> > On the other hand, in order to
>>satisfy the axioms, the 'model' must contain all
>>the structure that these axioms explicitly
>>describe, so in a very precise sense it cannot be
>>of lower fidelity than the axioms, but it can
>>well be of much higher fidelity.
>
>That statement could be very misleading, because the referent of
>"fidelity" is unclear. (01)
It was in reply to Charles' email comment: (02)
>Model, now matter how it is used, seems to be some sort of extended
>implementation from an original (whether it is an abstraction, like a
>mathematical, computer, or physical model - or some instantiation of a
>series of axioms). In doing this extension, something is lost from the
>original (in the case of the model plane, it is reduced in size,
>functionality, and fidelity - in the case of a Tarskian model it is reduced
>from an ideal state explained in axioms to something that can change and
>lose some of its adherence to those axioms). (03)
which I read as saying that the Tarskian 'model'
models axioms in the sense that it is an
abstraction or reduction from the "ideal state
explained in axioms", and in the passage from
that ideal state to the 'model' "something is
lost", and the 'model' is "reduced...in
fidelity". Note the relationship being described
here is between the (ideal state explained in)
axioms to a Tarskian 'model' of those axioms.
That is the sense and context in which my
statement was made. I did not say that anything
about the fidelity of wither axioms of Tarskian
'models' to the real world. (04)
>Tarski interpretations must be faithful to
>the axioms, but may be wildly unfaithful to the intention of the
>modeler, and to the world we are trying to represent with the axioms. (05)
Of course. The axioms may be unfaithful to the
world we are trying to represent. In fact, this
is exactly one of the primary utilities of model
theory in practice: it gives a way to show that a
set of axioms FAILS to capture some aspect of
reality, but exhibiting a 'model' of the axioms
in which this aspect is not displayed. I was once
quite startled to find that an axiomatization of
time I had written, which contained an axiom
which I had thought ensured that time was
infinite, had a 'model' in the open unit interval. (06)
>
>Let me give an example to make things concrete.
>
>Consider the set of axioms:
> All men are mortal.
> Pat is a man.
> John is a man.
> Kathy is a woman.
>
>There are Tarskian interpretations of this set of axioms in which
>Kathy is immortal. The axioms say men are mortal, but don't pin down
>whether or not women are mortal. There are also Tarskian
>interpretations of this set in which Kathy is a man. That's because
>the axioms don't say whether women and men are mutually exclusive
>categories. In the Tarskian interpretations in which Kathy is a man,
>though, she has to be mortal, because all men are mortal.
>
>All the Tarski interpretations are faithful to the axioms, in the
>sense that the axioms are true in every interpretation. The ones in
>which Kathy is immortal or Kathy is a man are very unfaithful to the
>actual world, and presumably to the intentions of anyone who would
>write down these axioms. There's a sense in which we might call them
>unfaithful to the axioms, because they define truth-values for things
>the axioms leave open. (07)
You could say that, although I think THAT would
be a rather misleading way of talking, myself.
Rather, one should realize that axioms only say
what they actually say, and do not come with a
kind of common-sense penumbra that we often apply
to assertions without noticing that we do so
(except, for example, when writing code, or
technical specs, or logic, or law, or
philosophy). The basic point is that a Tarskian
interpretation may well have a lot of 'extra'
structure which is not explicitly sanctioned by
the axioms it 'models'. It can be more
complicated than is actually needed to specify
the truth of the axioms. It is not a
simplification or reduction of an axiomatic
ideal, as Charles suggested: on the contrary, it
may well be an exotic elaboration of such an
ideal. (08)
>
>I think it is less misleading to say that Tarskian models are more
>specific than the axioms, in that any given Tarski model pins down
>the truth-value of every sentence. But this specificity comes at the
>cost of assigning definite truth-values to sentences whose
>truth-value is left open by the axioms. (09)
Yes, exactly. And that is precisely what I meant
by using 'fidelity' in my reply to Charles, who
was speaking of fidelity of a 'model' to the
*axioms*. (010)
Pat (011)
PS. Good to hear your email voice again. (012)
>
>Kathy
>
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