The Signs for Multiplication (cont.)

The associative principle does not hold in this counting of factors.  Because it does not hold, these subjacent numbers are frequently inconvenient in practice, and I therefore use also another mode of showing where the correlate of a term is to be found.  This is by means of the marks of reference, which are placed subjacent to the relative term and before and above the correlate.  Thus, giver of a horse to a lover of a woman may be written:

The asterisk I use exclusively to refer to the last correlate of the last relative of the algebraic term.

Now, considering the order of multiplication to be: — a term, a correlate of it, a correlate of that correlate, etc. — there is no violation of the associative principle.  The only violations of it in this mode of notation are that in thus passing from relative to correlate, we skip about among the factors in an irregular manner, and that we cannot substitute in such an expression as a single letter for

I would suggest that such a notation may be found useful in treating other cases of non‑associative multiplication.  By comparing this with what was said above [CP 3.55] concerning functional multiplication, it appears that multiplication by a conjugative term is functional, and that the letter denoting such a term is a symbol of operation.  I am therefore using two alphabets, the Greek and [Gothic], where only one was necessary.  But it is convenient to use both.

(Peirce, CP 3.71–72)