•  23
    Qua Solution, 0-Qua Has Problems
    Journal of Analytic Theology 8 (1): 405-411. 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 0-Qua view. This is because on their 0-Qua account, these claims are either acceptable both to Conciliar Christians as well as those who are not Conciliar Christians or because th…Read more
  •  14
    Correction to: Lambek Calculus with Conjugates
    Studia Logica 109 (3): 471-471. 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, 26:1865-1912.
  •  35
    How Much Propositional Logic Suffices for Rosser’s Essential Undecidability Theorem?
    with Guillermo Badia, Petr Cintula, and Petr Hajek
    Review of Symbolic Logic 1-18. 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 non-total non-functional relations. We present a proof of essential undecidability in a much weaker substructural logic and fo…Read more
  •  25
    On Structural Features of the Implication Fragment of Frege’s Grundgesetze
    Journal of Philosophical Logic 46 (4): 443-456. 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 Cut-like features of his Hypothetical Syllogism rule.
  •  13
    Lambek Calculus with Conjugates
    Studia Logica 109 (3): 447-470. 2020.
    We study an expansion of the Distributive Non-associative Lambek Calculus with conjugates of the Lambek product operator and residuals of those conjugates. The resulting logic is well-motivated, under-investigated 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.
  •  22
    On elimination of quantifiers in some non‐classical mathematical theories
    Mathematical Logic Quarterly 64 (3): 140-154. 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.
  •  28
    Axioms for finite collapse models of arithmetic
    Review of Symbolic Logic 8 (3): 529-539. 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 ill-suited 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
  •  52
    Currying Omnipotence: A Reply to Beall and Cotnoir
    Thought: A Journal of Philosophy 7 (2): 119-121. 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
  •  52
    No cause for collapse
    Asian Journal of Philosophy 2 (2): 1-19. 2023.
    We investigate a hitherto under-considered avenue of response for the logical pluralist to collapse worries. In particular, we note that standard forms of the collapse arguments seem to require significantorder-theoreticassumptions, namely that the collection of admissible logics for the pluralist should be closed undermeetsandjoins. We consider some reasons for rejecting this assumption, noting someprima facieplausible constraints on the class of admissible logics which would lead a pluralist a…Read more
  •  16
    Consistent Theories in Inconsistent Logics
    Journal of Philosophical Logic 52 (04): 1133-1148. 2023.
    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
  •  8
    This is an introduction to the special issue of the AJL on Val Plumwood's manuscript "False Laws of Logic" and her other work in logic.
  •  15
    Relevant Consequence Relations: An Invitation
    with Guillermo Badia, Libor Běhounek, and Petr Cintula
    Review of Symbolic Logic 1-31. forthcoming.
    We generalize the notion ofconsequence relationstandard in abstract treatments of logic to accommodate intuitions ofrelevance. The guiding idea follows theuse criterion, according to which in order for some premises to have some conclusion(s) as consequence(s), the premises must each beusedin some way to obtain the conclusion(s). This relevance intuition turns out to require not just a failure of monotonicity, but also a move to considering consequence relations as obtaining betweenmultisets. We…Read more
  •  4
    Consistency and Decidability in Some Paraconsistent Arithmetics
    Australasian Journal of Logic 18 (5): 473-502. 2021.
    The standard style of argument used to prove that a theory is unde- cidable relies on certain consistency assumptions, usually that some fragment or other is negation consistent. In a non-paraconsistent set- ting, this amounts to an assumption that the theory is non-trivial, but these diverge when theories are couched in paraconsistent logics. Furthermore, there are general methods for constructing inconsistent models of arithmetic from consistent models, and the theories of such inconsistent mo…Read more
  •  12
    A Note on R-Mingle and the Danger of Safety
    Australasian Journal of Logic 19 (1). 2022.
    Dunn has recently argued that the logic R-Mingle (or RM) is a good, and good enough, choice for many purposes in relevant and paraconsistent logic. This includes an argument that the validity of Safety principle, according to which one may infer an arbitrary instance of the law of excluded middle from an arbitrary contradiction, in RM is not a problem because it doesn’t allow one to infer anything new from a contradiction. In this paper, I argue that while Dunn’s claim holds for the logic, there…Read more
  •  10
    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
  •  28
    Neighbourhood Semantics for Modal Relevant Logics
    Journal of Philosophical Logic 52 (1): 145-181. 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.
  •  25
    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.
  •  13
    Relevant propositional dynamic logic
    with Marta Bilková
    Synthese 200 (3): 1-42. 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
  •  4
    We investigate an information based generalization of the incompatibility-frame treatment of logics with non-classical 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
  •  5
    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 non-monotonic update in substructural dynam…Read more
  •  17
    Neighbourhood Semantics for Quantified Relevant Logics
    Journal of Philosophical Logic 51 (3): 457-484. 2022.
    The Mares-Goldblatt semantics for quantified relevant logics have been developed for first-order 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 Mares-Goldblatt 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?
    with Guillermo Badia, Petr Cintula, and Petr Hajek
    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
  •  12
    Information Flow In Logics in the Vicinity of BB
    Australasian Journal of Logic 18 (1): 1-24. 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 non-classical logics. Among the constraints imposed by channel-theory, 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 non-classical logics, it is relatively easy to show that a certain such condition …Read more