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Neighborhood Semantics for the Exponentials
Folks, the abstract for an extended abstract for a paper I intend to submit
to a conference appears below. The Postscript or DVI file can be had by
anonymous ftp at ftp.cica.indiana.edu. Go to the directory
/pub/gtall/neighborhood. The Kripke model paper of Mike Dunn and myself is
in the other directory, /pub/gtall/relation. Send mail to gtall@cs.indiana.edu
if you have any problems getting the abstract or with the content.
The full paper will include the non-commutative, non-associative case. Space
considerations prevented me from including this here. The full paper
is symmetric with respect to <> (Girard's ?) working with + and -<, >- (the
cotensor product and the coimplication operators) in the same manner that
[] works with o and ->,<- (tensor product and implication operators). I didn't
do anything with the intensional analogues of Sheffer's stroke and dagger but
I may later on.
All axioms are added conservatively. One need not start with a negation or
commutation or association axioms.
Shortly, I put up for anonymous ftp a paper on the categorical duality
between the Kripke models and the algebras for LL. But now that I have the
exponentials done, I may want to go back and extend that work.
Gerry
\centerline{\bf Abstract}
{\narrower
This paper presents a neighborhood semantics for the ``exponentials'' of
Girard's Linear Logic.
In Allwein and Dunn, {\it Kripke Models for Linear Logic}, the lattice
representation work of Alasdair Urquhart, {\it A topological representation
theory for lattices}, is used to yield a Kripke style semantics for
propositional Linear Logic without the exponentials.
The theory presented there treated logics much weaker than Linear Logic and
allows one to add axioms to the Hilbert style system for the weakest Linear
Logic until the full propositional form is reached with soundness and
completeness following at each addition. This paper extends that theory to
cover the exponentials for Linear Logic using neighborhoods around each world
such as presented in Chellas' {\bf Modal Logic}. Familiarity with the Allwein
and Dunn paper will help to understand this paper but is not absolutely
necessary.
\smallbreak}