Consequence Relations with Real Truth Values

Abstract

Syntax and semantics in Łukasiewicz infinite-valued sentential logic Ł are harmonized by revising the Bolzano-Tarski paradigm of “semantic consequence,” according to which, $\theta$ follows from $\Theta$ iff every valuation v that satisfies all formulas in $\Theta$ also satisfies $\theta .$ For $\theta$ to be a consequence of $\Theta$, we also require that any infinitesimal perturbation of v that preserves the truth of all formulas of $\Theta$ also preserves the truth of $\theta$. An elementary characterization of Łukasiewicz implication shows that the Łukasiewicz axiom $( ( X \rightarrow Y ) \rightarrow Y ) \rightarrow ( (Y \rightarrow X) \rightarrow X )$ guarantees the continuity and the piecewise linearity of the implication operation $\rightarrow$, an appropriate fault-tolerance property of any logic of ${{\,\mathrm{[0,1]}\,}}$-valued observables. The directional derivability of the functions coded by all $\psi \in \Theta$ and by $\theta$ then provides a quantitative formulation of our refinement of Bolzano-Tarski consequence, which turns out to coincide with the time-honored syntactic Ł-consequence.