Chapter 27

THE DEMONSTRATION OF THE SYLLOGISTIC MOODS.--THE CANONS OF THE SYLLOGISM.

How do we know that the nineteen moods are the only possible forms of valid syllogism?

Aristotle treated this as being self-evident upon trial and simple inspection of all possible forms in each of his three Figures.

Granted the parity between predication and position in or out of a limited enclosure (term, [Greek: horos]), it is a matter of the simplest possible reasoning. You have three such terms or enclosures, S, P and M; and you are given the relative positions of two of them to the third as a clue to their relative positions to one another. Is S in or out of P, and is it wholly in or wholly out or partly in or partly out? You know how each of them lies toward the third: when can you tell from this how S lies towards P?

We have seen that when M is wholly in or out of P, and S wholly or partly in M, S is wholly or partly in or out of P.

Try any other given positions in the First Figure, and you find that you cannot tell from them how S lies relatively to P. Unless the Major Premiss is Universal, that is, unless M lies wholly in or out of P, you can draw no conclusion, whatever the Minor Premiss may give. Given, _e.g._, All S is in M, it may be that All S is in P, or that No S is in P, or that Some S is in P, or that Some S is not in P.

Again, unless the Minor Premiss is affirmative, no matter what the Major Premiss may be, you can draw no conclusion. For if the Minor Premiss is negative, all that you know is that All S or Some S lies somewhere outside M; and however M may be situated relatively to P, that knowledge cannot help towards knowing how S lies relatively to P. All S may be P, or none of it, or part of it. Given all M is in P; the All S (or Some S) which we know to be outside of M may lie anywhere in P or out of it.

Similarly, in the Second Figure, trial and simple inspection of all possible conditions shows that there can be no conclusion unless the Major Premiss is universal, and one of the premisses negative.

Another and more common way of eliminating the invalid forms, elaborated in the Middle Ages, is to formulate principles applicable irrespective of Figure, and to rule out of each Figure the moods that do not conform to them. These regulative principles are known as The Canons of the Syllogism.

_Canon I._ In every syllogism there should be three, and not more than three, terms, and the terms must be used throughout in the same sense.

It sometimes happens, owing to the ambiguity of words, that there seem to be three terms when there are really four. An instance of this is seen in the sophism:--

He who is most hungry eats most. He who eats least is most hungry. [.'.] He who eats least eats most.

This Canon, however, though it points to a real danger of error in the application of the syllogism to actual propositions, is superfluous in the consideration of purely formal implication, it being a primary assumption that terms are univocal, and remain constant through any process of inference.

Under this Canon, Mark Duncan says (_Inst. Log._, iv. 3, 2), is comprehended another commonly expressed in this form: There should be nothing in the conclusion that was not in the premisses: inasmuch as if there were anything in the conclusion that was in neither of the premisses, there would be four terms in the syllogism.

The rule that in every syllogism there must be three, and only three, propositions, sometimes given as a separate Canon, is only a corollary from Canon I.

_Canon II._ The Middle Term must be distributed once at least in the Premisses.

The Middle Term must either be wholly in, or wholly out of, one or other of the Extremes before it can be the means of establishing a connexion between them. If you know only that it is partly in both, you cannot know from that how they lie relatively to one another: and similarly if you know only that it is partly outside both.

The Canon of Distributed Middle is a sort of counter-relative supplement to the _Dictum de Omni_. Whatever is predicable of a whole distributively is predicable of all its several parts. If in neither premiss there is a predication about the whole, there is no case for the application of the axiom.

_Canon III._ No term should be distributed in the conclusion that was not distributed in the premisses.

If an assertion is not made about the whole of a term in the premisses, it cannot be made about the whole of that term in the conclusion without going beyond what has been given.

The breach of this rule in the case of the Major term is technically known as the Illicit Process of the Major: in the case of the Minor term, Illicit Process of the Minor.

Great use is made of this canon in cutting off invalid moods. It must be remembered that the Predicate term is "distributed" or taken universally in O (Some S is not in P) as well as in E (No S is in P); and that P is never distributed in affirmative propositions.

_Canon IV._ No conclusion can be drawn from two negative premisses.

Two negative premisses are really tantamount to a declaration that there is no connexion whatever between the Major and Minor (as quantified in the premisses) and the term common to both premisses; in short, that this is not a Middle term--that the condition of a valid Syllogism does not exist.

There is an apparent exception to this when the real Middle in an argument is a contrapositive term, not-M. Thus:--

Nobody who is not thirsty is suffering from fever. This person is not thirsty. [.'.] He is not suffering from fever.

But in such cases it is really the absence of a quality or rather the presence of an opposite quality on which we reason; and the Minor Premiss is really Affirmative of the form S is in not-M.

_Canon V._ If one premiss is negative, the conclusion must be negative.

If one premiss is negative, one of the Extremes must be excluded in whole or in part from the Middle term. The other must therefore (under Canon IV.) declare some coincidence between the Middle term and the other extreme; and the conclusion can only affirm exclusion in whole or in part from the area of this coincidence.

_Canon VI._ No conclusion can be drawn from two particular premisses.

This is evident upon a comparison of terms in all possible positions, but it can be more easily demonstrated with the help of the preceding canons. The premisses cannot both be particular and yield a conclusion without breaking one or other of those canons.

Suppose both are affirmative, II, the Middle is not distributed in either premiss.

Suppose one affirmative and the other negative, IO, or OI. Then, whatever the Figure may be, that is, whatever the order of the terms, only one term can be distributed, namely, the predicate of O. This (Canon II.) must be the Middle. But in that case there must be Illicit Process of the Major (Canon III.), for one of the premisses being negative, the conclusion is negative (Canon V.), and P its predicate is distributed. Briefly, in a negative mood, both Major and Middle must be distributed, and if both premisses are particular this cannot be.

_Canon VII._ If one Premiss is particular the conclusion is particular.

This canon is sometimes combined with what we have given as Canon V., in a single rule: "The conclusion follows the weaker premiss".

It can most compendiously be demonstrated with the help of the preceding canons.

Suppose both premisses affirmative, then, if one is particular, only one term can be distributed in the premisses, namely, the subject of the Universal affirmative premiss. By Canon II., this must be the Middle, and the Minor, being undistributed in the Premisses, cannot be distributed in the conclusion. That is, the conclusion cannot be Universal--must be particular.

Suppose one Premiss negative, the other affirmative. One premiss being negative, the conclusion must be negative, and P must be distributed in the conclusion. Before, then, the conclusion can be universal, all three terms, S, M, and P, must, by Canons II. and III., be distributed in the premisses. But whatever the Figure of the premisses, only two terms can be distributed. For if one of the Premisses be O, the other must be A, and if one of them is E, the other must be I. Hence the conclusion must be particular, otherwise there will be illicit process of the Minor, or of the Major, or of the Middle.

The argument may be more briefly put as follows:

In an affirmative mood, with one premiss particular, only one term can be distributed in the premisses, and this cannot be the Minor without leaving the Middle undistributed. In a negative mood, with one premiss particular, only two terms can be distributed, and the Minor cannot be one of them without leaving either the Middle or the Major undistributed.

Armed with these canons, we can quickly determine, given any combination of three propositions in one of the Figures, whether it is or is not a valid Syllogism.

Observe that though these canons hold for all the Figures, the Figure must be known, in all combinations containing A or O, before we can settle a question of validity by Canons II. and III., because the distribution of terms in A and O depends on their order in predication.

Take AEE. In Fig. I.--

All M is in P No S is in M No S is in P--

the conclusion is invalid as involving an illicit process of the Major. P is distributed in the conclusion and not in the premisses.

In Fig. II. AEE--

All P is in M No S is in M No S is in P--

the conclusion is valid (Camestres).

In Fig. III. AEE--

All M is in P No M is in S No S is in P--

the conclusion is invalid, there being illicit process of the Major.

In Fig. IV. AEE is valid (Camenes).

Take EIO. A little reflection shows that this combination is valid in all the Figures if in any, the distribution of the terms in both cases not being affected by their order in predication. Both E and I are simply convertible. That the combination is valid is quickly seen if we remember that in negative moods both Major and Middle must be distributed, and that this is done by E.

EIE is invalid, because you cannot have a universal conclusion with one premiss particular.

AII is valid in Fig. I. or Fig. III., and invalid in Figs. II. and IV., because M is the subject of A in I. and III. and predicate in II. and IV.

OAO is valid only in Fig. III., because only in that Figure would this combination of premisses distribute both M and P.

Simple exercises of this kind may be multiplied till all possible combinations are exhausted, and it is seen that only the recognised moods stand the test.

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_.