> 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. (01)
That statement could be very misleading, because the referent of
"fidelity" is unclear. 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. (02)
Let me give an example to make things concrete. (03)
Consider the set of axioms:
All men are mortal.
Pat is a man.
John is a man.
Kathy is a woman. (04)
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. (05)
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. (06)
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. (07)
Kathy (08)
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