By tradition, the subject of quantum logic is the lattices of closed subspaces of Hilbert spaces or, more abstractly, orthomodular lattices. \textit{D. J. Foulis} showed in Can. J. Math. 17, 40-51 (1965; Zbl 0146.029) that an orthomodular lattice can be coordinatized by a Baer \(*\)- semigroup. On the other hand, \textit{C. J. Mulvey} [Rend. Circ. Math. Palermo, II. Ser., Suppl. 12, 99-104 (1986; Zbl 0633.46065)] has introduced the concept of a quantale -- a complete lattice with an additional non-commutative operation preserving suprema. Quantales are known as possible models of linear logic, proposed by \textit{J.-Y. Girard} [J. Theor. Comput. Sci. 50, 1-102 (1987; Zbl 0625.03037)]. (By the way, linear logic is also related to calculi of binary relations and to relevant logic.) The author shows that various recently considered kinds of quantales can be viewed as involution semigroups, and tries to unify on this ground the concepts of an orthomodular lattice and a quantale, as well as to clarify the relations between quantum and linear logics.