1,720,988 research outputs found
Some Remarks on the Logic of Probabilistic Relevance
No full textIn 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
On Contextuality and Unsharp Quantum Logic
No full textIn this paper we provide a preliminary investigation of subclasses of bounded posets with antitone involution which are "pastings" of their maximal Kleene sub-lattices. Specifically, we introduce super-paraorthomodular lattices, namely paraothomodular lattices whose order determines, and it is fully determined by, the order of their maximal Kleene sub-algebras. It will turn out that the (spectral) paraorthomodular lattice of effects over a separable Hilbert space can be considered as a prominent example of such. Therefore, it arguably provides an algebraic/order theoretical rendering of complementarity phenomena between unsharp observables. A number of examples, properties and characterization theorems for structures we deal with will be outlined. For example, we prove a forbidden configuration theorem and we investigate the notion of commutativity for modular pseudo-Kleene lattices, examples of which are (spectral) paraorthomodular lattices of effects over finite-dimensional Hilbert spaces
Considerations on Everett J. Nelson’s connexive logic
No full textThis 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 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
Percorsi di logica
No full textLo scopo di Percorsi di Logica è guidare il lettore alla scoperta di alcuni tra i temi più importanti della logica formale classica. Il volume mira a esporre in modo rigoroso ed esauriente, agevolando la costruzione di percorsi tematici personalizzati, i concetti e i risultati di base impiegati in questa disciplina, senza trascurarne l’accessibilità da parte di un pubblico non specialista. A tal fine, il testo è corredato di una sezione riservata ai preliminari matematici e alle tecniche dimostrative più comuni. Il volume presenta la logica proposizionale e predicativa, considerandole da una prospettiva semantica e sintattica. Particolare attenzione è riservata alla loro metateoria. Percorsi di Logica è pensato per le esigenze dei nuovi corsi di laurea in discipline umanistiche e scientifiche. Tuttavia, il volume è accessibile anche ai lettori autodidatti che intendono approfondire autonomamente i temi classici e alcuni degli sviluppi più recenti della logica formale
The generalized orthomodularity property: configurations and pastings
No full textIn this paper, we consider a generalization of the notion of orthomodularity for posets to the concept of the generalized orthomodularity property (GO-property) by considering the LU-operators. This seemingly mild generalization of orthomodular posets and its order theoretical analysis yield rather strong application to effect algebras and orthomodular structures. Also, for several classes of orthoalgebras, the GO-property yields a completely order-theoretical characterization of the coherence law, and, in turn, of proper orthoalgebras
On Finch's conditions for the completion of orthomodular posets
No full textIn this paper, we aim at highlighting the significance of the Aand B- properties introduced by P.D. Finch in [16]. These conditions turn out to capture interesting structural features of lattices of closed subspaces of complete inner vector spaces. Moreover, we generalise them to the context of effect algebras, establishing a novel connection between quantum structures (orthomodular posets, orthoalgebras, effect algebras) arising from the logicoalgebraic approach to quantum mechanics
Connexive implications in Substructural Logics
Full text linkThis paper is devoted to the investigation of term-definable connexive implications in substructural logics with exchange and, on the semantical perspective, in sub-varieties of commutative residuated lattices (FLe-algebras). In particular, we inquire into sufficient and necessary conditions under which generalizations of the connexive implication-like operation defined in [6] for Heyting algebras still satisfy connexive theses. It will turn out that, in most cases, connexive principles are equivalent to the equational Glivenko property with respect to Boolean algebras. Furthermore, we provide some philosophical upshots like e.g., a discussion on the relevance of the above operation in relationship with G. Polya\u27s logic of plausible inference, and some characterization results on weak and strong connexivity
Editorial Introduction
No full textAfter providing an overview of the algebraic investigations into substructural logics in a historical perspective, with a special focus on their relationships with quantum logics, we summarise the contents of the subsequent chapters of this volume
- …
