Scientific Explanation and Modified Semantic Tableaux

Abstract

Standard semantic tableaux method and a modification of its δ-rule, due to [Boolos, 1984] and [Díaz, 1993], allows us to obtain δ′-tableaux and apply that to abduction problem, paying attention to some versions given in [Aliseda, 1997]. Our approach of abduction in semantic tableaux faces up to the problem of the existence of infinite branches. Defined Cn, a basic logical operation, a new operation Cn* is obtained. An abduction problem 〈θ, ϕ〉 can be seen as the problem of choosing the appropriated sentence of the set Ab(〈θ, ϕ〉), defined from Cn, or, taking into account Cn*, Ab(〈θ,ϕ〉). The main results are: (i) a (finite) set Γ of L-sentences is n-satisfiable iff the δ′-tableau of Γ has an open branch in which only n constants occur; (ii) if a finite set Γ of L-sentences is satisfiable, then the δ′-tableau provides a minimal interpretation that satisfies Γ; (iii) for any abduction problem 〈θ,ϕ〉 such that θ is n-satisfiable, if there is a δ′-tableau of θ ∪ ¬ϕ different from its standard tableau, then there is a solution a for 〈θ,ϕ〉; (iv) given an abduction problem 〈θ,ϕ〉 such that θ is n-satisfiable, if a is a solution and θ ∪ a is consistent, then a ∈ Ab*(〈θ,ϕ〉) and it is an explanatory solution (with respect to Ab*); (v) given an abduction problem 〈θ,ϕ〉 such that θ is n-satisfiable, if a is an explanatory solution, then θ ∈ Th*(θ ∪ a). They are Theorem 3, Corollary 4, Theorems 14 and 15 and Corollary 17, respectively. Several examples to illustrate that are given.