[FOM] On models of Intuitionistic Logic and SILs

[A. Mani]

Hello,
        A consequence of a recent result of mine in ["Di-algebraic Semantics 
of Logics" Fund. Informat. 70(4) 2006, 333-350] is that 
#"Some models of intuitionistic propositional logic (IPL) are representable as 
models of logics with a pair of classical consequences  and some models of 
super-intuitionistic propositional logics are also so representable". 

The result # can be extended as a semantics IPL if we allow multiple classical 
consequences (getting it with two classical consequences is open).  This is 
possible by representing heyting algebras as 'convex amalgams' of boolean 
algebras (*).  This does not relate easily to Kripke semantics.

So that it would be admissible to reason (about propositions) 
intuitionistically with multiple classical consequence relations set in a way 
(*) and in many cases with two classical consequence relations alone. 

In the light of the above, the typical Intuitionist argument of rejecting the 
law of excluded middle as an admissible law in any context requires 
modification.  The new intuitionist position may be that the "law of excluded 
middle" must not be used unless the context permits it. The 'context' might 
actually be definable to the extent necessary.  The other way of seeing this 
from a model-theoretic viewpoint can be that  the LEM is an 
intuitionistically coherent rule to X extent in class Y of models of the 
logic- but this requires some development.  From the philosophical viewpoint 
'superintuitionism' is all part of intuitionism, but there are of course no 
such concepts for LEM in the models of superintuitionistic logics. 
Comments please.

All this is from a classical perspective of model theory. So it cannot be 
wrong from an intuitionistic perspective and some weaker versions of this must 
be correct. I mean this from the point of view of the restricted concept 
lattices associated with classical first order logic and intuitionistic first 
order logic respectively. There exists a natural many-valued correspondence, 
apart from the embedding.    

My other question is : Are there direct ways of transforming classical models 
into intuitionistic models ? 


A. Mani
Member, Cal. Math. Soc

PS. Thanks to the FOM editor for useful suggestions for improving this mail.  

     
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