Boolean Threshold Representations of Finite and Rational Łukasiewicz Truth Values: Minimality, Compositional Translations, and Direct Limits

Can many-valued truth be constructed from classical Boolean truth? For the finite Łukasiewicz chain Lₙ = {0, 1/n, …, 1}, the answer is yes. Each truth value k/n is uniquely determined by the n Boolean answers to the threshold questions "is x ≥ i/n?", and these answers form a monotone Boolean string—the thermometer encoding of k/n. The collection of all such strings, with coordinatewise operations for the lattice connectives and with complemented reversal and threshold convolution for negation and truncated sum, is isomorphic to Lₙ as an MV-algebra. Among all order-preserving representations of Lₙ inside a Boolean cube, this encoding is dimensionally minimal; moreover, every minimal representation is obtained from it by permuting the Boolean coordinates. The isomorphism extends compositionally to formulas, yielding one classical Boolean formula for each truth threshold and a reduction of finite-valued validity to classical validity. The construction extends to the rational MV-algebra ℚ ∩ [0,1]. We organise the finite profile algebras into a directed system indexed by divisibility: when m divides n, a refinement map repeats each threshold bit n/m times, and these maps are MV-homomorphisms. The direct limit of the system is isomorphic to ℚ ∩ [0,1], so a rational truth value is faithfully represented by an equivalence class of finite Boolean threshold profiles at compatible resolutions. As n grows, the Boolean construction approximates the standard MV-algebra [0,1] with uniform error O(1/n), connecting the discrete Boolean picture to the continuous case.