each premise; (7) and (d) are contradicted by (1), and (8) and () by (2), so that there remain only

To describe any term under the conditions of the premises (1) and (2), we have only to draw out the proper combinations from this list; thus-A is represented only by ABC or

similarly

A = ABC,

c = abc.

For B we have two alternatives thus stated,

and for b we have

B = ABC + a BC;

babC+ abc.

When we have a problem involving four distinct terms we need to double the number of combinations, and as we add each new term the combinations become twice as

I

propose to call

thirty-two

sixty-four

to call any such series of combinations the Logical Abecedarium. It holds in logical science a position of importance which cannot be exaggerated. As we proceed from logical to mathematical considerations it will

become apparent that there is a close connection between these combinations and the most fundamental theorems of mathematical science. For the convenience of the reader who may wish to employ the abecedarium in logical questions, I have had printed on the next page a complete series of the combinations up to those of six terms. At the very commencement in the first column is placed a single letter X which might seem to be superfluous. This letter serves to denote that it is always some higher class which is divided up. Thus the combination AB really means ABX, or that part of some larger class, say X, which has the qualities of A and B present. The letter X is omitted in the greater part of the table merely for the sake of brevity and clearness. In a later chapter on Combinations it will become apparent that the introduction of this unit class is requisite in order to complete the analogy with the Arithmetical Triangle there described.

The reader ought to bear in mind that though the abecedarium seems to give mere lists of combinations, these combinations are intended in every case to constitute the development of a term of a proposition. Thus the four combinations AB, Ab, aB, ab really mean that any class X is described by the following proposition, X = X (AB + Ab + aBab).

If we select the A's, we obtain the following proposition A X = X(AB+ Ab).

Thus whatever group of combinations we treat must be conceived as part of a higher class, summum genus or universe symbolised in the term X; but bearing this in mind, it is needless to complicate our formulæ by always introducing the letter. All inference consists in passing from propositions to propositions, and combinations per se have no meaning. They are consequently to be regarded in all cases as forming parts of propositions.

In a theoretical point of view we may conceive that the abecedarium is always extended indefinitely. Every new quality or circumstance which can belong to an object, subdivides each combination or class, so that the number of such combinations when unrestricted by logical conditions is represented by an indefinitely high power of two. The extremely rapid increase in the number of subdivisions obliges us to confine our attention to a few circumstances at a time.

When contemplating the properties of this abecedarium, I am often inclined to think that Pythagoras perceived the deep logical importance of duality; for while unity was the symbol of identity and harmony, he described the number two as the origin of contrasts, or the symbol of diversity, division and separation. The number four or the Tetractys was also regarded by him as one of the chief elements of existence, for it represented the generating virtue whence come all combinations.

In one of the golden verses ascribed to Pythagoras, he conjures his pupil to be virtuouse:

'By him who stampt The Four upon the Mind,

The Four, the fount of Nature's endless stream.'

Now four and the higher powers of duality do represent in this logical system the variety of combinations which can be generated in the absence of logical restrictions. The followers of Pythagoras may have shrouded their master's doctrines in mysterious and superstitious notions, but in many points these doctrines seem to have some basis in logical philosophy.

The Logical Slate.

To a person who has once comprehended the extreme agroficance and utility of the Logical Abecedarium, the

་

Whowell, History of the Inductive Sciences,' vol. i. p. 222.

indirect process of inference becomes reduced to the repetition of a few uniform operations of classification, selection, and elimination of contradictories. Logical deduction even in the most complicated questions becomes a matter of mere routine, and the amount of labour required is the only impediment when once the meaning of the premises is rendered clear. But the amount of labour is often found to be considerable. The mere writing down of sixty-four combinations of six letters each is no small task, and, if we had a problem of five premises, each of the sixty-four combinations would have to be examined in connection with each premise. The requisite comparison is often of a very tedious character and considerable chance of errors thus arises.

I have given much attention therefore to reducing both the manual and mental labour of the process, and I shall describe several devices which may be adapted for saving trouble and risk of mistake.

In the first place, as the same sets of combinations occur over and over again in different problems, we may avoid the labour of writing them out by having the sets of letters ready printed upon small sheets of writing paper. It has also been suggested by a correspondent that, if any one series of combinations were marked upon the margin of a sheet of paper, and a slit cut between each pair of combinations, it would be easy to fold down any particular combination, and thus strike it out of view. The combinations consistent with the premises would then remain in a broken series. This method answers sufficiently well for occasional use.

A more convenient mode, however, is to have the series of letters shown on p. 109, engraved upon a common school writing slate, of such a size, that the letters may occupy only about a third of the space on the left hand side of the slate. The conditions of the problem can then