Online Appendix: Fuzzy Equilibrium Logic: Declarative Problem Solving in Continuous Domains

Let α1 be a formula from N5 logic, and let α2 be the formula that is obtained from α1 by replacing conjunction, disjunction and implication by respectively a t-norm ⊗, a t-conorm ⊕ and an implicator →. Noting that V − 2 (w, a) ∈ {0, 1} and V + 2 (w, a) ∈ {0, 1} for all a ∈ At and w ∈ {h, t}, it follows easily from the definitions (17)–(23) that V1(w,α1) = 1 iff V2(w,α2) = [1, 1], V1(w,α1) = 1 iff V2(w,α2) = [1, 1], V1(w,α1) = −1 iff V2(w,α2) = [0, 0] and V1(w,α1) = 0 iff V2(w,α2) = [0, 1] (using structural induction). From this, we immediately find that V1 is an N5 model of Θ1 iff V2 is a fuzzy N5 model of Θ2.