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On Some Constructive Fuzzy Logics

### Andrew Lewis-Smith

We present a generalization of the Kripke semantics of intuitionistic
logic (IL) appropriate for intuitionistic Łukasiewicz logic IŁL – a
logic in the intersection between IL and (classical) Łukasiewicz logic.
This generalised Kripke semantics is based on the poset-sum construction
of Bova and Montagna, developed to show the decidability (and PSPACE
completeness) of the quasi-equational theory of commutative, integral
and bounded GBL-algebras.

The main idea is that "w forces ψ", which for IL is a relation between
worlds w and formulas ψ, i.e. (w forces ψ) ∈ Bool, becomes a function
taking values on the unit interval (w forces ψ) ∈ [0, 1]. An appropriate
monotonicity restriction (which we call sloping functions) needs to be
put on such functions in order to ensure soundness and completeness of
the semantics. We are currently extending these insights to
Intuitionistic Affine Logic. This is based on joint work of A.
Lewis-Smith, P. Oliva, and E. Robinson.