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Logic for rough truth. (English) Zbl 1094.03015
Summary: Z. Pawlak had proposed the notion of rough truth in 1987 [Bull. Pol. Acad. Sci., Tech. Sci. 35, 253–258 (1987; Zbl 0645.03019)]. The present article takes a fresh look at this “soft” truth, and presents a formal system ${ℒ}_{ℛ}$ that is shown to be sound and complete with respect to a semantics determined by this notion. ${ℒ}_{ℛ}$ is based on the modal logic S5. Notable are the rough consequence relation defining ${ℒ}_{ℛ}$, and rough consistency (introduced in [M. K. Chakraborty and M. Banerjee, Bull. Pol. Acad. Sci., Math. 41, 299–304 (1993; Zbl 0795.03036)]), used to prove the completeness result. The former is defined in order to be able to derive roughly true propositions from roughly true premisses in an information system. The motivation for the latter stems from the observation that a proposition and its negation may well be roughly true together. A characterization of ${ℒ}_{ℛ}$-consequence shows that the paraconsistent discussive logic $J$ of Jaśkowski is equivalent to ${ℒ}_{ℛ}$. So, ${ℒ}_{ℛ}$, developed from a totally independent angle, viz. that of rough set theory, gives an alternative formulation to this well-studied logic. It is further observed that pre-rough logic [M. Banerjee and M. K. Chakraborty, Ann. Soc. Math., Ser. IV, Fundam. Inf. 28, 211–221 (1996; Zbl 0864.03041)] and 3-valued Łukasiewicz logic are also embeddable into ${ℒ}_{ℛ}$.
##### MSC:
 03B60 Other nonclassical logic 03B53 Paraconsistent logics 68T37 Reasoning under uncertainty