Re: On Formal IS-A definition

On May 10, 5:54 pm, David BL <davi..._at_iinet.net.au> wrote:
> On May 10, 1:34 pm, Keith H Duggar <dug..._at_alum.mit.edu> wrote:
>
>
>
>
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> > On May 9, 9:29 am, David BL <davi..._at_iinet.net.au> wrote:
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> > > On May 9, 11:38 am, Bob Badour <bbad..._at_pei.sympatico.ca> wrote:
>
> > > > My set of three variables and a dog fully complies with ZFC.
>
> > > Here is a quote from (http://en.wikipedia.org/wiki/Zermelo
> > > %E2%80%93Fraenkel_set_theory)
>
> > > "ZFC has a single primitive ontological notion, that of a hereditary
> > > well-founded set, and a single ontological assumption, namely that all
> > > individuals in the universe of discourse are such sets. Thus, ZFC is a
> > > set theory without urelements (elements of sets which are not
> > > themselves sets)."
>
> > > and this (fromhttp://en.wikipedia.org/wiki/Hereditary_set)
>
> > > "In set theory, a hereditary set (or pure set) is a set all of whose
> > > elements are hereditary sets. That is, all elements of the set are
> > > themselves sets, as are all elements of the elements, and so on."
>
> > > I wonder whether Bob enjoys putting a leash on a set and taking it for
> > > a walk.
>
> > I wonder if you know what a variable is? Or more specifically I
> > wonder if you can prove that a variable is not a set? Well, that
> > is a rhetorical question really because I already know that a
> > variable /is/ a set. Or rather, because the word "is" is vacuous
> > most of the time, a variable can be represented by a set. Since
> > you enjoy wikipedia so much (since when did wikipedia become an
> > authoritative source?) try reading this (thoughtfully):
>
> > http://en.wikipedia.org/wiki/Variable_(mathematics)
>
> > and see if you can figure out how it is that variables can be
> > represented by sets. Hint, a variable is a /symbol/.
>
> You are using "variable" in the sense that a logician would use it.
> This discussion actually began with variables accessed by programs
> that support imperative assignment statements. Let's be sure we
> don't confuse these.
>
> In any case you are still wrong. I believe you are suggesting one
> can
>
> 1) Have a symbol x
>
> 2) Form a set {x}
>
> 3) Have a logic formula where symbol x is a variable, such as
>
> for all x, x+0 = x
>
> 4) Deduce that variables can appear in sets.
>
> I accept 1), 2) and 3) but not 4). You make the mistake of thinking
> that symbols represent variables outside the context of the formula
> they appear in - even when the variable is bound. If that were true
> that would be remarkably bad!

I've been reading the following Stanford articles:

    http://plato.stanford.edu/entries/types-tokens/

(note well section 8 on occurrences), and

    http://plato.stanford.edu/entries/logic-classical/

I'm going to eat my words. I see now that I was wrong. I was associating the term "variable" with occurrences of symbols, whereas the Stanford articles go to the trouble to distinguish between a variable and an occurrence of a variable.

Therefore assuming this terminology it is indeed valid to have a set of variables.

However it is very curious that if one says:

    Let {x,y,z} be a set of variables

then what we appear to have is an expression {x,y,z} containing free variables. According to Section 4 (Semantics) in the SEP article on classical logic, an interpretation M = <d,I> assigns denotations to constants. E.g. For constant c, I(c) is an element of d, whereas a variable-assignment function s on M is required to assign a denotation to a free variable. The denotation of variable v is s(v) which is an element of d, not the variable itself. It seems that although one can have sets of variables, it is rather difficult to denote them!

> > DBL knows that in formal semantics /variables/ are /interpreted/
>
> Wrong (assuming "interpreted" means mapped by an interpretation
> function). Only function symbols and predicate symbols are
> interpreted.

I was correct there.

>
> > DBL knows that an interpretation is formally a /relation/ mapping
> > variables (and all other symbols) to elements of the /domain of
> > interpretation/ also sometimes called a "universe"
>
> Wrong. FOL variables are not mapped to anything. They are *only*
> used to express quantification in logic.
>
> A sentence (i.e. formula where all variables are bound) is interpreted
> according to the semantics of existential or universal quantification
> on the bound variables that are assumed to range over the universe of
> discourse.

I will qualify that. An interpretation function I doesn't map a variable to anything. Rather a variable assignment function s defined on an interpretation M is used to assign denotations to free variables.

Keith's comment was incorrect. A variable assignment function s is not part of an interpretation M, and therefore it is incorrect to say that an interpretation M assigns a denotation to a free variable. Received on Tue May 18 2010 - 04:57:53 CEST