Re: On Formal IS-A definition

On May 10, 1:34 pm, Keith H Duggar <dug..._at_alum.mit.edu> wrote:
> On May 9, 9:29 am, David BL <davi..._at_iinet.net.au> wrote:
>
>
>
>
>
> > 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 agree that a symbol can and should be regarded as a value, and one can indeed have a set of symbols. When one says that a variable is a symbol one really means that the symbol takes on a role of being a variable within a logic formula. For example, in a formula where the variable is bound, it may either be existentially or universally quantified. The only purpose of the symbol is to express quantification within the scope of the formula in which it appears, and it would be possible to replace the symbol value with any other (being careful to avoid "name clashes") without changing the meaning of the formula.

In a formula or term where the variable is free, there is little to be said, other than the fact that it is "waiting" to be bound by putting it in some containing formula.

I can imagine you disliking this subtle distinction between variable=symbol and variable=role of symbol. I suggest you keep in mind that within a large number of formulas you may see a symbol appear as a variable many times. Ask yourself whether these occurrences represent the same variable or distinct variables. If you feel that they are the same variable then I suggest you are confusing a symbol with its role as a variable within a formula.

Variables are only used to express quantification in FOL formulas. That gives them both a narrow purpose and scope. In particular, they have no role to play in the elements of the domain of discourse of some interpretation. So even though the axioms of set theory are expressed in FOL, there is no risk of variables "leaking" into the sets in the domain of discourse about which logic statements are being made. There isn't even a concept of the interpretation of a variable being an element of the domain of discourse. Instead variables are said to "range over the domain of discourse".

Below you are confusing variables with function symbols with arity 0. Only the latter are mapped to elements of the domain of discourse under an interpretation.

> Once you've got that it should be easy to understand how one can
> have sets of variables.
>
> Now honestly I would have thought, from what you've posted in the
> past, that you would already have known this. Taken from previous
> discussions we have the following:
>
> 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.

> 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. Received on Mon May 10 2010 - 11:54:06 CEST