Pat, (01)
I apologize for violating one of my own rules: (02)
CM>>> If you simply mean a *list*, then obviously no infinite set can
>>> be specified extensionally; but that's not very interesting. (03)
JFS>> I agree. But Pat seemed to object. (04)
PH> I did not object to this, which is obvious. (05)
I always complain when somebody attributes positions to me without
quoting my exact words, and I'm sorry for doing that to you. (06)
JFS>> That [CM's example] is what I was calling a specification
> by a finite statement of some rule or axiom for determining
> the elements of the set. (07)
PH> This does not specify what the elements are of the *smallest*
> set satisfying the conditions are. Chris' description does not give
> any way to determine the elements of that set, it only describes
> conditions on the set itself, of the form... (08)
I agree that a constructive specification is preferable to one
that makes existential claims without an explicit construction. (09)
CM>>> If one were to formalize the grammar above as a first-order
>> theory, even an infinite, undecidable theory (which of course
>> wouldn't be much of a theory), there would be no way to rule out
>> nonstandard interpretations of the constant "N". (010)
JFS>> I agree. But I did *not* formalize it in FOL. What I did was
>> to *formalize* it as a grammar. A formal grammar is just as formal
>> as any statement in any version of logic. But nobody talks about
>> nonstandard models of a grammar. (011)
PH> They could, and some do. The grammar is a set of assertions,
> and these assertions have non-standard models, AKA non-minimal
> fixpoints. For example, your grammar can be satisfied by the
> minimal interpretation N = {0, s0, ss0, sss0, ...}, which you have
> in mind; but also by an infinite number of non-standard solutions,
> such as the 'transfinite' solution {0, s0, ss0, sss0, ...., A, sA,
> ssA, sssA, ...} or the 'intermediate-point' solution {0, H, s0, sH,
> ss0, ssH, sss0, sssH, ...}, etc.. What rules out these
> interpretations? (012)
I admit that some grammars can have nonstandard models, which are
usually ruled out by assuming the least fixed point. That is a
perfectly reasonable and very formal way to solve the problem,
but it requires a higher-order or at least a metalevel statement. (013)
But for the grammar I gave for 0, s0, ss0, ..., the nonstandard
models in your example are ruled out for a much simpler reason: (014)
PH> but also by an infinite number of non-standard solutions, such
> as the 'transfinite' solution {0, s0, ss0, sss0, ...., A, sA, ssA,
> sssA, ...} or the 'intermediate-point' solution {0, H, s0, sH, ss0,
> ssH, sss0, sssH, ...}, etc.. What rules out these interpretations? (015)
I agree that those nonstandard models can't be ruled out when the
grammar is translated to FOL. But that is because FOL does not
have a built-in convention that is included in formal grammar
theory: the requirement for stating finite sets of terminal
and nonterminal symbols. (016)
In the note to Chris, I just quoted the two grammar rules by
themselves. But in an earlier note to you, I explicitly
stated the permissible terminal and nonterminal symbols: (017)
JFS> Terminal symbols: {S, 0}
>
> Nonterminal symbols: {N}
>
> Grammar rules:
>
> N -> 0
> N -> S N (018)
The nonstandard solutions you suggested cannot be generated
because they use symbols that aren't in the above sets. (019)
PH> You rely on the same intuition that we rely on when we use
> the numerals and arithmetic operators in the usual way, and
> believe that we are talking about the standard model. (020)
No. In addition to the requirement for specifying all permissible
symbols, generative grammars have a formal convention that rules
out those nonstandard interpretations. They require one of the
nonterminal symbols to be designated as the start symbol. (For
the above grammar, that is N.) (021)
For each sentence in the language, there must exist at least one
derivation that begins with the start symbol and ends with only
terminal symbols. With this convention, every legal sentence has
an explicit construction, and there is no need for any assumption
or intuition about a least fixed point. (022)
In summary, formal grammar rules can be translated to FOL, but
the FOL translation does not include the generative constraint
that requires each sentence in each model to have an explicit
derivation from the start symbol. (023)
John (024)
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