The concept of evidence

We construct an explicatum of the evidence concept, evidence logic (EL), wh ich is conceptually antecedent to classical logic (CL), having applications in knowledge representation and knowledge processing areas of artificial i ntelligence (AI), and satisfying Quinean concerns by not going beyond the s implicity (viz. Boolean algebraic structure) of CL, while implementing two requirements often present in Al domains: (1) that "absence of evidence" ma y well not be "evidence of absence" and (2) evidence gradations. For each i nteger n greater than or equal to 2, let size n - 1 evidence space be linea r order E-n = {i/(n - 1): i = 1,...,n - 1}. For each n greater than or equa l to 2 and each logical similarity type tau, evidence logic ELn, tau is equ ipped with both confirmatory and refutatory predicate symbols R-c and R-r f or each tau(i)-ary predicate, as well as evidence space E-n of evidence ann otations for atomic formulas, while added to a usual set of logical axioms are axioms which assure that "stronger evidence strictly entails weaker evi dence"; models of ELn, tau are similarly equipped, providing annotated conf irmatory and refutatory relations interpreting each tau(i)-ary predicate. T rivializing all refutatory predicates in ELn, tau yields CELn, tau, confirm atory evidence logic; also EL2, tau is viewed as absolute evidence logic AE L(tau), while CEL2, tau is both confirmatory and absolute and is exactly cl assical logic CLtau. Let mu be monadic, stipulating p propositions, k const ants, and u unary predicates; let mu' be functional, obtained by adding to mu the stipulation of one unary function; and let nu be undecidable, stipul ating a finite number of predicates and functions including at least one pr edicate or function which is at least binary or at least two unary function s. For theory T, let BSA(T) be the Boolean sentence algebra of T, let BA(si gma) be the Boolean algebra with ordered basis of order type sigma, and let congruent to denote "recursive isomorphism." Consequences of previous work yield the characterization of EL: (1) BSA(ELn, mu) congruent to BA(omega(n2u) . n(2p) . Sigma(i = 1)(k) s(ki) . n(2ui)) and BSA(CELn, mu) congruent to BA(omega(nu) . n(p) . Sigma(i = 1)(k) s(ki) . n( ui)), where Sigma tau(i) = 1 if k = 0, and s(ki) are Stirling numbers of th e second kind; (2) BSA(ELn, mu') congruent to BA((omega(omega) . eta(0) + eta(0)) . eta(0) ) and BSA(CELn, mu') congruent to BA((omega(omega) . eta(0) + eta(0)) . eta(0)); (3) BSA(ELn, nu) congruent to BSA(CL[2]) and BSA(CELn, nu) congruent to BSA (CL[2]). (C) 2000 John Wiley & Sons, Inc.