Interval-valued logics
Bart Van Gasse, Chris Cornelis, Glad Deschrijver, Etienne E. Kerre · Ghent University Academic Bibliography (Ghent University) · 2010
In [5], the authors introduced Interval-Valued Monoidal Logic (IVML). Its language is the language of Hohle’s Monoidal Logic (ML,[3]) enriched with two unary connectives and ♦, and a constant u. Its axioms are those of ML plus 15 new ones describing the behaviour of , ♦ and u. The deduction rules are modus ponens (MP, from φ and φ→ ψ infer ψ), generalization (G, from φ infer φ) and monotonicity of ♦ (M♦, from φ→ ψ infer ♦φ→ ♦ψ). In some way, ML can be seen as a special case of IVML. Indeed, it can be proven [7] that for all sets T ∪ {φ} of ML-formulae, T ⊢ML φ iff {χ |χ ∈ T} ⊢IV ML φ ′ (where ψ is the IVML-formula obtained by substituting p in ψ for every proposition variable p in ψ). IVML is sound and complete with respect to the variety of triangle algebras. These are algebraic structures that describe interval-valued residuated lattices (IVRLs): (closed) interval-valued bounded lattices endowed with a product and implication that satisfy the residuation principle, such that the sublattice of exact intervals (i.e., intervals consisting of one element) is closed under product and implication. Table 1 shows to which mappings and interval in an IVRL the connectives and constant in IVML correspond. Also the notations in triangle algebras are included, in the second column. The soundness and completeness