Abstract

Boolean logic deals with {0, 1}-observables and yes–no events, as many-valued logic does for continuous ones. Since every measurement has an error, continuity ensures that small measurement errors on elementary observables have small effects on compound observables. Continuity is irrelevant for {0, 1}-observables. Functional completeness no longer holds when n-ary connectives are understood as [0, 1]-valued maps defined on [0, 1] n . So one must envisage suitable selection criteria for [0, 1]-connectives. Łukasiewicz implication has a well known characterization as the only continuous connective \ satisfying the following conditions: \= y\Rightarrow}\) and \ . Then syntactic consequence can be defined purely algorithmically using the Łukasiewicz axioms and Modus Ponens. As discussed in this paper, to recover a strongly complete semantics one may use differential valuations