Chapter 26

FIGURES AND MOODS OF THE SYLLOGISM.

I.--The First Figure.

The forms (technically called MOODS, _i.e._, modes) of the First Figure are founded on the simplest relations with the Middle that will yield or that necessarily involve the disputed relation between the Extremes.

The simplest type is stated by Aristotle as follows: "When three terms are so related that the last (the Minor) is wholly in the Middle, and the Middle wholly either in or not in the first (the Major) there must be a perfect syllogism of the Extremes".[1]

When the Minor is partly in the Middle, the Syllogism holds equally good. Thus there are four possible ways in which two terms ([Greek: oroi], plane enclosures) may be connected or disconnected through a third. They are usually represented by circles as being the neatest of figures, but any enclosing outline answers the purpose, and the rougher and more irregular it is the more truly will it represent the extension of a word.

Conclusion A. All M is in P. All S is in M. All S is in P.

Conclusion E. No M is in P. All S is in M. No S is in P.

Conclusion I. All M is in P. Some S is in M. Some S is in P.

Conclusion O. No M is in P. Some S is in M. Some S is not in P.

These four forms constitute what are known as the moods of the First Figure of the Syllogism. Seeing that all propositions may be reduced to one or other of the four forms, A, E, I, or O, we have in these premisses abstract types of every possible valid argument from general principles. It is all the same whatever be the matter of the proposition. Whether the subject of debate is mathematical, physical, social or political, once premisses in these forms are conceded, the conclusion follows irresistibly, _ex vi formae, ex necessitate formae_. If an argument can be analysed into these forms, and you admit its propositions, you are bound in consistency to admit the conclusion--unless you are prepared to deny that if one thing is in another and that other in a third, the first is in the third, or if one thing is in another and that other wholly outside a third, the first is also outside the third.

This is called the AXIOM OF SYLLOGISM. The most common form of it in Logic is that known as the _Dictum_, or _Regula de Omni et Nullo:_ "Whatever is predicated of All or None of a term, is predicated of whatever is contained in that term". It has been expressed with many little variations, and there has been a good deal of discussion as to the best way of expressing it, the relativity of the word best being often left out of sight. _Best_ for what purpose? Practically that form is the best which best commands general assent, and for this purpose there is little to choose between various ways of expressing it. To make it easy and obvious it is perhaps best to have two separate forms, one for affirmative conclusions and one for negative. Thus: "Whatever is affirmed of all M, is affirmed of whatever is contained in M: and whatever is denied of all M, is denied of whatever is contained in M". The only advantage of including the two forms in one expression, is compendious neatness. "A part of a part is a part of the whole," is a neat form, it being understood that an individual or a species is part of a genus. "What is said of a whole, is said of every one of its parts," is really a sufficient statement of the principle: the whole being the Middle Term, and the Minor being a part of it, the Major is predicable of the Minor affirmatively or negatively if it is predicable similarly of the Middle.

This Axiom, as the name imports, is indemonstrable. As Aristotle pointed out in the case of the Axiom of Contradiction, it can be vindicated, if challenged, only by reducing the challenger to a practical absurdity. You can no more deny it than you can deny that if a leaf is in a book and the book is in your pocket, the leaf is in your pocket. If you say that you have a sovereign in your purse and your purse is in your pocket, and yet that the sovereign is not in your pocket: will you give me what is in your pocket for the value of the purse?

II.--THE MINOR FIGURES OF THE SYLLOGISM, AND THEIR REDUCTION TO THE FIRST.

The word Figure ([Greek: schema]) applies to the form or figure of the premisses, that is, the order of the terms in the statement of the premisses, when the Major Premiss is put first, and the Minor second.

In the First Figure the order is

M P S M

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,

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 conclusion All S is in M. If then the original conclusion is denied, it follows that All S is in M. But this contradicts the Minor Premiss, which has been admitted to be true. It is thus shown that an opponent cannot admit the premisses and deny the conclusion without contradicting himself.

The same process may be applied to Bokardo.

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

Deny the conclusion, and you must admit that All S is in P. Syllogised in Barbara with All M is in S, this yields the conclusion that All M is in P, the contradictory of the Major Premiss.

The beginner may be reminded that the argument _ad absurdum_ is not necessarily confined to Baroko and Bokardo. It is applied to them simply because they are not reducible by the ordinary processes to the First Figure. It might be applied with equal effect to other Moods, DI_m_A_r_I_s_, _e.g._, of the Third.

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

Let Some S is in P be denied, and No S is in P must be admitted. But if No S is in P and All M is in S, it follows (in Celarent) that No M is in P, which an opponent cannot hold consistently with his admission that Some M is in P.

The beginner sometimes asks: What is the use of reducing the Minor Figures to the First? The reason is that it is only when the relations between the terms are stated in the First Figure that it is at once apparent whether or not the argument is valid under the Axiom or _Dictum de Omni_. It is then undeniably evident that if the Dictum holds the argument holds. And if the Moods of the First Figure hold, their equivalents in the other Figures must hold too.

Aristotle recognised only two of the Minor Figures, the Second and Third, and thus had in all only fourteen valid moods.

The recognition of the Fourth Figure is attributed by Averroes to Galen. Averroes himself rejects it on the ground that no arguments expressed naturally, that is, in accordance with common usage, fall into that form. This is a sufficient reason for not spending time upon it, if Logic is conceived as a science that has a bearing upon the actual practice of discussion or discursive thought. And this was probably the reason why Aristotle passed it over.

If however the Syllogism of Terms is to be completed as an abstract doctrine, the Fourth Figure must be noticed as one of the forms of premisses that contain the required relation between the extremes. There is a valid syllogism between the extremes when the relations of the three terms are as stated in certain premisses of the Fourth Figure.

III.--THE SORITES.

A chain of Syllogisms is called a Sorites. Thus:--

All A is in B. All B is in C. All C is in D. : : : : All X is in Z. [.'.] All A is in Z.

A Minor Premiss can thus be carried through a series of Universal Propositions each serving in turn as a Major to yield a conclusion which can be syllogised with the next. Obviously a Sorites may contain one particular premiss, provided it is the first; and one universal negative premiss, provided it is the last. A particular or a negative at any other point in the chain is an insuperable bar.

[Footnote 1: [Greek: Hotan oun horoi treis autos echosi pros allelous oste ton eschaton en holo einai to meso, kai ton meson en holo to kroto e einai e me einai, ananke ton akron einai syllogismon teleion.] (Anal. Prior., i. 4.)]