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LOCATION:60 Evans Hall
SEQUENCE:0
DTEND:20090925T180000
DTSTART:20090925T161000
UID:philosophy.berkeley.edu:events:575
DTSTAMP:20091124T190103
DESCRIPTION:Orlov first [1928] and Gödel later [1933] pointed out the conne
 ction between the Lewis System S4 and Intuitionistic Logic.  McKinsey and T
 arski gave an algebraic formulation and proved completeness theorems for pr
 opositional systems using as models topological spaces with the interior op
 erator corresponding to the necessitation modality.  Earlier\, Tarski and S
 tone had each shown that the lattice of open subsets of a topological space
  models intuitionistic propositional logic.  Expanding on a suggestion of M
 ostowski about interpreting quantifiers\, Rasiowa and Sikorski used the top
 ological models to model many first-order logics. After the advent of Solov
 ey'’s recasting of Cohen'’s independence proofs as using Boolean-valued mod
 els\, topological models for modal higher- order logic were studied by Gall
 in and others.  For Boolean-valued logic\, the complete Boolean algebra M =
  Meas([0\,1])/Null of measurable subsets of the unit interval modulo sets o
 f measure zero gives every proposition a probability.  Perhaps not as well 
 known is the observation that the measure algebra also carries a nontrivial
  S4 modality defined with the aid of the sublattice Open([0\,1])/Null of op
 en sets modulo null sets.  This sublattice is closed under arbitrary joins 
 and finite meets in the measure algebra\, but it is not the whole of the me
 asure algebra.  By working by analogy to the construction of Boolean-valued
  models for ZF\, we can construct over M a model for a modal ZF (MZF) where
  membership and equality predicates have interesting and natural modal prop
 erties.  In such a universe the real numbers correspond to random variables
 \, and --- following a suggestion of Alex Simpson (Edinburgh) --- there is 
 also an well motivated modeling of random reals. A modal set theory\, howev
 er\, requires a reexamination of comprehension principles\, and much work r
 emains to be done to organize methods of proof to take account of the new d
 istinctions encountered.  It is also possible to recast well known theorems
  of Ergodic Theory as principles about this modal universe\, and the questi
 on to be considered is whether the new perspective can lead to some new res
 ults.\n
SUMMARY:Logic Colloquium\nDana S. Scott\nMixing Modality and Probability
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