This paper investigates a non-associative generalization, more exactly, a weak t-associative (wta) generalization, of continuous t-norm-based logics. First, the wta-uninorm logic WAtBL and its axiomatic extensions WAtŁ, WAtΠ, and WAtG are introduced as [0, e]-continuous wta-uninorm-based analogues of the most famous continuous t-norm-based logics, BL (Basic fuzzy logic), Ł (Łukasiewicz logic), Π (Product logic), and G (Gödel logic), respectively. Their corresponding algebraic structures and algebraic completeness results are considered. Next, completeness with respect to the algebras whose lattice reduct is [0, 1], known as standard completeness, is established for these systems by a construction in the style of Jenei–Montagna.
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