Universal Properties of Lukasiewicz Consequence
Daniele Mundici · 2014
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 ele- mentary observables have small effects on compound observables. Con- tinuity 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. � Lukasiewicz implication has a well known characterization as the only continuous connective ⇒ satisfying the following conditions: (i) x ⇒ (y ⇒ z )= y ⇒ (x ⇒ z) and (ii) x ⇒ y =1 if fx ≤ y .T hen syntac- tic consequence can be defined purely algorithmically using the � wicz axioms and Modus Ponens. As discussed in this paper, to recover a strongly complete semantics one may use differential valuations.