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LFCS Seminar: Jamie Gabbay

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Stone duality for first-order logic (a nominal approach)

What
  • LFCS Seminar
When Dec 06, 2011
from 04:00 PM to 05:00 PM
Where IF 4.31-4.33
Contact Name
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Abstract:

 

I will present a brand new paper in which we prove a Stone duality between a nominal algebra axiomatisation of first-order logic (FOL-algebras), and a notion of topological space (forall-Stone spaces).

 

We are familiar with Boolean algebras being sets with conjunction and negation actions satisfying certain axioms. We are also familiar with the fact that powersets naturally have a Boolean algebra structure, given by interpreting conjunction as sets intersection and negation as sets complement.

 

Using nominal techniques we can axiomatise substitution and first-order logic, so we can try to extend the Stone duality theorem from Boolean algebras to FOL-algebras, and to some class of nominal topological spaces.

 

If we can answer this question, then we obtain a nominal representation of first-order logic without Tarski-style valuations. A variable populates the denotation directly as nominal atoms, and substitution acts on variables directly in that denotation. This is a very different view of variables in logical meaning than the one which the reader is most likely accustomed to.

 

The proofs contain a wealth of interesting structure and they give a sense in which variables really can directly inhabit denotations in logic and topology. The paper will be online by the time of my talk and will be found at my papers page. See also a precursor paper on Stone duality for the NEW quantifier.
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