SUO: Re: Numbrance Of Times Perished
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Pat Hayes wrote:
>
> [John Sowa wrote:]
> >
> > Pat,
> >
> > We went round and round on this issue before,
> > so I don't want to repeat the experience.
>
> I didnt go round and round on this issue.
> I wrote a review of Burch's thesis which
> I believe settled the matter conclusively.
> (J. Man-Machine Studies, 1995)
>
...
>
> Had Peirce or Whitehead lived a little longer maybe they would have
> become aware of the fact that any n-ary relation can be defined in
> terms of binary relations, with the aid of the existential quantifier.
> The translation, as I know you know, John, is this:
>
> R(t1, ..., tn)
> --->
> (exists e)(R(e) & first(e, t1) & second(e, t2) & ... & nth(e, tn))
What this translation says is just this:
| t in R iff there is an e in R such that e = t.
I think that the total lack of any analysis in this ought to be evident --
it is like that all too frequent case of "divide and conquer" where one
starts with a problem of complexity 10 and splits into two problems of
complexity 10 and 0, respectively -- but just in case, ...
After all, the binary relations 1st, 2nd, ..., nth,
are just the projection functions Proj<1>, Proj<2>, ..., Proj<n>
that are used to define the cartesian product P such that R c P.
The assertion "R(t<1>, ..., t<n>)", written extensionally,
becomes "<t<1>, ..., t<n>> in R". Using the definition of
t<j> as Proj<j>(t), this can be written more succinctly as:
| t in R.
That is what the translandum says.
The following are equivalent:
1. There exists an e in R and Proj<j>(e) = t<j> for j = 1 to n.
2. There exists an e in R and Proj<j>(e) = Proj<j>(t) for j = 1 to n.
3. There exists an e in R and e = t.
That is what the translans says.
Hence, all that this translation says is just this:
| t in R iff there is an e in R such that e = t.
Jon Awbrey
> where 'first', 'second', etc., are some fixed set of binary relations.
> (In case grammar these correspond to cases such as 'agent', 'subject',
> and so on, and the 'e' is something like an event or a situation, of
> type R, corresponding to the verb of the simple sentence, as in:
>
> Gave(John, Book, Mary, yesterday)
> --->
> (exists e)
> (Giving (e) &
> agent (e, John) &
> subject (e, book) &
> recipient (e, Mary) &
> time (e, yesterday)
> ).
>
> Since a binary relation has a name and relates two other things,
> it is conventionally called a 'triple' in data structure terminology
> (eg in RDF), and this ghost of trinaricity is where Peirce's simple
> graph-theoretic result can be glimpsed. His error was to conclude
> that this implied that a trinary *relation* was necessary; and that
> in turn was because he made the error of thinking that the assertion
> of identity between n things involved an irreducibly n-ary identity
> relation.
>
> Pat Hayes
>
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