But there are three other possible orders or figures, namely:--

Fig. ii. Fig. iii. Fig. iv.

PM MP PM SM MS MS.

It results from the doctrines of Conversion that valid arguments may be stated in these forms, inasmuch as a proposition in one order of terms may be equivalent to a proposition in another. Thus No M is in P is convertible with No P is in M: consequently the argument

No P is in M All S is in M,

in the Second Figure is as much valid as when it is stated in the First--

No M is in P All S is in M.

Similarly, since All M is in S is convertible into Some S is in M, the following arguments are equally valid:--

Fig. iii. Fig. i.

All M is in P All M is in P = All M is in S Some S is in M.

Using both the above Converses in place of their Convertends, we have--

Fig. iv. Fig. i.

No P is in M No M is in P = All M is in S Some S is in M.

It can be demonstrated (we shall see presently how) that altogether there are possible four valid forms or moods of the Second Figure, six of the Third, and five of the Fourth. An ingenious Mnemonic of these various moods and their reduction to the First Figure by the transposition of terms and premisses has come down from the thirteenth century. The first line names the moods of the First, Normal, or Standard Figure.

BA_rb_A_r_A, CE_l_A_r_E_nt_, DA_r_II, FE_r_IO_que_ prioris; CE_s_A_r_E, CA_m_E_str_E_s_, FE_st_I_n_O, BA_r_O_k_O, secundae; Tertia DA_r_A_pt_I, DI_s_A_m_I_s_, DA_t_I_s_I, FE_l_A_pt_O_n_, BO_k_A_rd_O, FE_r_I_s_O_que_, habet; quarta insuper addit, B_r_A_m_A_nt_IP, CA_m_E_n_E_s_, DI_m_A_r_I_s_, FE_s_A_p_O, F_r_E_s_I_s_O_n_.

The vowels in the names of the Moods indicate the propositions of the Syllogism in the four forms, A E I O. To write out any Mood at length you have only to remember the Figure, and transcribe the propositions in the order of Major Premiss, Minor Premiss, and Conclusion. Thus, the Second Figure being

PM SM

FE_st_I_n_O is written--

No P is in M.

Some S is in M.

Some S is not in P.

The Fourth Figure being

PM MS

DI_m_A_r_I_s_ is

Some P is in M.

All M is in S.

Some S is in P.

The initial letter in a Minor Mood indicates that Mood of the First to which it may be reduced. Thus Festino is reduced to Ferio, and Dimaris to Darii. In the cases of Baroko and Bokardo, B indicates that you may employ Barbara to bring any impugner to confusion, as shall be afterwards explained.

The letters _s_, _m_, and _p_ are also significant. Placed after a vowel, _s_ indicates that the proposition has to be simply converted.

Thus, FE_st_I_n_O:--

No P is in M.

Some S is in M.

Some S is not in P.

Simply convert the Major Premiss, and you get FE_r_IO, of the First.

No M is in P.

Some S is in M.

Some S is not in P.

_m_ (_muta_, or _move_) indicates that the premisses have to be transposed. Thus, in CA_m_E_str_E_s_, you have to transpose the premisses, as well as simply convert the Minor Premiss before reaching the figure of CE_l_A_r_E_nt_.

All P is in M No M is in S = No S is in M All P is in M.

From this it follows in CE_l_A_r_E_nt_ that No P is in S, and this simply converted yields No S is in P.

A simple transposition of the premisses in DI_m_A_r_I_s_ of the Fourth

Some P is in M All M is in S

yields the premisses of DA_r_II

All M is in S Some P is in M,

but the conclusion Some P is in S has to be simply converted.

Placed after a vowel, _p_ indicates that the proposition has to be converted _per accidens_. 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,