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Re: [ontolog-forum] Logic, Datalog and SQL

To: "[ontolog-forum] " <ontolog-forum@xxxxxxxxxxxxxxxx>, Charles D Turnitsa <CTurnits@xxxxxxx>, Chris Menzel <chris.menzel@xxxxxxxxx>
Cc: "[ontolog-forum] " <ontolog-forum@xxxxxxxxxxxxxxxx>
From: Kathryn Blackmond Laskey <klaskey@xxxxxxx>
Date: Fri, 09 Feb 2007 18:22:15 -0500
Message-id: <p06110435c1f2af167cd6@[]>
>  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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