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.