Pat, (01)
Some responses: (02)
PH> Nothing useful is served by throwing around vaguely insulting remarks... (03)
I apologize. There are many very useful subsets of FOL, and I agree
that DLs (and even smaller subsets, such as Aristotle's syllogisms)
have abundantly proved their value. (04)
JFS>> But I believe that making an explicit distinction between the
>> model and reality is important for any application of formal
>> languages... (05)
PH> And for the record, I believe that this is a basic mistake.
> The 'model' here (ie in Tarski's sense) can *be* a part of reality. (06)
I agree, but the word I would emphasize is *part*. There is an
open-ended number of selections and subdivisions that can be made.
The choice of the domain D and relations R is always done for some
purpose, and what is ignored may be as significant or even more
significant than what is selected. (07)
PH> Tarski himself seems to have taken his view by his choice
> of example: "Snow is white" is true when snow, in fact, is white.
> No "model" there: he is talking about (real) snow and (really being)
> white. (08)
Many people have pointed out that his choice of example was
unfortunate, since that sentence raises two thorny issues that
Tarski did not intend to represent in his introductory paper:
snow as a continuous substance, and the use of a singular noun
for making a generic statement about all snow. (09)
JFS>> The mapping of D and R to reality is often unstated, and
>> for any important application, it's usually far from obvious. (010)
PH> I disagree. It's often blindingly obvious... (011)
We'd need a corpus of data to settle the question about which
kinds occur usually or often. For starters, Tarski's actual
example about snow is definitely not obvious. (012)
JFS>> I agree that formal model theorists tend to talk that way, but
>> that is not how scientists work. The model comes first, and
>> the axioms are derived from the model. (013)
PH> First, I wonder how you can possibly know this. (014)
From reading what scientists actually report about how they do science. (015)
PH> Isn't one way that scientists work the simple noting of
> observational facts in lab journals? Aren't such notations sentences.
> and don't they describe reality? (016)
They certainly do. But note that observational reports require
*only* the operators of conjunction and the existential quantifier.
(Any other operators that may occur in such reports must always
be suitably restricted -- e.g., "Every sample tested was..." --
and they can always be replaced by a list of ex-con statements.) (017)
Those statements can be, as you correctly noted, directly mapped
to some part of reality that may be regarded as the model. (018)
> But more seriously, HOW does one 'derive' axioms from a model? (019)
By induction (introducing universal quantifiers and implications),
abduction (guessing, which may introduce any logical operators
that seem "reasonable"), and further testing of any predictions
derivable by deduction. (020)
This procedure provides a way to guarantee that the axioms are
consistent: just check that every hypothesis derived by induction
or abduction is consistent with the observational reports (i.e.,
those that describe a suitable model). (021)
John (022)
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