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Re: Is the meaning of order intrinsic ?

From: Seth Russell <seth@robustai.net>
Date: Fri, 25 May 2001 09:36:42 -0700
Message-ID: <009101c0e538$e27ac5c0$b17ba8c0@c1457248a.sttls1.wa.home.com>
To: "pat hayes" <phayes@ai.uwf.edu>
Cc: <www-rdf-logic@w3.org>
From: "pat hayes" <phayes@ai.uwf.edu>

> It depends what you mean by 'define'. There is no definition of the
> truth-value 'Truth', of course. But there is a (fairly elaborate)
> definition of satisfaction of a set of sentences in an
> interpretation. Being mathematics, it does not provide metaphysical
> definitions of its basic terminology; but then mathematics never does.

Actually I think it is quite easy to define the truth-value 'Truth'.  It's
simply an attitude adopted by an agent towards a particular statement.  Look
ma, no metaphysics :)

> >I suppose Tarski tried,
> >but I for one was not satisfied that he said anything.
>
> My goodness, I guess we will have to re-think the foundations of our
> subject in the face of such a devastating critique.

Cool !  I wouldn't have even imagined that my mere opinion would have had
such an impact.

> Actually, sequence can be defined in terms of functional application,
> or of set membership, using a well-known trick , the common idea of
> using trees to encode sequences. You need the notion of a pair and
> the ability to *distinguish* the two things that are paired, but that
> distinguishing need not be described in terms of an order. We
> conventionally use order in the syntax to indicate the two parts of a
> pair, eg by writing (A . B) to indicate the pair consisting of A and
> B, but that is really just a lexicographic accident arising from the
> linear notation which we use in writing text. Abstractly, a pair is
> just a structure with two distinguishable substructures. Then a
> sequence can be defined by iterating pair construction in the usual
> way, which would look like this using dotted-pair notation:
> (A1 . (A2 . (A3 . (..... (An .  END)...))) The same kind of thing can
> be done with functions, or sets, or whatever. Eg using functions, one
> encodes an n-ary function - whichis interchangeable with an n-ary
> sequence at this level of abstraction -  as a single-argument
> function whose value is an (n-1)-argument function (a trick invented
> by a logician called Curry, and hence known as currying.)

Yeah this is pretty much the way numbers are defined with set theory.   But
I still have a trouble with using this technique of defining a sequence.  My
thesis is that 'ordering' is a first principal, is prior, is axiomatic (at
least to humans and the systems they tend to construct).  It seems to me
that to disprove this thesis you would need to be able to define an order
without using anything that presupposes that order already.  Your syntactic
string (sequence of characters) above of course contains that
presupposition.  But let's forgive that trespass for a moment, because I
know what this string means and have interpreted it in a graph [1] quite
apart from the ASCII string you used.   But here again I still have problems
arriving at an unambiguous interpretation of the order without presupposing
some known order already.  In particular (1) we need to establish the
sequence {begin, end} ... iow try explaining that sequence to an alien life
form that doesn't understand time; and then (2) we need to know the order or
each pair (A . B) ... in Example 3 of my diagram I have perversely
interpreted the sequence to be {A2, A3, A1}.

[1] http://robustai.net/mentography/order.gif

... thanks for the dialogue ...

Seth
Received on Saturday, 26 May 2001 03:19:14 UTC

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