<p>Super paraorthomodular lattices (sp-orthomodular lattices) coincide with pseudo-Kleene lattices whose order determines, and it is fully determined by, the order of their maximal Kleene sub-algebras. It turns out that the (spectral) paraorthomodular lattice of effects over a separable Hilbert space can be considered as a prominent example of such. This work is aimed at investigating sp-orthomodular lattices from a logical perspective. First, we show that any sp-orthomodular lattice can be regarded as an algebra in the language of left-residuated <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(\ell \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>ℓ</mi> </math></EquationSource> </InlineEquation>-groupoids augmented with a binary operator/modality, and satisfying a “modalized” version of left-residuation. Therefore, structures we deal with and algebras, like e.g.&#xa0;residuated lattices, which have received great attention over the past years might be somehow regarded as “siblings”. Furthermore, we investigate the logic of order as well as two generalizations of Priest’s Logic of Paradox induced by sp-orthomodular lattices. A preliminary inquiry into their properties, some upshots about their relevance for logic and the foundation of quantum mechanics, as well as two Gentzen style calculi axiomatizing algebraizable Gentzen relations whose internal and external logics coincide with logics investigated in this paper will be outlined.</p>