http://hdl.handle.net/11089/58258 2026-05-01T11:34:36Z http://hdl.handle.net/11089/58262 Classical Logic, Uniformity, and Weak Excluded Middle in Non-Monotonic Proof-Theoretic Semantics Piccolomini d’Aragona, Antonio Non-monotonic base-extension semantics (nB-eS), a kind of non-monotonic proof-theoretic semantics (nPTS), is known to validate classical logic when its meta-logic is classical. Schroeder-Heister has remarked that classical meta-logic is as problematic for the project of modelling intuitionistic logic, as an intuitionistic proof of incompleteness would be. It may be unclear, though, whether Schroeder-Heister’s remark holds for non-monotonic proof-theoretic validity (nP-tV) as well, i.e., for Prawitz’s original version of nPTS. We only know that, with classical meta-logic again, classical logic is sound over a variant of nP-tV, which I shall call liberal non-monotonic proof-theoretic validity (LnP-tV). The latter, in turn, differs from nP-tV in that reductions for the rewriting of proof-structures are not required to be uniform. After drawing attention to a number of divergences between nB-eS, nP-tV and LnP-tV, I show that Schroeder-Heister’s remark might after all apply to nP-tV too. In particular, Weak Excluded Middle (WEM) is logically valid via uniform reductions (with a meta-logic which is non-intuitionistic, but non-classical either). 2026-04-24T00:00:00Z http://hdl.handle.net/11089/58263 Bridging Classical and Modern Approaches to Thales' Theorem Błaszczyk, Piotr; Petiurenko, Anna In this paper, we reconstruct Euclid’s theory of similar triangles, as developed in Book VI of the Elements, along with its 20th-century counterparts, formulated within the systems of Hilbert, Birkhoff, Borsuk and Szmielew, Millman and Parker, as well as Hartshorne. In the final sections, we present recent developments concerning non-Archimedean fields and mechanized proofs.Thales’ theorem (VI.2) serves as the reference point in our comparisons. It forms the basis of Euclid’s system and follows from VI.1 – the only proposition within the theory of similar triangles that explicitly applies the definition of proportion.Instead of the ancient proportion, modern systems adopt the arithmetic of line segments or real numbers. Accordingly, they adopt other propositions from Euclid’s Book VI, such as VI.4, VI.6, or VI.9, as a basis.In §10, we present a system that, while meeting modern criteria of rigor, reconstructs Euclid’s theory and mimics its deductive structure, beginning with VI.1. This system extends to automated proofs of Euclid’s propositions from Book VI.Systems relying on real numbers provide the foundation for trigonometry as applied in modern mathematics. In §9, we prove Thales’ theorem in geometry over the hyperreal numbers. Just as Hilbert managed to prove Thales’ theorem without referencing the Archimedean axiom, so do we by applying the arithmetic of the non-Archimedean field of hyperreal numbers. 2026-04-24T00:00:00Z http://hdl.handle.net/11089/58261 An Elementary Proof of the Characterization Theorem for Conjunctive Multiple-Conclusion Consequence Relations Taşdelen, İskender We give a characterization theorem for multiple-conclusion consequence relations with the conjunctive reading of conclusions. As in the case of disjunctive multiple-conclusion consequence relations, we define consequence relations in terms of sets of two-set partitions of formulae. We see that a binary relation between sets of formulae is a conjunctive multiple-conclusion consequence relation if it is closed under the properties of inclusion, transitivity and reducibility. To prove this result we use only the definition and some basic properties of conjunctive multiple-conclusion consequence relations. 2026-04-15T00:00:00Z http://hdl.handle.net/11089/58259 The Amalgamation Property in the Variety of Regular Double Stone Algebras: A Constructive View Ledda, Antonio; Sankappanavar, Hanamantagouda P.; Vergottini, Gandolfo In this paper we give a constructive proof that the variety of Boolean algebras has the strong amalgamation property by describing constructively the strong amalgams in the variety. Then, capitalizing on this construction, we investigate several forms of amalgamation, such as the strong amalgamation property and Maksimova super-amalgamation for the varieties of regular double Stone algebras and centered regular double Stone algebras. In fact, we prove that the amalgamation property holds for the variety RDS. Then, we introduce the variety RDSk of centered regular double Stone algebras and prove that RDSk enjoys the strong amalgamation property. It is also proved that the varieties of Boolean algebras and centered regular double Stone algebras have the super-amalgamation property. We close the paper by providing a number of concrete examples and applications to illustrate the theory developed in the paper. 2026-04-24T00:00:00Z