Spanish translation & annotations 
3 Bedford Place, Bloomsbury Sq. Dear Sir
I have to thank you very much for copies of two interesting papers on logic and also for your kind notice of my "Logic of Relatives" [of which I have succeeded in finding] a copy of which I will send you. My views on the subject have naturally undergone some modification since 1869 [and as I have published nothing since then I will just give a brief outline of the subject as it seems to me now and] I think now that I somewhat cramped and warped my presentation of logic by the effort to make the notation for it as analogous as possible to that of the algebra of quantity. I first described algebraic notation without reference to its signification and then applied it to logic, but it seems to me now that it is impossible to have a more general algebra than logical algebra and that the proper way is to begin with studying logic and then to apply the results to algebra. Algebra indeed is clearly a branch

of logic. For what is algebraic notation? It is a system of arbitrary signs fit for representing the relations of any objects or of certain classes of objects, and for representing them in such a manner as to facilitate reasoning about them.
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

propositional forms with = for the copula rather than the more philosophical < is that the rule for its use is simpler. For x<y signifies that in some cases x may be substituted for y and in others that y may be substituted for x but x=y signifies simply that either may in every case be substituted for the other.
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 termsigns 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.

Transcription by Max Fisch (Peirce Edition Project), revised by Sara Barrena
Una de las ventajas de los textos en formato electrónico respecto de los textos impresos es que pueden corregirse con gran facilidad mediante la colaboración activa de los lectores que adviertan erratas, errores o simplemente mejores transcripciones. En este sentido agradeceríamos que se enviaran todas las sugerencias y correcciones a sbarrena@unav.es
Proyecto de investigación "Charles S. Peirce en Europa (187576): comunidad científica y correspondencia" (MCI: FFI201124340)
Fecha del documento: 19 de junio 2012
Última actualización: 14 de septiembre 2017