The concept of evidence

Abstract

We construct an explicatum of the evidence concept, evidence logic (EL), which is conceptually antecedent to classical logic (CL), having applications in knowledge representation and knowledge processing areas of artificial intelligence (AI), and satisfying Quinean concerns by not going beyond the simplicity (viz. Boolean algebraic structure) of CL, while implementing two requirements often present in AI domains: (1) that “absence of evidence” may well not be “evidence of absence” and (2) evidence gradations. For each integer n≥2, let size n−1 evidence space be linear order En={i/(n−1): i=1,…,n−1}. For each n≥2 and each logical similarity type τ, evidence logic ELn, τ is equipped with both confirmatory and refutatory predicate symbols Rc and Rr for each τ(i)-ary predicate, as well as evidence space En of evidence annotations for atomic formulas, while added to a usual set of logical axioms are axioms which assure that “stronger evidence strictly entails weaker evidence”; models of ELn, τ are similarly equipped, providing annotated confirmatory and refutatory relations interpreting each τ(i)-ary predicate. Trivializing all refutatory predicates in ELn, τ yields CELn, τ, confirmatory evidence logic; also EL2, τ is viewed as absolute evidence logic AELτ, while CEL2, τ is both confirmatory and absolute and is exactly classical logic CLτ. Let μ be monadic, stipulating p propositions, k constants, and u unary predicates; let μ′ be functional, obtained by adding to μ the stipulation of one unary function; and let ν be undecidable, stipulating a finite number of predicates and functions including at least one predicate or function which is at least binary or at least two unary functions. For theory T, let BSA(T) be the Boolean sentence algebra of T, let BA(σ) be the Boolean algebra with ordered basis of order type σ, and let ≅ denote “recursive isomorphism.” Consequences of previous work yield the characterization of EL: (1) BSA(ELn, μ)≅BA(ω⋅n2p⋅∑ski⋅n2ui) and BSA(CELn, μ)≅BA(ω⋅np⋅∑ski⋅nui), where ∑τi=1 if k=0, and ski are Stirling numbers of the second kind; (2) BSA(ELn, μ′)≅BA((ωω⋅η0+η0)⋅η0) and BSA(CELn, μ′)≅BA((ωω⋅η0+η0)⋅η0); (3) BSA(ELn, ν)≅BSA(CL〈2〉) and BSA(CELn, ν)≅BSA(CL〈2〉). ©2000 John Wiley & Sons, Inc.