http://hdl.handle.net/11089/49155 2026-04-05T12:52:15Z 2026-04-05T12:52:15Z Adlešić, Tin Čačić, Vedran http://hdl.handle.net/11089/49162 2024-01-05T02:18:04Z 2023-09-28T00:00:00Z

The Cardinal Squaring Principle and an Alternative Axiomatization of NFU Adlešić, Tin; Čačić, Vedran In this paper, we rigorously prove the existence of type-level ordered pairs in Quine’s New Foundations with atoms, augmented by the axiom of infinity and the axiom of choice (NFU + Inf + AC). The proof uses the cardinal squaring principle; more precisely, its instance for the (infinite) universe (VCSP), which is a theorem of NFU + Inf + AC. Therefore, we have a justification for proposing a new axiomatic extension of NFU, in order to obtain type-level ordered pairs almost from the beginning. This axiomatic extension is NFU + Inf + AC + VCSP, which is equivalent to NFU + Inf + AC, but easier to reason about.

2023-09-28T00:00:00Z Coniglio, Marcelo Esteban de Toledo, Guilherme Vicentin http://hdl.handle.net/11089/49161 2024-01-05T02:18:00Z 2023-08-16T00:00:00Z

A Category of Ordered Algebras Equivalent to the Category of Multialgebras Coniglio, Marcelo Esteban; de Toledo, Guilherme Vicentin It is well known that there is a correspondence between sets and complete, atomic Boolean algebras (\(\textit{CABA}\)s) taking a set to its power-set and, conversely, a complete, atomic Boolean algebra to its set of atomic elements. Of course, such a correspondence induces an equivalence between the opposite category of \(\textbf{Set}\) and the category of \(\textit{CABA}\)s.We modify this result by taking multialgebras over a signature \(\Sigma\), specifically those whose non-deterministic operations cannot return the empty-set, to \(\textit{CABA}\)s with their zero element removed (which we call a \(\textit{bottomless Boolean algebra}\)) equipped with a structure of \(\Sigma\)-algebra compatible with its order (that we call \(\textit{ord-algebras}\)). Conversely, an ord-algebra over \(\Sigma\) is taken to its set of atomic elements equipped with a structure of multialgebra over \(\Sigma\). This leads to an equivalence between the category of \(\Sigma\)-multialgebras and the category of ord-algebras over \(\Sigma\).The intuition, here, is that if one wishes to do so, non-determinism may be replaced by a sufficiently rich ordering of the underlying structures.

2023-08-16T00:00:00Z Martini, Simone Masini, Andrea Zorzi, Margherita http://hdl.handle.net/11089/49159 2024-01-05T02:18:01Z 2023-09-25T00:00:00Z

Cut Elimination for Extended Sequent Calculi Martini, Simone; Masini, Andrea; Zorzi, Margherita We present a syntactical cut-elimination proof for an extended sequent calculus covering the classical modal logics in the \(\mathsf{K}\), \(\mathsf{D}\), \(\mathsf{T}\), \(\mathsf{K4}\), \(\mathsf{D4}\) and \(\mathsf{S4}\) spectrum. We design the systems uniformly since they all share the same set of rules. Different logics are obtained by “tuning” a single parameter, namely a constraint on the applicability of the cut rule and on the (left and right, respectively) rules for \(\Box\) and \(\Diamond\). Starting points for this research are 2-sequents and indexed-based calculi (sequents and tableaux). By extending and modifying existing proposals, we show how to achieve a syntactical proof of the cut-elimination theorem that is as close as possible to the one for first-order classical logic. In doing this, we implicitly show how small is the proof-theoretical distance between classical logic and the systems under consideration.

2023-09-25T00:00:00Z Walendziak, Andrzej http://hdl.handle.net/11089/49160 2024-01-05T02:18:02Z 2023-09-25T00:00:00Z

On Implicative and Positive Implicative GE Algebras Walendziak, Andrzej GE algebras (generalized exchange algebras), transitive GE algebras (tGE algebras, for short) and aGE algebras (that is, GE algebrasverifying the antisymmetry) are a generalization of Hilbert algebras. Here some properties and characterizations of these algebras are investigated. Connections between GE algebras and other classes of algebras of logic are studied. The implicative and positive implicative properties are discussed. It is shown that the class of positive implicative GE algebras (resp. the class of implicative aGE algebras) coincides with the class of generalized Tarski algebras (resp. the class of Tarski algebras). It is proved that for any aGE algebra the property of implicativity is equivalent to the commutative property. Moreover, several examples to illustrate the results are given. Finally, the interrelationships between some classes of implicative and positive implicative algebras are presented.

2023-09-25T00:00:00Z