2026-03-12
Agustina Borzi, Martina Zirattu
In a multiple-premise and multiple-conclusion setting, logicians typically use the metalinguistic expression "," to gather premises and conclusions. In most logical systems, this comma can be freely interchanged with the object-language conjunction on the premise side and with the object-language disjunction on the conclusion side. However, this is not the case for Weak Kleene Logics (WK logics), which include Paraconsistent Weak Kleene ( PWK ) and its paracomplete counterpart ( K 3 w ). The aim of this work is to define two new substructural systems called uniform Weak Kleene logics (uWK logics), uPWK and uK 3 w , which re-establish what we call "Uniformity": the correspondence between the structural comma and the object-language WK operators. We begin by providing some philosophical arguments advocating for it as a desirable logical feature. We then introduce a novel two-sided sequent calculus free of linguistic restrictions both for PWK and K 3 w and show that, with minor adjustments, we can achieve sound and complete sequent calculi for uPWK and uK 3 w . Finally, we generalize our results for WK logics and their uniform counterparts to other infectious Tarskian logics that meet specific criteria.
2026-03-12
Alasdair Urquhart
This note corrects an error in an earlier paper (Urquhart 2023) claiming that the algebra of three-variable formulas in the logic KR is finite. Here we show that the algebra of one-variable formulas in this logic is infinite.
2026-03-12
Jie Fan
In a seminal work Fine (2018), Fine classifies several forms of ignorance, among which are Fitchean ignorance, first-order ignorance, Rumsfeld ignorance, and second-order ignorance. It is shown that there are some relationships among some of them, including that in S4 , all higher-order ignorance are reducible to second-order ignorance. This is thought of as a disadvantage by some researchers. It is then natural to ask how to avoid this consequence. We deal with this issue in a much more general framework. In details, we treat the forms of Fitchean ignorance and first-order ignorance as primitive modalities and study them as first-class citizens under neighborhood semantics, in which Rumsfeld ignorance and second-order ignorance are definable. The main contributions include model-theoretical results such as expressivity and frame definability, and axiomatizability. Last but not least, by updating the neighborhood models via the intersection semantics, we extend the results to the dynamic case of public announcements, which gives us some applications to successful formulas. We also show that the result in Fine (2018) stating all higher-order ignorance are reducible to second-order ignorance does not hold in any of our logical systems.
2026-03-12
Mario Piazza, Andrea Sabatini
In this paper, we investigate certain intricacies and peculiarities of the proof theory of deduction-refutation systems (D-R systems, henceforth), namely systems integrating theorems and antitheorems of a given logic. Our primary focus is on establishing the general conditions under which anticut rules (the contrapositive versions of the familiar cut rule) can be eliminated from D-R sequent calculi, while distinguishing between two main variants of these systems. This is a topic that has recently been highlighted in the context of D-R sequent calculi for first-degree entailment-based logics. Finally, we examine the relationship between anticut rules and the completeness of D-R sequent calculi with respect to deducibility and refutability (Ł-completeness).