ONT Re: Logic Of Relatives
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LOR. Discussion Note 10
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BM = Bernard Morand
JA = Jon Awbrey
BM: Thanks for your very informative talk. There
is a point that I did not understand in note 35:
JA: If we operate in accordance with Peirce's example of `g`'o'h
as the "giver of a horse to an owner of that horse", then we
may assume that the associative law and the distributive law
are by default in force, allowing us to derive this equation:
JA: 'l','s'w = 'l','s'(B +, D +, E) = 'l','s'B +, 'l','s'D +, 'l','s'E
BM: May be because language or more probably my lack of training in logic, what
does mean that "associative law and distributive law are by default in force"?
Those were some tricky Peirces,
and I was trying to dodge them
as artful as could be, but now
you have fastly apprehended me!
It may be partly that I left out the initial sections of this paper where Peirce
discusses how he will regard the ordinarily applicable principles in the process
of trying to extend and generalize them (CP 3.45-62), but there may be also an
ambiguity in Peirce's use of the phrase "absolute conditions" (CP 3.67-68).
Does he mean "absolutely necessary", "indispensable", "inviolate", or
does he mean "the conditions applying to the logic of absolute terms",
in which latter case we would expect to alter them sooner or later?
We lose the commutative law, xy = yx, as soon as we extend to 2-adic relations,
but keep the associative law, x(yz) = (xy)z, as the multiplication of 2-adics
is the logical analogue of ordinary matrix multiplication, and Peirce like
most mathematicians treats the double distributive law, x(y + z) = xy + xz
and (x + y)z = xz + yz, and as something that must be striven to preserve
as far as possible.
Strictly speaking, Peirce is already using a principle that goes beyond
the ordinary associative law, but that is recognizably analogous to it,
for example, in the modified Othello case, where (J:J:D)(J:D)(D) = J.
If it were strictly associative, then we would have the following:
1. (J:J:D)((J:D)(D)) = (J:J:D)(J) = 0?
2. ((J:J:D)(J:D))(D) = (J)(D) = 0?
In other words, the intended relational linkage would be broken.
However, the type of product that Peirce is taking for granted
in this situation often occurs in mathematics in just this way.
There is another location where he comments more fully on this,
but I have the sense that it was a late retrospective remark,
and I do not recall if it is in CP or in the microfilm MS's
that I read it.
By "default" conditions I am referring more or less to what
Peirce says at the end of CP 3.69, where he use an argument
based on the distributive principle to rationalize the idea
that 'A term multiplied by two relatives shows that the same
individual is in the two relations'. This means, for example,
that one can let "`g`'o'h", without subjacent marks or numbers,
be interpreted on the default convention of "overlapping scopes",
where the two correlates of `g` are given by the next two terms
in line, namely, 'o' and h, and the single correlate of 'o' is
given by the very next term in line, namely, h. Thus, it is
only when this natural scoping cannot convey the intended
sense that we have to use more explicit mark-up devices.
BM: About another point: do you think that the LOR could be of some help to solve
the puzzle of the "second way of dividing signs" where CSP concludes that 66
classes could be made out of the 10 divisions (Letters to lady Welby)?
(As I see them, the ten divisions involve a mix of relative terms,
dyadic relations and a triadic one. In order to make 66 classes
it is clear that these 10 divisions have to be stated under some
linear order. The nature of this order is at the bottom of the
disagreements on the subject).
This topic requires a longer excuse from me
than I am able to make right now, but maybe
I'll get back to it later today or tomorrow.
Cheers,
Jon
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