OT: SUO: peircean triadicity proofs
The interest on the SUO list in Pat Hayes' Burch review and related
discussion, prompts me to share, with the permission of the author, two
emails from Paul Chernoff, who has long taught mathematical analysis at
Berkeley, on proofs of Peirce's triadicity theorems. Interested members
of the SUO list will now have been made aware that are really three prongs
to Peirce's problem:
1) to show that all n-ary relations of >3 are reducible to ternary
relations without loss of logical content;
2) to show that there are genuine ("non-degenerate") ternary relations
which are not reducible to relations of fewer than 3;
3) to show that *for logical purposes* monadic and dyadic relations
can be adequately translated into ternary form.
Chernoff's emails address 1) and 3). He emphasizes that he is not a
logician.
Paul Chernoff welcomes direct correspondence on this matter, and I,
too would like to be included without further burdening the entire
SUO list. Hence I have marked this OT (off topic).
On Tue, 30 Jan 2001, Paul R. Chernoff wrote:
>
> Date: Tue, 30 Jan 2001 19:15:44 -0800 (PST)
> From: Paul R. Chernoff <chernoff@math.berkeley.edu>
> To: lee@sabre.org
> Subject: Re: C.S. Peirce "threeness" conjecture
>
>
> ---------------------------------
> As for my (probably mistaken ) notions about Peirce, threeness, and
n-ary
> relations -- I posted a query about this on the Usenet newsgroup sci.math.
> research, and received the following reply:
>
>
>
> >From ger@tzi.de Thu Jan 25 08:00:19 PST 2001
> Article: 15716 of sci.math.research
> From: George Russell <ger@tzi.de>
> Newsgroups: sci.math.research
> Subject: Re: C.S. Peirce "threeness" conjecture
>
> "Paul R. Chernoff" wrote:
> >
> > The 19th century American philosopher/mathematician Charles Sanders
> > Peirce had notions of "twoness", "threeness", etc. As near as I can
> > figure out what he meant by this, "n-ness" is that property which
> > characterizes n-ary relations. According to Peirce, threeness is the
> > most fundamental. I interpret this to mean that any n-ary relation
> > can be built up out of ternary relations.
> >
> > Is this actually true? Is there a reasonably short proof?
G. Russell responded:
> [snip]
> I don't understand the question, or if I do it is trivially true.
> Let T(A,B,C) be the ternary relation of the form "C is the ordered
> pair (A,B)". Then I claim that inductively we can encode any n-ary
> relation as a ternary relation. Firstly n=1,n=2,n=3 are trivally
> true (for n=1 or 2 we replace R(A) by R'(A,A,A) and R(A,B) by
> R'(A,A,A)).
>
> [Lee: Here I am beginning to have second thoughts. At any rate, R'(A,A,A)
> is clearly a typo-- I think G.Russel means R'(A,B,C), where I believe
> the R' notation means "C is the ordered pair (A,B) and R(A,B) is true"...
> but this worries me, as it implicitly involves the binary relation R.]
>
> Suppose we can encode any n-ary relation (n>=3) by
> ternary relations, then let R be an n+1'ary relation. We encode
> R(A1,...,A_{n+1}) by the conjunction of R'(A1,...,A_{n-1},C) and
> T(A_n,A_{n+1},C), where R' is defined to be true iff C is the ordered pair
> (A_n,A_{n+1}) and R(A1,...,A_{n+1}).
>
> ---------------------------------
>
> Anyhow, I thought at first that George Russell's argument was correct, but now
> Ihave my doubts.
>
>
On Fri, 2 Feb 2001, Paul R. Chernoff wrote:
>
> Date: Fri, 2 Feb 2001 13:06:36 -0800 (PST)
> From: Paul R. Chernoff <chernoff@math.berkeley.edu>
> To: lee@sabre.org
> Subject: reducing binary relations to ternary relations
>
>
> Dear Lee--
>
> In connection with what I think of as Peirce's "threeness conjecture"
> (or should it be "thirdness"?) I think I can fill the gap I thought
> might be present in the short proof I e-mailed you recently that
> n-ary relations can be reduced to a composition of ternary relations.
>
> (Of course, this may *not* be what Peirce was claiming; perhaps he
> was claiming something far more difficult to prove -- witness the
> monograph on so-called 'topological logic' whose author/title
> you sent me. So I am notat all sure what Peirce was conjecturing.
> What is your impression?)
>
> First of all, introduce the notation <a,b> for the ordered pair whose
> first member is a and whose second member is b. (This is to be contrasted
> with the *set* {a,b} whose members are a and b -- {b,a} = {a,b},
> but, unless a = b, <b,a> is not the same as <a,b>. )
>
>
> Now a binary relation R is a set of ordered pairs. We write a R b if and
> only if <a,b> is a member of the set R. Or we may write R(a,b) to mean
> the same thing: "a is R-related to B". For example, if R is the relation
> of fastherhood, R(a,b) is interpreted to mean 'a is the father of b'
> So <a,b> is a member of R, but certainly <b,a> is then *not* a member of
> R.
>
> Similarly, a ternary relation S is a set of ordered triples <a,b,c>,
> and the analogous definition holds for n-ary relations. (For n = 1,
> <a> is the same as a. So a unary relation U is simply a set of objects,
> and U(a) means simply that a is a member of U. This is the same
> as a *property*. E,g. if C is the set of all cats, C(x) means that
> x is a cat.)
>
>
> A unary relation U may be represented by a ternary relation U',
>
> where U'(x,y,z) means x = y = z and x is a member of the set U.
>
>
> I was worried about binary relations. But now I think this is OK.
> Let B be a binary relation, and let B' be the ternary relation
> such that B'(x,y,z) means that z = <x,y> and z is a member of the set B.
> So B(x,y) holds if & only if B'(x,y,<x,y>) holds.
>
> The rest of the argument procedes by mathematicalinduction,as indicated
> in my earlier message to you.
>
> ----------
> Now, this is a *very* simple argument, so I can't believe it escaped
> Peirce. Do you know what Peirce actually wrote? My feeling, to reiterate,
> is that Peirce was after something far deeper - but of course I
> don't know what it was. Presdumably the book you have (which I did glance
> at last time I visited you) begins by stating clearly what the problem
> was. I wonder if you have time to take a brief look at that book and
> see if you can decipher what Peirce was really after.
>
> Best regards--
>
> Paul
>
JLA response:
Paul,
The distinction between the set a,b and the ordered pair a,b is indeed
simple, but not simple-minded. It does clarify a crucial point.
My sense of Peirce is that he was driving toward establishing a formal,
logical, non-algebraic notation that would better reflect (in Peircespeak:
be "perspicuous" with respect to) the insight that in inflected languages
is broached in special endings for the dative case.
In English, which has only residual case endings, we have not yet
abolished the dative case, and it helps us to grasp quickly that "John
gives Mary a book," or alternatively "John gives a book to Mary" is a
triadic relation in which important logical content is lost by the
reduction to pairs.
The sign relation may be approached pedagogically as a generalization of
the same form: R (for "representamen") represents O (for "object") to I
(for "interpretant"): or R is O to I. Though there may be cases in which
triadic relations are resolvable into dyads ("degenerate thirdness" in
Peircespeak), *this* triadic relation, the semiotic one, is not resolvable
into dyads without losing something important, but n-ary relations of
greater than 3 are resolvable into triads without loss of logical content.
Thus Peirce's irreducibility conjecture. Further, Peirce would argue that
relations of less than 3 may be represented triadically without loss of
logical content-- we might call this his conjecture of translational
adequacy.
"Proving" Peirce's triadic theorems makes me uneasy, since it is never
really possible to prove the postulates of a system within the system
bounded by them, and Peirce's larger argument is that the semiotic triad
is postulated in all discourse, including the discourse with ourselves
that we call thought. If every sign is viewed as a three-in-one, then the
appropriate proof is more heuristic (or "pragmatic") than formal, and this
may explain why Peirce wanted to *do *things with the idea rather than
prove it mathematically. It leads to a theory of sign types that enables
us to specify the limitations and power of mathematical thought itself.
Peirce is really proceeding philosophically from an elucidation of
recursive patterns postulated in all discourse ("the categories").
As with Burch (the book you saw), the attempt to render Peirce in standard
mathematical logic is bound to miss something essential in Peirce's
non-standard approaches to such issues as relation, predication,
intension/extension, quantification, notation, infinitesimals, and
information.
Still, one does want to show that the working out of a system does not
land one into hopeless contradiction and anomaly, and perhaps if Peirce
had thought of the device you use (from Norbert Weiner, as you have taught
me), he would have brought it forward.
I haven't looked at the literature on the triadicity issue for quite a few
years, but if my memory serves me right, the main critique of Peirce from
mathematical quarters has come from the school of Tarski, within which I
would include the young Quine, whose review of the Collected Papers of C.
S. Peirce in the late 30s (in Isis if I remember correctly) suggested that
genuine quadratic relations might be irreducible to triads. I seem to
remember that a similar suggestion was made in response to Burch's book
from two of Tarski's later followers in a collection of essays edited by
Nathan Houser.
On a metaphysical level, the attempt to posit an irreducible "fourth"
will be found in Paul Weiss, who called this "God" and described his
indebtedness to Peirce in an extended appreciation.
Lee