Raffaele Mascella

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 $$\ell $$ -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. 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.

This work explores Everett John Nelson’s connexive logic, outlined in his PhD thesis and partially summarized in his 1930 paper _Intensional Relations_, which is obtained by extending the system $$\mathsf{NL}\,\,$$ reconstructed by E. Mares and F. Paoli with a weak conjunction elimination rule explicitly assumed in the former but not in the latter. After a preliminary analysis of Nelson’s philosophical ideas, we provide an algebraic-relational semantics for his logic and we investigate possible extensions thereof which are able to cope with Nelson’s ideas with much more accuracy than the original system. For example, we will inquire into extensions whose algebraic-relational models are endowed with irreflexive incompatibility relations, or determine a “weakly” transitive entailment. Such an investigation will allow us to establish relationships between some of the trademarks of Nelson’s thought and concepts of prominent importance for connexive logic, as e.g. Kapsner’s strong connexivity and superconnexivity, as well as between the algebraic-relational semantics of Nelsonian logics and ordered structures that have gained great attention over the past years, namely partially ordered involutive residuate groupoids and (non-orthomodular) orthoposets.

Projects concerning the participation of citizens to the scientific research have become very popular in recent years. However, the literature seems not offer much foundational investigations into the epistemological conditions that make citizens’ contribution to science a fruitful and rigorous activity. Research institutions and citizens, understood as entities bearing generally distinct requests, practices and goals, knowledge and skills, are totally misaligned. In order to participate to collaborative research, citizens need an epistemic and methodological literacy to overcome common sense ideas, towards an ever adequate vision of phenomena under investigation. Besides understanding the social advantages introduced by the participatory dimension, it is therefore necessary to analyze the related epistemological problems, to understand to what extent and in what respect to the participation of citizens to research processes might be regarded as truly advantageous from an epistemological perspective.

In this paper we deepen some aspects of the statistical approach to relevance by providing logics for the syntactical treatment of probabilistic relevance relations. Specifically, we define conservative expansions of Classical Logic endowed with a ternary connective ⇝ - indeed, a constrained material implication - whose intuitive reading is “x materially implies y and it is relevant to y under the evidence z”. In turn, this ensures the definability of a formula in three-variables R(x, z, y) which is the representative of relevance in the object language. We outline the algebraic semantics of such logics, and we apply the acquired machinery to investigate some termdefined weakly connexive implications with some intuitive appeal. As a consequence, a further motivation of (weakly) connexive principles in terms of relevance and background assumptions obtains.

All logic instruments and tools in possession of, and used by researchers are generally considered as the results of bivalent logic. A common error to people interested in science is that, usually, they don’t known with certainty which things are true and which are false. But they are sure that things are true or false. No ways in the middle. The fuzzy principle asserts that this is completely a question of measure. Fuzziness is the opposite concept to bivalency, while fuzzy quality means polyvalency. Fuzzy logic is able to manage uncertain knowledge and, more precisely, to manage those rules difficult to formalize. In many cases, such rules haven’t got, or cannot have, the necessary precision, required for a whole formalization