On poly-logistic natural-deduction for finitely-valued propositional logics 1 Nissim Francez and Michel Kaminski Computer Science dept., the Technion-IIT, Haifa, Israel {francez@cs.technion.ac.il, kaminski@cs.technion.ac.il} Abstract The paper presents a systematic construction of natural-deduction proof-systems for multi-valued log- ics from the truth-tables for the connectives. The construction is based on poly-sequents of the form Γ 1 |···|Γ n :Δ 1 |···|Δ n ,n ≥ 2, improving on a previous approach by Baaz et. al. Poly-sequents al- low to speak explicitly about the truth-value of a formula, and have in I/E-rules both assumptions and conclusion that have any truth-value. Soundness and strong completeness are proved. The generality of the construction is exemplified by retrieving within the constructed ND-system a host of well-known ND-systems for multi-valued logics. 1. Introduction Our point of departure is the construction of a natural-deduction (ND) proof-system for many-valued logics out of the truth-tables for the connectives, as presented in Baaz et. al. [1]. In contrast to [1], our construction relies directly on the truth-tables only, while Baaz et. al. rely on consequences of the truth-table which are not easy to establish in general (see [1]) as it appeals to all interpretations. This 5 reliance on consequences of the truth-table is then shown to be eliminable. Our central purpose is to present simpler and more intuitive construction, based on a more advantageous extension of Gentzen’s logistic-ND to poly-logistic ND, an extension referred to as poly-sequents, where both contexts of a sequent are poly-contexts, a structure originating from sequent calculi. The central idea of using poly-contexts is to allow for different kinds of assumptions and conclusions in an ND-system, 10 having arbitrary truth-values. In more detail, our main aim is to devise a uniform construction of natural-deduction proof-systems, denoted by N n , for any n-valued logic, n ≥ 2, constructed in a systematic way (only!) from the truth- tables for the connectives in the object language. In particular, unlike [1], the construction is direct, without an intermediate passage through a mediating sequent-calculus. This directness of construction 15 is facilitated by the following two characteristics of poly-sequents: 1. While only disjunction is present in the meta-language defining satisfaction in Baaz et. al., we have in the meta-language all the classical connectives: disjunction, conjunction, implication and (in a certain form) negation. 2. Our poly-sequents provide a clear separation of assumptions and conclusions, thereby being more 20 faithful to the general concept of natural deduction and to Gentzen’s original sequents. 1 A talk based on this paper was presented at ISRALOG17, Haifa, October 2017. Preprint submitted to Journal of Applied Logic October 2, 2018