The Principles of science

proved indirectly, we may say that the process is a necessary and sufficient one, and the question of its comparative excellence or usefulness is not worth discussion. As a matter of fact I believe that nearly half our logical conclusions rest upon its employment.

Simple Illustrations.

In tracing out the powers and results of this method, we will begin with the simplest possible instance. Let us take a proposition of the very common form, A AB, say,

A Metal is an Element,

and let us investigate its full meaning. Any person who has had the least logical training, is aware that we can draw from the above proposition an apparently different one, namely,

A Not-element is a Not-metal.

While some logicians, as for instance De Morgan, have considered the relation of these two propositions to be purely self-evident, and neither needing nor allowing analysis, a great many more persons, as I have observed while teaching logic, are at first unable to perceive the close connection between them. I believe that a true and complete system of logic will furnish a clear analysis of this process which has been called Contrapositive Conversion; the full process is as follows :—

Firstly, by the Law of Duality we know that

Not-element is either Metal or Not-metal. Now if it be metal, we know that it is by the premise an element; we should thus be supposing that the very same thing is an element and a not-element, which is in opposition to the Law of Contradiction. According to the only other alternative, then, the not-element must be a not-metal.

e Philosophical Magazine,' December 1852, Fourth Series, vol. iv. p. 435, 'On Indirect Demonstration.'

To represent this process of inference symbolically we take the premise in the form

(1)

We observe that by the Law of Duality the term not-B is thus described

b = Ab + ab.

(2)

For A in this proposition we substitute its description as given in (1), obtaining

b = ABb + ab.

But according to the Law of Contradiction the term. ABb must be excluded from thought or

ABb = o.

Hence it results that b is either nothing at all, or it is ab; and the conclusion is

b = ab.

As it will often be necessary to refer to a conclusion of this kind I shall call it, as is usual, the Contrapositive Proposition of the original. The reader need hardly be cautioned to observe that from all A's are B's it does not follow that all not-A's are not-B's. For by the Law of Duality we have

α = aBab,

and it will not be found possible to make any substitution in this by our original premise A = AB. It still remains doubtful, therefore, whether not-metal is element or not-element.

The proof of the Contrapositive Proposition given above is exactly the same as that which Euclid applies in the case of geometrical notions. De Morgan describes Euclid's process as follows:- From every not-B is not-A he produces every A is B, thus-If it be possible, let this A be not-B, but every not-B is not-A, therefore this A is not-A, which is absurd: whence every A is B.' Now De Morgan thinks that this proof is entirely needless, because common

d Philosophical Magazine,' Dec. 1852; p. 437.

logic gives the inference without the use of any geometrical reasoning. I conceive however that logic gives the inference only by an indirect process. De Morgan claims 'to see identity in every A is B and every not-B is not-A, by a process of thought prior to syllogism.' But whether prior to syllogism or not, I claim that it is not prior to the laws of thought and the process of substitutive inference by which it may be undoubtedly demonstrated.

Employment of the Contrapositive Proposition.

We can frequently employ the contrapositive form of a proposition by the method of substitution; and certain moods of the ancient syllogism, which we have hitherto passed over, may thus be satisfactorily comprehended in our system. Take for instance the following syllogism in the mood Camestres :

'Whales are not true fish: for they do not respire water, whereas true fish do respire water.'

Now, by the process of contraposition we obtain from (2)

C = bc,

and we can substitute this expression for c in (1), obtaining

A = Abc,

or Whales are not true fish, not respiring water.'

The mood Cesare does not really differ from Camestres except in the order of the premises, and it could be exhibited in an exactly similar manner.

The mood Baroko gave much trouble to the old logicians who could not reduce it to the first figure in the same manner as the other moods, and were obliged to invent, specially for it and for Bokardo, a method of Indirect Reduction closely analogous to the Indirect proof of Euclid. Now these moods require no exceptional processes in this system. Let us take as an instance of Baroko, the argument

(1)

(2)

(3)

All heated solids give continuous spectra, Some nebulæ do not give continuous spectra; Therefore some nebulæ are not heated solids. Treating the little word some as an indeterminate adjective of selection, to which we assign a symbol like any other adjective, let

Now from (1) we obtain by the Indirect method the

and if we substitute this expression for c in (2) we have AB = ABcd;

the full meaning of which is that 'some nebulæ do not give continuous spectra and are not solids.'

We might similarly apply the contrapositive in many other instances. Take the argument-All fixed stars are self-luminous; but some of the heavenly bodies are not self-luminous, and are therefore not fixed stars.' Taking our terms