Considering the inference rules in generalized logics, J.C. Abbott arrives to the notion of orthoimplication algebra (see Abbott (1970) and Abbott (Stud. Logica. 2:173–177, XXXV)). We show that when one enriches the Abbott orthoimplication algebra with a falsity symbol and a natural\documentclass[12pt]{minimal}
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				\begin{document}$$\mathbb {XOR}$$\end{document}XOR-type operation, one obtains an orthomodular difference lattice as an enriched quantum logic (see Matoušek (Algebra Univers. 60:185–215, 2009)). Moreover, we find that these two structures endowed with the natural morphisms are categorically equivalent. We also show how one can introduce the notion of a state in the Abbott\documentclass[12pt]{minimal}
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				\begin{document}$$\mathbb {XOR}$$\end{document}XORalgebras strenghtening thus the relevance of these algebras to quantum theories.