Re: Exponentials in linear logic

Answer to Clemens H. CAP (mail of August 1st)

The question of duality of ! and ? is exactly the same as the  
question of the unicity of !. By nature ! cannot be unique :

1st reason : ! controls infinity in logic ; we get usual infinity by  
means of the familiar rules, but another choice of rules can yield  
weaker notions of infinity, eg polytime. In this respect, there is no  
a priori unicity of !, and this is far from being a weakness...

2nd reason : just write twice the same set of rules (eg in an  
intuitionistic style) for ! and -say- $ ; it will be impossible to  
prove the equivalence between !A and $A. If we want to prove $A -o !A  
then we must allow contexts $Gamma,!Delta in the !S-rule. In general  
we can handle several !, partially ordered, etc. But one should use  
this latter possibility with extreme care : multimodal logics are  
close relatives of non-monotonic "logics". 


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