|Spanish translation & annotations|
I received a few days ago your gratifying letter from England and as you are the only active worker now I suppose upon mathematical logic I wish very much to set my views before you in their true colours & shall therefore reply to two of the points contained in your letter.
It appears that you do not accept my extended definitions of the mathematical operations. The validity of that generalization it will be for mathematicians to decide & I myself think they will accept my definitions. With regard to addition and multiplication, indeed, the only novelty in my views is that I do not regard it as essential to these operations that they should be “invertible.” Now it appears to me that the study of the calculus of functions does not lead us to regard this character as very significant, but on the contrary an operation’s being invertible is usually owing to a restriction of its application. Thus, when we take no account of negative
& imaginary quantities involution is invertible. If addition be considered as essentially invertible then the operation is applicable to some logical terms & not to others. Now I greatly mistake the spirit of modern algebra if it is not contrary to it not to extend the definition of addition under these circumstances. Observe, that you cannot shut addition out of logic altogether, for the moment you take = as the sign of identity, that is conceive of equality as a case of identity, you thereby make addition applicable to mutually exclusive terms.
Of course if addition is not essentially invertible multiplication is not either. I do not quite see how you can say that I use the term multiplication in a manner quite unconnected with its original meaning. Take my definition on page 3 of my paper. In no respect does this differ from the now universally admitted conceptions, except in regard to the “invertible” character. But take my usual logical multiplication. On p. 15 I think I have shown that the conception of multiplication in quaternions is exactly the same as mine & that numerical
multiplication is merely a case of my operation.
I believe that you hold that all reasoning is by substitutions (in which I agree with you); that all substitutions when algebraically denoted appear as the substitution of equals for equals, that, therefore, the copula signifies equality & the theory of the quantified predicate holds. But I fancy the second premise would be hard to make out. You will observe that in a note on p. 2 I have shown rigidly that according to admitted principles, the conception of = is compounded of those of and (or and ). This being so the substitution-syllogism
is a compound of the two
& the logician in analyzing inferences ought to represent it so. What reply is there to my note or to the conclusion I draw from it? Practically it is much easier to manipulate the logical calculus with as a copula than with .
I especially doubt
the possibility of a successful treatment of the substitutions of scientific
reasoning upon the principles of the theory of the
In a former paper I have endeavored to prove that all inference proceeds
by the substitution of one sign for another on the principle that a
sign of a sign is a sign & that reasoning differs according to the
different kinds of signs with which it deals, & that signs are of
three kinds, first, things similar to their objects, second, things
physically connected with their objects, as blushing is a sign of shame,
third general signs. The substitution of a general sign gives deductive
reasoning; the substitution of similars gives reasoning to an hypothesis,
as for example if I say this man’s coat hat mode of speaking etc.
are like a quaker therefore I suppose he is one; the substitution of
physical signs gives induction, as for example if I say all these samples
have been drawn at random from this collection & so the manner in
which have been drawn out physically necessitates their being a sign
of what the collection consists of. Since therefore these are red balls
all the collection are red balls.
Letters addressed to Robt. Thode & Co. Berlin will reach me.
Transcription by Sara Barrena (2008)
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 firstname.lastname@example.org
Proyecto de investigación "La correspondencia europea de C. S. Peirce: creatividad y ccooperación científica (Universidad de Navarra 2007-09)
Fecha del documento: 27 de mayo 2008
Última actualización: 14 de septiembre 2017