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.