ONT Re: Logic Of Relatives
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LOR. Discussion Note 25
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HC = Howard Callaway
JA = Jon Awbrey
HC: I might object that "teridentity" seems to come
to a matter of "a=b & b=c", so that a specific
predicate of teridentity seems unnecessary.
JA: I am presently concerned with expositing and interpreting
the logical system that Peirce laid out in the LOR of 1870.
It is my considered opinion after thirty years of study that
there are untapped resources remaining in this work that have
yet to make it through the filters of that ilk of syntacticism
that was all the rage in the late great 1900's. I find there
to be an appreciably different point of view on logic that is
embodied in Peirce's work, and until we have made the minimal
effort to read what he wrote it is just plain futile to keep
on pretending that we have already assimilated it, or that
we are qualified to evaluate its cogency.
JA: The symbol "&" that you employ above denotes a mathematical object that
qualifies as a 3-adic relation. Independently of my own views, there
is an abundance of statements in evidence that mathematical thinkers
from Peirce to Goedel consider the appreciation of facts like this
to mark the boundary between realism and nominalism in regard to
mathematical objects.
HC: I would agree, I think, that "&" may be thought of
as a function mapping pairs of statements onto the
conjunction of that pair.
Yes, indeed, in the immortal words of my very first college algebra book:
"A binary operation is a ternary relation". As it happens, the symbol "&"
is equivocal in its interpretation -- computerese today steals a Freudian
line and dubs it "polymorphous" -- it can be regarded in various contexts
as a 3-adic relation on syntactic elements called "sentences", on logical
elements called "propositions", or on truth values collated in the boolean
domain B = {false, true} = {0, 1}. But the mappings and relations between
all of these interpretative choices are moderately well understood. Still,
however many ways you enumerate for looking at a B-bird, the "&" is always
3-adic. And that is sufficient to meet your objection, so I think I will
leave it there until next time.
On a related note, that I must postpone until later:
We seem to congrue that there is a skewness between
the way that most mathematicians use logic and some
philosophers talk about logic, but I may not be the
one to set it adjoint, much as I am inclined to try.
At the moment I have this long-post-poned exponency
to carry out. I will simply recommend for your due
consideration Peirce's 1870 Logic Of Relatives, and
leave it at that. There's a cornucopiousness to it
that's yet to be dreamt of in the philosophy of the
1900's. I am doing what I can to infotain you with
the Gardens of Mathematical Recreations that I find
within Peirce's work, and that's in direct response
to many, okay, a couple of requests. Perhaps I can
not hope to attain the degree of horticultural arts
that Gardners before me have exhibited in this work,
but then again, who could? Everybody's a critic --
but the better ones read first, and criticize later.
Jon Awbrey
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