If a more systematic way of demonstrating the valid moods is desired, the simplest method is to deduce from the Canons special rules for each Figure. Aristotle arrived at these special rules by simple inspection, but it is easier to deduce them.

I. In the First Figure, the Major Premiss must be Universal, and the Minor Premiss affirmative.

To make this evident by the Canons, we bear in mind the Scheme or Figure--

M in P S in M--

and try the alternatives of Affirmative Moods and Negative Moods.

Obviously in an affirmative mood the Middle is undistributed unless the Major Premiss is Universal. In a negative mood, (1) If the Major Premiss is O, the Minor must be affirmative, and M is undistributed; (2) if the Major Premiss is I, M may be distributed by a negative Minor Premiss, but in that case there would be an illicit process of the Major--P being distributed in the conclusion (Canon V.) and not in the Premisses. Thus the Major Premiss can neither be O nor I, and must therefore be either A or E, _i.e._, must be Universal.

That the Minor must be affirmative is evident, for if it were negative, the conclusion must be negative (Canon V.) and the Major Premiss must be affirmative (Canon IV.), and this would involve illicit process of the Major, P being distributed in the conclusion and not in the Premisses.

These two special rules leave only four possible valid forms in the First Figure. There are sixteen possible combinations of premisses, each of the four types of proposition being combinable with itself and with each of the others.

AA EA IA OA AE EE IE OE AI EI II OI AO EO IO OO

Special Rule I. wipes out the columns on the right with the particular major premisses; and AE, EE, AO, and EO are rejected by Special Rule II., leaving BA_rb_A_r_A, CE_l_A_r_E_nt_, DA_r_II and FE_r_IO.

II. In the Second Figure, only Negative Moods are possible, and the Major Premiss must be universal.

Only Negative moods are possible, for unless one premiss is negative, M being the predicate term in both--

P in M S in M--

is undistributed.

Only negative moods being possible, there will be illicit process of the Major unless the Major Premiss is universal, P being its subject term.

These special rules reject AA and AI, and the two columns on the right.

To get rid of EE and EO, we must call in the general Canon IV.; which leaves us with EA, AE, EI, and AO--CE_s_A_r_E, CA_m_E_str_E_s_, FE_st_I_n_O BA_r_O_k_O.

III. In the Third Figure, the Minor Premiss must be affirmative.

Otherwise, the conclusion would be negative, and the Major Premiss affirmative, and there would be illicit process of the Major, P being the predicate term in the Major Premiss.

M in P M in S.

This cuts off AE, EE, IE, OE, AO, EO, IO, OO,--the second and fourth rows in the above list.

II and OI are inadmissible by Canon VI.; which leaves AA, IA, AI, EA, OA, EI--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--three affirmative moods and three negative.

IV. The Fourth Figure is fenced by three special rules. (1) In negative moods, the Major Premiss is universal. (2) If the Minor is negative, both premisses are universal. (3) If the Major is affirmative, the Minor is universal.

(1) Otherwise, the Figure being

P in M M in S,

there would be illicit process of the Major.

(2) The Major must be universal by special rule (1), and if the Minor were not also universal, the Middle would be undisturbed.

(3) Otherwise M would be undistributed.

Rule (1) cuts off the right-hand column, OA, OE, OI, and OO; also IE and IO.

Rule (2) cuts off AO, EO.

Rule (3) cuts off AI, II.

EE goes by general Canon IV.; and we are left with AA, AE, IA, EA, EI--B_r_A_m_A_nt_I_p_, 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_.

CHAPTER IV.

THE ANALYSIS OF ARGUMENTS INTO SYLLOGISTIC FORMS.

Turning given arguments into syllogistic form is apt to seem as trivial and useless as it is easy and mechanical. In most cases the necessity of the conclusion is as apparent in the plain speech form as in the artificial logical form. The justification of such exercises is that they give familiarity with the instrument, serving at the same time as simple exercises in ratiocination: what further uses may be made of the instrument once it is mastered, we shall consider as we proceed.

I.--FIRST FIGURE.

Given the following argument to be put into Syllogistic form: "No war is long popular: for every war increases taxation; and the popularity of anything that touches the pocket is short-lived".

The simplest method is to begin with the conclusion--"No war is long popular"--No S is P--then to examine the argument to see whether it yields premisses of the necessary form. Keeping the form in mind, Celarent of Fig. I.--

No M is P All S is M No S is P-- we see at once that "Every war increases taxation" is of the form All S is M. Does the other sentence yield the Major Premiss No M is P, when M represents the increasing of taxation, _i.e._, a class bounded by that attribute? We see that the last sentence of the argument is equivalent to saying that "Nothing that increases taxation is long popular"; and this with the Minor yields the conclusion in Celarent.

Nothing that increases taxation is long popular.

Every war increases taxation.

No war is long popular.

Observe, now, what in effect we have done in thus reducing the argument to the First Figure. In effect, a general principle being alleged as justifying a certain conclusion, we have put that principle into such a form that it has the same predicate with the conclusion.

All that we have then to do in order to inspect the validity of the argument is to see whether the subject of the conclusion is contained in the subject of the general principle. Is war one of the things that increase taxation? Is it one of that class? If so, then it cannot long be popular, long popularity being an attribute that cannot be affirmed of any of that class.

Reducing to the first figure, then, amounts simply to making the predication of the proposition alleged as ground uniform with the conclusion based upon it. The minor premiss or applying proposition amounts to saying that the subject of the conclusion is contained in the subject of the general principle. Is the subject of the conclusion contained in the subject of the general principle when the two have identical predicates? If so, the argument falls at once under the _Dictum de Omni et Nullo_.

Two things may be noted concerning an argument thus simplified.

1. It is not necessary, in order to bring an argument under the _dictum de omni_, to reduce the predicate to the form of an extensive term. In whatever form, abstract or concrete, the predication is made of the middle term, it is applicable in the same form to that which is contained in the middle term.

2. The quantity of the Minor Term does not require special attention, inasmuch as the argument does not turn upon it. In whatever quantity it is contained in the Middle, in that quantity is the predicate of the Middle predicable of it.