. Thus in FE_l_A_pt_O_n_ of the Third (MP, MS)

No M is in P
All M is in S
Some S is not in P

you have to substitute for All M is in S its converse by limitation to
get the premisses of FE_r_IO.

Two of the Minor Moods, Baroko of the Second Figure, and Bokardo
of the Third, cannot be reduced to the First Figure by the ordinary
processes of Conversion and Transposition. It is for dealing with
these intractable moods that Contraposition is required. Thus in
BA_r_O_k_O of the Second (PM, SM)

All P is in M.
Some S is not in M.

Substitute for the Major Premiss its Converse by Contraposition, and
for the Minor its Formal Obverse or Permutation, and you have FE_r_IO
of the First, with not-M as the Middle.

No not-M is in P.
Some S is in not-M,
Some S is not in P.

The processes might be indicated by the Mnemonic FA_cs_O_c_O, with
_c_ indicating the contraposition of the predicate term or Formal
Obversion.

The reduction of BO_k_A_rd_O,

Some M is not in P
All M is in S
Some S is not in P,

is somewhat more intricate. It may be indicated by DO_cs_A_m_O_sc_.
You substitute for the Major Premiss its Converse by Contraposition,
transpose the Premisses and you have DA_r_II.

All M is in S.
Some not-P is in M.
Some not-P is in S.

Convert now the conclusion by Contraposition, and you have Some S is
not in P.

The author of the Mnemonic apparently did not recognise
Contraposition, though it was admitted by Boethius; and, it being
impossible without this to demonstrate the validity of Baroko and
Bokardo by showing them to be equivalent with valid moods of the First
Figure, he provided for their demonstration by the special process
known as _Reductio ad absurdum_. B indicates that Barbara is the
medium.

The rationale of the process is this. It is an imaginary opponent that
you reduce to an absurdity or self-contradiction. You show that it
is impossible with consistency to admit the premisses and at the same
time deny the conclusion. For, let this be done; let it be admitted as
in BA_r_O_k_O that,

All P is in M
Some S is not in M,

but denied that Some S is not in P. The denial of a proposition
implies the admission of its Contradictory. If it is not true that
Some S is not in P, it must be true that All S is in P. Take this
along with the admission that All P is in M, and you have a syllogism
in BA_rb_A_r_A,

All P is in M
All S is in P,

yielding the conclusi