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.)]

CHAPTER III.

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.

[Illustration:

Circles of M and P, overlapping, with 3 instances of a circle of S: 1. S in M, but not in P; 2. S in the overlap of M and P; 3. S in M, some S 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.

[Illustration:

Concentric circles of P and M, M in center, with 5 instances of circle of S: 1. S wholly outside P and M; 2. S partly overlapping both P and M, and partly outside both; 3. S overlapping P, but outside M; 4. S wholly within P, but wholly outside M; 5. S touching circle of P, but outside both circles.

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:--