The results of applying logic to the formation of an algebraic notation seem to me to be as follows:__

1. The first signs required are signs of inference. If we restrict ourselves to deduction reasoning we need have but one of these. Let ∴ for example be the symbol ergo.

2. The next sign required is the copula. Let -< be the natural copula. It signifies some class of relations such that:

2nd x-<x, whatever x may be.

1st If x-<y and y-<z then x-<z whatever x, y and z may be.

3rd If x-<y and y-<x then x and y may be everywhere substituted for one another.

Of the various classes of relations for which these conditions hold good the sign -< may be used to denote any one or we may distinguish -<, -<', -<'', affixing accents to distinguish different significations of the sign. Corresponding to every signification there is a second which we may omite -< such that x-<y is the same as y-<x.

No other copula sign is neccesary, but as algebra is an art of facilitating reasoning another sign = is called for. To say x=y signifies that x-<y and y-<x. The reason for preferring proposi

3. The next signs required are signs of operations. The first of these which logic calls for is the sign of combination & the conditions of the application of which are

1st x-<x &y

2nd y-<x &y

3rd If x-<z and y-<z then x & y-<z.

The sign & is applicable to any operation fulfilling these conditions. Its different signification may be distinguished by accents &', &'', etc. Corresponding to each signification of & there is another in the conditions of which -< is substituted for -<. We may denote this by x,.

Corresponding to the operation of combination is the inverse operation of exception which may be denoted by —, the definition of which is that x—, y<z signifies the same as x-<y +, z. It may be noticed that x—,y is the same as x x, (∞—,y).

When terms are taken in the vague sense it is useful to employ the operation of addition which differ from conjunction in as much as x+x is not the same as x as x+, x is; and in short the defining conditions

1st x+y same x+,y if x//y

2nd x+x not the same as x

Algebraic notation is a system of arbitrary signs fit to represent the relations of objects in such a manner as to facilitate reasoning about them.

I.

The first class of algebraic signs are signs of inference. This will be of various kinds and must indicated the probability of the inference. But this branch of algebraic notation is as yet undeveloped and we have only one inferential sign ∴ which indicates perfect demonstration.

II

The second class of algebraic signs are copulas. The first and most important of these is -<. It signifies that whatever can be denoted by that which precedes it has all the common characters of all those objects which can be denoted by that which follows it. But according to the manner of the denotation the copula may have various meanings. Thus "man -< animal"

Algebraic notation is a system of arbitrary signs fit to represent the relations of objects in such a manner as to facilitate reasoning about them.

The first class of algebraic signs consists of signs of inference. These must indicate the probability of the inference. As yet only one such sign that of demonstrative illation ∴ has been used.

The second class of algebraic signs are copulas. The first of these is -<. It signifies that whatever corresponds to that which precedes it has all the common characters of all the objects which correspond to what follows it. According to the manner of the correspondence the signification of the sign may be varied. Thus "man-<sinner" may be used to denote that every man is a sinner or that every quality of all men is a quality of all sinners or that the number of men is as great or as small as the number of sinners, etc. Accents may be affixed to the general sign when it is necessary to distinguish the different meanings. The definition just given of this sign is precisely equivalent to three algebraical conditions: --viz.

1st If x-<y and y-<z then x-<z whatever x, y, and z may be.

2nd x-<x

3rd If x-<y and y-<x then x and y may everywhere be substituted for one another.

Every signification of the copula of inclusion -< has an [sic] converse signification -<' such that x-<y is the same as y-<'x.

Two logical term-signs are suggested and defined by the copula. They are 0, and ∞ infinity. Their definitions are as follows: --

Zero is such a term that 0-<x whatever x may be;

Infinity is such that x-<∞ whatever x may be.

Of course the zero of one progression may not be the zero of another progression; and therefore I have greatly enlarged the usual meanings of the words in these definitions.

The copula of inclusion is subject to rather complicated rules of algebraic manipulation. To simplify this process we may use the copula of identity =. The definition of this is that x=y is the same as x-<y and y-<x. The third class of algebraic signs consists of signs of operation. Signs of operation are of three orders according as they are applicable to all terms or only to relative and conjugative terms or to conjugative terms alone. The sign of conjunction +, is the first sign of the first order. It is defined by three conditions, as follows:

1st x-<x +,y 2nd y-<x +, y

3rd If x-<z and y-<z then x+,y-<z.