Re: SUO: Re: Parse Of Things Remembered
>Pat,
>
>Peirce and Whitehead were both familiar with the point
>you keep repeating endlessly:
I only ever say it as a response to the endless repetition of a
mantra about 'irreducible triadicity' by Peircian cultists.
> >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))
> >
> >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:
>
>Now let me use your example to explain the point which both
>CSP and ANW were trying to make, which and you keep missing:
>
> >Gave(John, Book, Mary, yesterday)
> > --->
> >(exists e)(Giving(e) & agent(e, John) & subject(e, book) &
> >recipient(e, Mary) & time(e, yesterday)) ).
>
>This is the transformation that Ernst Schroeder used in his 1890
>book for replacing triadic and higher relations with dyadics.
>Both Peirce and Whitehead were very familiar with it, and they
>both rejected it because it is *not* a decomposition into
>dyads. It is merely a relabeling of the arguments 1, 2, 3, 4
>with labels "agent", "subject", "recipient", and "time".
And such relabelling IS a reduction to binary relations. Whatever
metaphysical significance y'all think it doesn't have, the fact
remains that this is a perfectly well-defined transliteration of FOL
into the sublanguage of FOL which uses only binary relation symbols
(and if that isn't a reduction to dyads, I have no idea what a dyad
is.)
It has a perfectly satisfactory extensional semantic interpretation,
in which 'e' denotes the n-tuple in the extension of R which contains
the arguments to which the relation is applied; it has a plausible
interpretation in ordinary language usage; and it preserves
provability.
>What CSP and ANW were trying to explain is that the verb
>"give" and many other irreducible triads *cannot* be reduced
>to conjunctions of dyads of the following form:
>
> give(x,y,z) = part1(x,y) & part2(y,z) & part3(y,z).
True. So what? Do you propose to outlaw the use of the existential
quantifier? But in any case, there is no need to use explicit
quantification. One only needs to introduce new names:
part1(x,e1) & part2(y,e2) & part3(z, e3) & =(e1, e2) & =(e2, e3)
Peirce's 'triadicity' result is relevant here, in an odd way. It
shows that there is no such reduction to a conjunction of binary
relations in which no name is *used* more than twice. One does need
to use some name at least three times (e2 in the above example.)
True; but, I would suggest, profoundly unimportant. What is the point
of counting the numbers of symbol tokens which are used in our
logical formulae? That is not even part of the syntax of the language.
>In this decomposition, part1, part2, and part3 are pure dyads
>that only relate two arguments at a time. In Schroeder's
>decomposition, which you keep repeating, you have to introduce
>a new argument e, which links the monad named "giving" to
>each of the other arguments: agent(e, John), subject(e, book),
>recipient(e, Mary), and time(e, yesterday).
>
>Notice that you have *not* performed a reduction of a triad
>(or tetrad in this case) to a conjunction of dyads because
>your new argument e links "giving" to every one of the other
>arguments. All the triadicity (or tetradicity) is still buried
>in the monad named "giving".
I have never understood what your point is, what 'triadicity' is
supposed to really amount to, or why you pay an almost religious
homage to what seems to be a trivial matter of numerology; but
whatever it is, if I can 'bury' it in a monad so effectively, why
should I give a damn about it?
The only issue of importance here is the intertranslateability
between various sublanguages of FOL. Peirce's graph-theoretic result
has no utility in this regard; in particular, it does not show that
triadic relations are required. This has apparently been known since
1890 (thanks for the historical pointer). We are now in 2001, so can
we move on?
Pat Hayes
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