SUO: Re: Policy On Substitutions
Robert Meersman wrote:
>
> At 11-02-01 12:12, Jon Awbrey wrote:
>
> > ¤
> > º
> > .°
> > \_/
> > |
> > -^-
>
> *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"
Well, more like, let's demote variables from their status
as formal or logical or mathematical primitives and treat
them as syntactic sugar -- it rots your teeth, I know! --
or a convenient substitute for a more careful analysis of
what's really going on, sign-theoretically speaking, when
we practice the particular form of deceit that is invoked
via the common, crude, but everyday use of their conceits.
> > 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... :-)
This is more or less a combinatorial and a formal language theoretical
take on the situation, where all the pictures in my head are basically
graph-theoretic and incidence-geometric. From this POV, I see the set
of expressions suggested by {"0 + 0", "1 + 1", "2 + 2", ...} as just a
particular formal language, and no, I am not forced to countenance any
infinite alphabet, as you would have seen if you had taken the time to
extend it as far as {..., "9 + 9", "10 + 10", "11 + 11", ...}, that is,
regarding it as a "two-level formal language" over the finite alphabet
A = {" ", "+", "0", "1", "2", "3", "4", "5", "6", "7", "8", "9"}, else
I could have been more clever and used a binary coding, but who thinks
ahead on a day like this!?
So the point is that the particular mechanics of substitution in
open expressions is of secondary interest, amounting to just one
style of grammar formalism for generating the formal language of
primary interest.
> ### 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.
Well, thanks for the compliment, but nothing that fancy just yet.
I have not even worked my way up to the logical level at present.
> 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.
The Horror! The Horror!
> But my memory definitely fails me on the usefulness
> of all this for our ontological pursuits.
Remember what I said somewhere in the middle of that bout of logorrhea --
I found this to be a very effective remedy, in my case, at any rate, for
resolving my "ontological insecurity" (OI) about the status of variables.
It also serves to place all of those ad hoc, hokey, syntactic mechanisms
for substitution in a more general context, one that is quite a bit more
elegant from a formal, combinatorial POV, if you like that sort of thing.
> > 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 >-)
I am trying not to think about it ...
Cheers, Hic Et Nunc,
Jon Awbrey
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