
4Consistent Theories in Inconsistent LogicsJournal of Philosophical Logic 116. forthcoming.The relationship between logics with sets of theorems including contradictions (“inconsistent logics”) and theories closed under such logics is investigated. It is noted that if we take “theories” to be defined in terms of deductive closure understood in a way somewhat different from the standard, Tarskian, one, inconsistent logics can have consistent theories. That is, we can find some sets of formulas the closure of which under some inconsistent logic need not contain any contradictions. We pr…Read more

7Kapsner Complementation: An Algebraic Take on Kapsner Strong LogicsStudia Logica 111 (2): 321352. 2023.Kapsner strong logics, originally studied in the context of connexive logics, are those in which all formulas of the form \(A\rightarrow \lnot A\) or \(\lnot A\rightarrow A\) are unsatisfiable, and in any model at most one of \(A\rightarrow B, A\rightarrow \lnot B\) is satisfied. In this paper, such logics are studied algebraically by means of algebraic structures in which negation is modeled by an operator \(\lnot \) s.t. any element _a_ is incomparable with \(\lnot a\). A range of properties w…Read more

17Neighbourhood Semantics for Modal Relevant LogicsJournal of Philosophical Logic 52 (1): 145181. 2023.In this paper, we investigate neighbourhood semantics for modal extensions of relevant logics. In particular, we combine the neighbourhood interpretation of the relevant implication (and related connectives) with a neighbourhood interpretation of modal operators. We prove completeness for a range of systems and investigate the relations between neighbourhood models and relational models, setting out a range of augmentation conditions for the various relations and operations.

13Incorporating the Relation into the Language?Logic and Logical Philosophy 30 (4). 2021.In this paper we discuss whether the relation between formulas in the relating model can be directly introduced into the language of relating logic, and present some stances on that problem. Other questions in the vicinity, such as what kind of functor would be the incorporated relation, or whether the direct incorporation of the relation into the language of relating logic is really needed, will also be addressed.

5Relevant propositional dynamic logicSynthese 200 (3): 142. 2022.Relevant propositional dynamic logics have been sporadically discussed in the broader context of modal relevant logics, but have not come up for sustained investigation until recently. In this paper, we develop a philosophical motivation for these systems, and present some new results suggested by the proposed motivation. Among these, we’ll show how to adapt some recent work to show that the extensions of relevant logics by the extensional truth constants \ are complete with respect to a natural…Read more

1Disjunction and Negation in Information Based SemanticsIn Alexandra Silva, Renata Wassermann & Ruy de Queiroz (eds.), Logic, Language, Information, and Computation: 27th International Workshop, Wollic 2021, Virtual Event, October 5–8, 2021, Proceedings, Springer Verlag. pp. 355371. 2021.We investigate an information based generalization of the incompatibilityframe treatment of logics with nonclassical negation connectives. Our framework can be viewed as an alternative to the neighbourhood semantics for extensions of lattice logic by various negation connectives, investigated by Hartonas. We set out the basic semantic framework, along with some correspondence results for extensions. We describe three kinds of constructions of canonical models and show that double negation law …Read more

Situated Epistemic UpdatesIn Sujata Ghosh & Thomas Icard (eds.), Logic, Rationality, and Interaction: 8th International Workshop, Lori 2021, Xi’an, China, October 16–18, 2021, Proceedings, Springer Verlag. pp. 192200. 2021.One way to model epistemic states of agents more realistically is to represent these states by sets of situations rather than possible worlds. In this paper we discuss representations of epistemic update in terms of situations. After linking epistemic update based on deleting epistemic accessibility arrows with update of situations, we discuss two specific kinds of public epistemic update; monotonic update in intuitionistic dynamic epistemic logic, and nonmonotonic update in substructural dynam…Read more

10Neighbourhood Semantics for Quantified Relevant LogicsJournal of Philosophical Logic 51 (3): 457484. 2022.The MaresGoldblatt semantics for quantified relevant logics have been developed for firstorder extensions of R, and a range of other relevant logics and modal extensions thereof. All such work has taken place in the the ternary relation semantic framework, most famously developed by Sylvan and Meyer. In this paper, the MaresGoldblatt technique for the interpretation of quantifiers is adapted to the more general neighbourhood semantic framework, developed by Sylvan, Meyer, and, more recently, …Read more

How Much Propositional Logic Suffices for Rosser's Essential Undecidability Theorem?Review of Symbolic Logic. forthcoming.In this paper we explore the following question: how weak can a logic be for Rosser’s essential undecidability result to be provable for a weak arithmetical theory? It is well known that Robinson’s Q is essentially undecidable in intuitionistic logic, and P. Hájek proved it in the fuzzy logic BL for Grzegorczyk’s variant of Q which interprets the arithmetic operations as nontotal nonfunctional relations. We present a proof of essential undecidability in a much weaker substructural logic and for …Read more

4Information Flow In Logics in the Vicinity of BBAustralasian Journal of Logic 18 (1): 124. 2021.Situation theory, and channel theory in particular, have been used to provide motivational accounts of the ternary relation semantics of relevant, substructural, and various nonclassical logics. Among the constraints imposed by channeltheory, we must posit a certain existence criterion for situations which result from the composites of multiple channels (this is used in modeling information flow). In associative nonclassical logics, it is relatively easy to show that a certain such condition …Read more

11Correction to: Lambek Calculus with ConjugatesStudia Logica 109 (3): 471471. 2020.We, the authors, would like to thank Guillaume Aucher for informing us of his “Displaying Updates in Logic”, published in the Journal of Logic and Computation, 26:18651912.

14Qua Solution, 0Qua Has ProblemsJournal of Analytic Theology 8 (1): 405411. 2020.We present an objection to Beall & Henderson’s recent paper defending a solution to the fundamental problem of conciliar Christology using qua or secundum clauses. We argue that certain claims the acceptance/rejection of which distinguish the Conciliar Christian from others fail to so distinguish on Beall & Henderson’s 0Qua view. This is because on their 0Qua account, these claims are either acceptable both to Conciliar Christians as well as those who are not Conciliar Christians or because th…Read more

39Currying Omnipotence: A Reply to Beall and CotnoirThought: A Journal of Philosophy 7 (2): 119121. 2018.Beall and Cotnoir (2017) argue that theists may accept the claim that God's omnipotence is fully unrestricted if they also adopt a suitable nonclassical logic. Their primary focus is on the infamous Stone problem (i.e., whether God can create a stone too heavy for God to lift). We show how unrestricted omnipotence generates Curry‐like paradoxes. The upshot is that Beall and Cotnoir only provide a solution to one version of the Stone problem, but that unrestricted omnipotence generates other prob…Read more

20On elimination of quantifiers in some nonclassical mathematical theoriesMathematical Logic Quarterly 64 (3): 140154. 2018.Elimination of quantifiers is shown to fail dramatically for a group of well‐known mathematical theories (classically enjoying the property) against a wide range of relevant logical backgrounds. Furthermore, it is suggested that only by moving to more extensional underlying logics can we get the property back.

28How Much Propositional Logic Suffices for Rosser’s Essential Undecidability Theorem?Review of Symbolic Logic 118. forthcoming.In this paper we explore the following question: how weak can a logic be for Rosser's essential undecidability result to be provable for a weak arithmetical theory? It is well known that Robinson's Q is essentially undecidable in intuitionistic logic, and P. Hajek proved it in the fuzzy logic BL for Grzegorczyk's variant of Q which interprets the arithmetic operations as nontotal nonfunctional relations. We present a proof of essential undecidability in a much weaker substructural logic and fo…Read more

21On Structural Features of the Implication Fragment of Frege’s GrundgesetzeJournal of Philosophical Logic 46 (4): 443456. 2017.We set out the implication fragment of Frege’s Grundgesetze, clarifying the implication rules and showing that this system extends Absolute Implication, or the implication fragment of Intuitionist logic. We set out a sequent calculus which naturally captures Frege’s implication proofs, and draw particular attention to the Cutlike features of his Hypothetical Syllogism rule.

8Lambek Calculus with ConjugatesStudia Logica 109 (3): 447470. 2020.We study an expansion of the Distributive Nonassociative Lambek Calculus with conjugates of the Lambek product operator and residuals of those conjugates. The resulting logic is wellmotivated, underinvestigated and difficult to tackle. We prove completeness for some of its fragments and establish that it is decidable. Completeness of the logic is an open problem; some difficulties with applying the usual proof method are discussed.

16Axioms for finite collapse models of arithmeticReview of Symbolic Logic 8 (3): 529539. 2015.The collapse models of arithmetic are inconsistent, nontrivial models obtained from ℕ and set out in the Logic of Paradox (LP). They are given a general treatment by Priest (Priest, 2000). Finite collapse models are decidable, and thus axiomatizable, because finite. LP, however, is illsuited to normal axiomatic reasoning, as it invalidates Modus Ponens, and almost all other usual conditional inferences. I set out a logic, A3, first given by Avron (Avron, 1991), and give a first order axiom syst…Read more