Re: On Formal IS-A definition

From: Keith H Duggar <duggar_at_alum.mit.edu>
Date: Sat, 22 May 2010 12:53:46 -0700 (PDT)
Message-ID: <ad1c2142-724e-4d85-b43d-9662ddee91ec_at_q13g2000vbm.googlegroups.com>



On May 17, 10:57 pm, David BL <davi..._at_iinet.net.au> wrote:
> 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:
> > > 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'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.

Good. We've come full circle jerk in yet another extravagant DBL pose fest. You should study how your profound ignorance and lame attraction to fallacies (context shifts, strawmen, etc) required nearly 30 posts across multiple days and posters to correct.

> 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!

"not the variable itself" and "it is rather difficult to denote them!" are just more nonsense meaningless drivel.

> > > 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.

No, you were and remain wrong because the context was not FOL and never has been! These ignorant context shifts you keep trying to impose on the discussion are plain stupid.

The context of my statements was and continues to be mathematics and formal languages in general and formal semantics in general. In that broader context variable assignment functions are simply a "part" of an interpretation. Read section 1 of the following:

   http://plato.stanford.edu/entries/model-theory/

Do you understand now? An "interpretation" is the totality of the "added information". Or as wikipedia concisely puts it

   http://en.wikipedia.org/wiki/Interpretation_(logic)

"an interpretation is an assignment of meaning to the symbols of a language." Across a variety of formal languages and semantics this is extra information is formalized as a /relation/. Sometimes that relation is thought of in parts (for various reasons) such as the "denotation assigment function" and the "variable assignment function" etc.

But of course, you are near totally ignorant of this broader context. As evidenced by this post

   http://groups.google.com/group/comp.databases.theory/msg/05f51dba4d854934?hl=en

you were even ignorant of the field of formal semantics until I told you about it a month ago. Now, after a month, you think you are qualified to pronounce yourself right and your teacher wrong?? This has got to be one of the clearest examples of an idiotic vociferous ignorant poser we've seen in a long while.

That what the rest of formal semantics calls a "model" (which is exactly why it is commonly represented by the letter M even in FOL) is often called the "interpretation" in classic first order interpretation, is completely irrelevant to the more general context of model theory applied to mathematics and formal languages as a whole.

As already demonstrated you were nearly ignorant of all this even just days ago. Had you been aware of variable assignment functions you would have understood that my general point made in a more general context, applied equally well to FOL because a function (the variable assignment function) is a relation!

> > > 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.

Obviously the above is flat wrong because free variables are not used to express quantification. This is part of the Dense Bullshit and Lies (DBL) that is so time consuming to respond to.

Also, we see yet another example of you trying to impose a context shift (from languages in general to FOL only). A dishonest "tactic" that is so blatantly easy to spot for those trained to do so and yet so annoying and time consuming to repeatedly correct.

> > 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.

Wrong. See above. In the general context of formal semantics and model theory the variable assignment functions discussed in FOL are just one part of what formal semantics calls "interpretation".

The problem was and remains that DBL is nearly completely ignorant of formal semantics. He's never sat in a class for it, never worked through examples of interpretation, never heard a professor warn you of some common ambiguities and overloaded terminology and to explain the history behind them. In other words, DBL is ignorant of the whole and worse is vociferously arrogant in that ignorance.

KHD Received on Sat May 22 2010 - 14:53:46 CDT

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