Re: SUO: Re: Policy On Substitutions
At 11-02-01 12:12, Jon Awbrey wrote:
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*tchin* *tchin*
Robert,
Let's see if we can cure you of that 'very' curiosity!
Once upon a time, a very long time ago, back when
I was deep in the muddle of my very first, and my
most extenuous "crisis of foundations" (COF, COF),
I was very sorely afflicted, among the variety of
other affectations that flesh and grass are heirs
to, with an anxious "ontological insecurity" (OI),
focussed on the "ontological status of variables",
the eventual resolution of which difficulty, over
too long a time to believe for the obviousness of
its ultimate form, was simply to banish variables
altogether from the realm of our finely canonical
& our most finally, commonly, sans-critical koine.
Henceforward, and for the rest of time, let us then say
that there are actually no "open sentences" at all, not
in our most wholy sanctioned languages, but that such a
common nickname or vulgar form of words as "Bla, bla X"
is really just a general name, a pleonastic periphrasis,
or perhaps just a colloquial circumlocution for a whole
collection of signs or sentences of a similar configure,
except with the "X" respliced by a fully concrete piece
of a fixed signal, or a fitful token of a sentence
part.
### can I summarize the above, at the risk of trivializing your
logorrheic talents, as "let's throw out variables from
FOL"
It needs to be noted that the specification of
this collection of signs
in no way relies on the particular artifices or the specialized
devices
of anybody's rude mechanics for performing substitutions, but since
the
collection of signs is conceived to exist, in some sense of
"existence",
prior to its secular and timely indications by the likes of me and
thee,
well, then we note, ours is not to question why, ours is but to
specify.
The point is that any way of specifying the desired collection of
signs
is equally as good as any other, to any of which indiscernible
outcomes,
assuming that they do indeed meet any one of many reasonably
equivalent
specifications, one might affix any convenient nickname for the
species.
So, for example, the so-called "open expression" of the form
"x + x",
where the values that one associates with the variable token
"x" are
the elements of the non-negative numbers N = {0, 1, 2, ...}, is
just
a nickname for the set of constant expressions that begins like
this:
"0 + 0", "1 + 1", "2 + 2", ...
And it is species of perfectly constant signs like this
that are the actual subjects of our generic discussions
in this regard, thereby removing every last trace of my
onetime "ontological insecurity" (OI) about
"variables".
### yes but I note your convenient ellipsis "..." and so I'd
like to see you do the same for a variable over a non-countable domain,
Jon... :-)
### I'd like to humor you on your b'day and I don't have my logic books
at hand but you seem well on your way to re-invent Herbrand models. If I
remember well (but I definitely can be wrong) this
"construction" (which in fact it is not) also
"worked" for domains of arbitrary cardinality but this involved
something like maximally consistent Henkin theories. But my memory
definitely fails me on the usefulness of all this for our ontological
pursuits
And that, my friends, is something to
celebrate!
So leave me to my illusions, at least for a day!
### ah, but life and science can be cruel sometimes >-)
*tchin* *tchin*
--Robert Meersman
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Prof Dr Robert A
Meersman
VUB STARLab
Department of Computer Science Vrije
Universiteit Brussel
Bldg. G-10, Pleinlaan
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phn (+32|0) 2 629
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