Re: The word "symbol"

From: VC <boston103_at_hotmail.com>
Date: Sat, 13 Aug 2005 10:22:08 -0400
Message-ID: <LNednYjy4K6Rn2PfRVn-2Q_at_comcast.com>


"David Cressey" <david.cressey_at_earthlink.net> wrote in message news:oEkLe.4898$RZ2.4896_at_newsread3.news.atl.earthlink.net...
>
> "vc" <boston103_at_hotmail.com> wrote in message
> news:1123862288.790137.272990_at_f14g2000cwb.googlegroups.com...
>>
>> David Cressey wrote:
>> > "Paul" <paul_at_test.com> wrote in message
>> > news:42fc7a2d$0$17487$ed2e19e4_at_ptn-nntp-reader04.plus.net...
>> >
>> > VC, if I read him right, views everything at two levels of
>> > abstraction:
>> > the logical level and the physical level.
>>
>> Well, no. The two things I see are a formal structure and its model.
>> E.g. Peano axioms as a formal stucture and integers as the standard
>> model of the PA. A formal structure has a language which is used to
>> talk about the model. The language vocabulary( a set) contains
>> names(constants, function names, connectives, etc) for various things
>> in the model. Sometimes, those names are called symbols. I have no
>> objection to such use of the word "symbol" although this use does not
>> occur in the modern math practice very frequently. In other words, if
>> you said: " by a symbol I undestand an element from the first-order
>> language vocabulary", there would have been no objection on my part.
>>
>
> I can accept the above, from the point of view of a math centered person.
> I'm not a math centered person,
> although I took some math in college, some 40 years ago. I'm a data
> modeler
> (retired). And I've found the word "symbol" useful enough to keep it
> part
> of my active vocabulary.
>
> The above definition of symbol makes sense to me, in the context of this
> discussion. But I never would have arrived at it in a million years, left
> to my own devices. "An element of the first-order language vocabulary"
> is
> not a phrase from my active vocabulary.
>

Well, the problem with the word "symbol" is not whether you have math inclinations or not. The word's ambiguity is a real issue. E.g. it's unclear what exactly one might be talking about: does a symbol mean a thing name, a model of a thing, some magical entity, or something else. There is no definition for the symbol as, say, there is for the word "force" in physics which is clear and unambiguous.

> The difference that arose between you and me appears to be about the way
> each of us uses language, rather than a substantive one. And I'm content
> to
> let the matter rest there.
>

OK.

>> OK, please define "a/the? consistent meaning" (not multiple meanings)
>> of the word symbol.
>
> Let's just go with "an element of the first-order language vocabulary",
> for
> now.
> But first, maybe you can give me a definition of "first-order language".

A first order languagy is a language of the first-order, also known as predicate, logic. ( "a " implies that there are slightly varying dialects of the language). The language is defined as consisting of:

  1. connectives (A, E, ^, &, ->,..)
  2. a union of disjoint sets of constants, function names, relation names, variables

The above is a f.o. language vocabulary. Then, there are rules(syntax) telling us how to construct terms and formulas from the vocabulary elements. E.g. E(person)Age(person)=20, where E is a connective, person is a variable name, Age is a function name, = is a relation name, 20 is a constant. The expression "first order" means that quantification (A and E) can be applied only to individuals. Second-order means that we can quantify over collections of individuals as well.
>
>
>
Received on Sat Aug 13 2005 - 16:22:08 CEST

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