular: for A will belong to no C, and C to some of the Bs. But
if the negative statement is particular, no resolution will be
possible, e.g. if B belongs to all C, and A not belong to some C:
convert the statement BC and both premisses will be particular.
    It is clear that in order to resolve the figures into one
another the premiss which concerns the minor extreme must be
converted in both the figures: for when this premiss is altered,
the transition to the other figure is made.
    One of the syllogisms in the middle figure can, the other
cannot, be resolved into the third figure. Whenever the universal
statement is negative, resolution is possible. For if A belongs to
no B and to some C, both B and C alike are convertible in relation
to A, so that B belongs to no A and C to some A. A therefore is
middle term. But when A belongs to all B, and not to some C,
resolution will not be possible: for neither of the premisses is
universal after conversion.
    Syllogisms in the third figure can be resolved into the middle
figure, whenever the negative statement is universal, e.g. if A
belongs to no C, and B to some or all C. For C then will