canine, bicuspid, or molar,' it follows that a not-incisor tooth is either canine, bicuspid, or molar.' The general rule is that from the denial of any of the alternatives the affirmation of the remainder can be inferred. Now this result clearly follows from our process of substitution; for if we have the proposition

A=BCD,

and insert this expression for A on one side of the selfevident identity

we obtain

Ab = Ab,

and, as the first of the three alternatives is self-contradictory, we strike it out according to the law of contradiction there remains

Thus our system fully includes and explains that mood of the Disjunctive Syllogism technically called the modus tollendo ponens.

But the reader must carefully observe that the Disjunctive Syllogism of the mood ponendo tollens, which affirms one alternative, and thence infers the denial of the rest, cannot be held true in this system. If I say, indeed, that Water is either salt or fresh water,

it seems evident that water which is salt is not fresh.' But this inference really proceeds from our knowledge that water cannot be at once salt and fresh. This inconsistency of the alternatives, as I have fully shown, will not always hold. Thus, if I say

Gems are either rare stones or beautiful stones,

(1)

Our symbolic method gives only true conclusions; for if

we take

but these inferences are not equivalent to the false ones (2) and (3).

We can readily represent such disjunctive reasoning, when it is valid, by expressing the inconsistency of the alternatives explicitly. Thus if we resort to our instance of Water is either salt or fresh,

and take

A = Water

B = salt

C = fresh,

then the premise is apparently of the form

A = AB AC;

but in reality there are the unexpressed conditions that 'what is salt is not fresh,' and 'what is fresh is not salt; or, in letter-terms,

Now, if we substitute

proposition, we obtain

A

B = Be

C = bc.

these descriptions in the original

uniting B to each side we infer

that is,

ABc + AbС;

Water which is salt is water salt and not fresh.

I should weary the reader if I attempted to illustrate the multitude of forms which disjunctive reasoning may take; and as in the next chapter we shall be constantly treating the subject, I must here restrict myself to a single

instance. A very common process of reasoning consists in the determination of the name of a thing by the successive exclusion of alternatives, a process called by the old name abscissio infiniti. Take the case:

Red-coloured metal is either copper or gold (1)
Copper is dissolved by nitric acid

This specimen is red-coloured metal

This specimen is not dissolved by nitric acid

Therefore this specimen consists of gold.

Assigning our letter-symbols thus

E =

(2)

(3)

(4)

(5)

dissolved by nitric acid,

the premises may be stated in the form

Substituting for C in (1) by means of (2) we get

BBCαE + BCD.
BCdEBcD.

From (3) and (4) we may infer likewise

A = ABe,

and if in this we substitute for B its equivalent just stated, it follows that

A = ABCdEe + ABcDe.

The first of the alternatives being contradictory, the result A = ABcDe

is

6

which contains a full description of this specimen,' as furnished in the premises, but by ellipsis indicates that it is gold. It will be observed that in the symbolic expression (1) I have explicitly stated what is certainly implied, that copper is not gold, and gold not copper, without which condition the inference would not hold good.

CHAPTER VI.

THE INDIRECT METHOD OF INFERENCE.

THE forms of deductive reasoning as yet considered, are mostly cases of Direct Deduction as distinguished from those which we are now about to treat. The method of Indirect Deduction may be described as that which points out what a thing is, by showing that it cannot be anything else. We can define a certain space upon a map, either by colouring that space, or by colouring all except the space; the first mode is positive, the second negative. The difference, it will be readily seen, is exactly analogous to that between the direct and indirect proof in geometry. Euclid often shows that two lines are equal, by showing that they cannot be unequal, and the proof rests upon the known number of alternatives, greater, equal or less, which are alone conceivable. In other cases, as for instance in the seventh proposition of the first book, he shows that two lines must meet in a particular point, by showing that they cannot meet elsewhere.

In logic we can always define with certainty the utmost number of alternatives which are conceivable. The Law of Duality (pp. 6, 88) enables us always to assert that any quality or circumstance whatsoever is either present or absent in anything. Whatever may be the meaning and nature of the terms A and B it is certainly true that

These are universal though tacit premises which may be employed in the solution of every problem, and which

are such invariable and necessary conditions of all thought, that they need not be specially laid down. The Law of Contradiction is a further condition of all thought and of all logical symbols; it enables, and in fact obliges, us to reject from further consideration all terms which imply the presence and absence of the same quality. Now, whenever we bring both these Laws of Thought into explicit action by the method of substitution, we employ the Indirect Method of Inference. It will be found that we can treat not only those arguments already exhibited according to the direct method, but we can also include an infinite multitude of other arguments which are incapable of solution by any other means.

Some philosophers, especially those of France, have held that the Indirect Method of Proof has a certain inferiority to a direct method, which should prevent our using it except when obliged. But there are an unlimited number of truths which we can prove only indirectly. We can prove that a number is a prime only by the purely indirect method of showing that it is not any of the numbers which have divisors, and the remarkable process known as Eratosthenes' Sieve is the only mode by which we can select the prime numbers a. It bears a strong analogy to the indirect method here to be described. We can also prove that the side and diameter of a square are incommensurable, but only in the negative or indirect manner, by showing that the contrary supposition constantly and inevitably leads to contradiction b. Many other demonstrations in various branches of the mathematical sciences rest upon a like method. Now if there is only one important truth which must be, and can only be

a See Horsley, Philosophical Transactions,' 1772; vol. lxii. p. 327. Montucla, Histoire des Mathematiques,' vol. i. p. 239. 'Penny Cyclopædia,' article Eratosthenes.

b Euclid, Book x. Prop. 117.