
Routes to relevance: Philosophies of relevant logicsPhilosophy Compass 19 (2). 2024.Relevant logics are a family of nonclassical logics characterized by the behavior of their implication connectives. Unlike some other nonclassical logics, such as intuitionistic logic, there are multiple philosophical views motivating relevant logics. Further, different views seem to motivate different logics. In this article, we survey five major views motivating the adoption of relevant logics: Use Criterion, sufficiency, meaning containment, theory construction, and truthmaking. We highligh…Read more

65VariableSharing as RelevanceIn Igor Sedlár, Shawn Standefer & Andrew Tedder (eds.), New Directions in Relevant Logic. forthcoming.

66Hyperintensionality in Relevant LogicsIn Natasha Alechina, Andreas Herzig & Fei Liang (eds.), Logic, Rationality, and Interaction: 9th International Workshop, LORI 2023, Jinan, China, October 26–29, 2023, Proceedings, Springer Nature Switzerland. pp. 238250. 2023.In this article, we present a definition of a hyperintensionality appropriate to relevant logics. We then show that relevant logics are hyperintensional in this sense, drawing consequences for other nonclassical logics, including HYPE and some substructural logics. We further prove results concerning extensionality in relevant logics. We close by discussing related concepts for classifying formula contexts and potential applications of these results.

24Logical methodsThe MIT Press. 2022.An advancedlevel logic textbook that presents proof construction on equal footing with model building. Potentially relevant to students of mathematics and computer science as well.

19Getting some (nonclassical) closure with justification logicAsian Journal of Philosophy 2 (2): 125. 2023.Justification logics provide frameworks for studying the fine structure of evidence and justification. Traditionally, these logics do not impose any closure requirements on justification. In this paper, we argue that for some applications they should subject justification to closure under some variety of logical consequence. Specifically, we argue, building on ideas from Beall, that the nonclassical logic FDE offers a particularly attractive notion of consequence for this purpose and define a j…Read more

9A Substructural Approach to Explicit Modal LogicJournal of Logic, Language and Information 32 (2). 2023.In this paper, we build on earlier work by Standefer (Logic J IGPL 27(4):543–569, 2019) in investigating extensions of substructural logics, particularly relevant logics, with the machinery of justification logics. We strengthen a negative result from the earlier work showing a limitation with the canonical model method of proving completeness. We then show how to enrich the language with an additional operator for implicit commitment to circumvent these problems. We then extend the logics with …Read more

26Weak relevant justification logicsJournal of Logic and Computation 33 (7). 2023.This paper will develop ideas from [44]. We will generalize their work in two directions. First, we provide axioms for justification logics over the base logic B and show that the logic permits a proof of the internalization theorem. Second, we provide alternative frames that more closely resemble the standard versions of the ternary relational frames, as well as a more general approach to the completeness proof. We prove that soundness and completeness hold for justification logics over a wide …Read more

62Collection Frames for Distributive Substructural LogicsReview of Symbolic Logic 16 (4): 11201157. 2023.We present a new frame semantics for positive relevant and substructural propositional logics. This frame semantics is both a generalisation of Routley–Meyer ternary frames and a simplification of them. The key innovation of this semantics is the use of a single accessibility relation to relate collections of points to points. Different logics are modeled by varying the kinds of collections used: they can be sets, multisets, lists or trees. We show that collection frames on trees are sound and c…Read more

199Varieties of Relevant S5Logic and Logical Philosophy 32 (1). 2023.In classically based modal logic, there are three common conceptions of necessity, the universal conception, the equivalence relation conception, and the axiomatic conception. They provide distinct presentations of the modal logic S5, all of which coincide in the basic modal language. We explore these different conceptions in the context of the relevant logic R, demonstrating where they come apart. This reveals that there are many options for being an S5ish extension of R. It further reveals a …Read more

74What is a Relevant Connective?Journal of Philosophical Logic 51 (4): 919950. 2022.There appears to be few, if any, limits on what sorts of logical connectives can be added to a given logic. One source of potential limitations is the motivating ideology associated with a logic. While extraneous to the logic, the motivating ideology is often important for the development of formal and philosophical work on that logic, as is the case with intuitionistic logic. One family of logics for which the philosophical ideology is important is the family of relevant logics. In this paper, …Read more

33Revisiting Semilattice SemanticsIn Ivo Düntsch & Edwin Mares (eds.), Alasdair Urquhart on Nonclassical and Algebraic Logic and Complexity of Proofs, Springer Verlag. pp. 243259. 2022.The operational semantics of Urquhart is a deep and important part of the development of relevant logics. In this paper, I present an overview of work on Urquhart’s operational semantics. I then present the basics of collection frames. Finally, I show how one kind of collection frame, namely, functional set frames, is equivalent to Urquhart’s semilattice semantics.

31An Incompleteness Theorem for Modal Relevant LogicsNotre Dame Journal of Formal Logic 62 (4). 2021.In this paper, an incompleteness theorem for modal extensions of relevant logics is proved. The proof uses elementary methods and builds upon the work of Fuhrmann.

10Proofs and Models in Naive Property Theory: A Response to Hartry Field's ‘Properties, Propositions and Conditionals’Australasian Philosophical Review 4 (2): 162177. 2020.ABSTRACT In our response Field's ‘Properties, Propositions and Conditionals’, we explore the methodology of Field's program. We begin by contrasting it with a prooftheoretic approach and then commenting on some of the particular choices made in the development of Field's theory. Then, we look at issues of property identity in connection with different notions of equivalence. We close with some comments relating our discussion to Field's response to Restall’s [2010] ‘What Are We to Accept, and W…Read more

19Identity in MaresGoldblatt Models for Quantified Relevant LogicJournal of Philosophical Logic 50 (6): 13891415. 2021.Mares and Goldblatt, 163–187, 2006) provided an alternative frame semantics for two quantified extensions of the relevant logic R. In this paper, I show how to extend the MaresGoldblatt frames to accommodate identity. Simpler frames are provided for two zeroorder logics en route to the full logic in order to clarify what is needed for identity and substitution, as opposed to quantification. I close with a comparison of this work with the FineMares models for relevant logics with identity and …Read more

12Translations between linear and tree natural deduction systems for relevant logicsReview of Symbolic Logic 14 (2). 2021.Anderson and Belnap presented indexed Fitchstyle natural deduction systems for the relevant logics R, E, and T. This work was extended by Brady to cover a range of relevant logics. In this paper I present indexed tree natural deduction systems for the Anderson–Belnap–Brady systems and show how to translate proofs in one format into proofs in the other, which establishes the adequacy of the tree systems.

104Actual Issues for Relevant LogicsErgo: An Open Access Journal of Philosophy 7. 2020.In this paper, I motivate the addition of an actuality operator to relevant logics. Straightforward ways of doing this are in tension with standard motivations for relevant logics, but I show how to add the operator in a way that permits one to maintain the intuitions behind relevant logics. I close by exploring some of the philosophical consequences of the addition.

26Tracking reasons with extensions of relevant logicsLogic Journal of the IGPL 27 (4): 543569. 2019.In relevant logics, necessary truths need not imply each other. In justification logic, necessary truths need not all be justified by the same reason. There is an affinity to these two approaches that suggests their pairing will provide good logics for tracking reasons in a finegrained way. In this paper, I will show how to extend relevant logics with some of the basic operators of justification logic in order to track justifications or reasons. I will define and study three kinds of frames for…Read more

55Translations Between Gentzen–Prawitz and Jaśkowski–Fitch Natural Deduction ProofsStudia Logica 107 (6): 11031134. 2019.Two common forms of natural deduction proof systems are found in the Gentzen–Prawitz and Jaśkowski–Fitch systems. In this paper, I provide translations between proofs in these systems, pointing out the ways in which the translations highlight the structural rules implicit in the systems. These translations work for classical, intuitionistic, and minimal logic. I then provide translations for classical S4 proofs.

47Proof Theory for Functional Modal LogicStudia Logica 106 (1): 4984. 2018.We present some prooftheoretic results for the normal modal logic whose characteristic axiom is \. We present a sequent system for this logic and a hypersequent system for its firstorder form and show that these are equivalent to Hilbertstyle axiomatizations. We show that the question of validity for these logics reduces to that of classical tautologyhood and firstorder logical truth, respectively. We close by proving equivalences with a Fitchstyle proof system for revision theory.

62Review of Stewart Shapiro's Varieties of Logic (review)Notre Dame Philosophical Reviews 2015. 2015.

58Solovaytype theorems for circular definitionsReview of Symbolic Logic 8 (3): 467487. 2015.We present an extension of the basic revision theory of circular definitions with a unary operator, □. We present a Fitchstyle proof system that is sound and complete with respect to the extended semantics. The logic of the box gives rise to a simple modal logic, and we relate provability in the extended proof system to this modal logic via a completeness theorem, using interpretations over circular definitions, analogous to Solovay’s completeness theorem forGLusing arithmetical interpretations…Read more

423The Relevant Logic E and Some Close Neighbours: A ReinterpretationIfCoLog Journal of Logics and Their Applications 4 (3): 695730. 2017.This paper has two aims. First, it sets out an interpretation of the relevant logic E of relevant entailment based on the theory of situated inference. Second, it uses this interpretation, together with Anderson and Belnap’s natural deduc tion system for E, to generalise E to a range of other systems of strict relevant implication. Routley–Meyer ternary relation semantics for these systems are produced and completeness theorems are proven.

649Conditionals in Theories of TruthJournal of Philosophical Logic 46 (1): 2763. 2017.We argue that distinct conditionals—conditionals that are governed by different logics—are needed to formalize the rules of Truth Introduction and Truth Elimination. We show that revision theory, when enriched with the new conditionals, yields an attractive theory of truth. We go on to compare this theory with one recently proposed by Hartry Field.

236The Tarskian Turn: Deflationism and Axiomatic TruthPhilosophical Review 122 (1): 144147. 2013.

578Contraction and revisionAustralasian Journal of Logic 13 (3): 5877. 2016.An important question for proponents of noncontractive approaches to paradox is why contraction fails. Zardini offers an answer, namely that paradoxical sentences exhibit a kind of instability. I elaborate this idea using revision theory, and I argue that while instability does motivate failures of contraction, it equally motivates failure of many principles that noncontractive theorists want to maintain.

77Intersubstitutivity principles and the generalization function of truthSynthese 195 (3): 10651075. 2018.We offer a defense of one aspect of Paul Horwich’s response to the Liar paradox—more specifically, of his move to preserve classical logic. Horwich’s response requires that the full intersubstitutivity of ‘ ‘A’ is true’ and A be abandoned. It is thus open to the objection, due to Hartry Field, that it undermines the generalization function of truth. We defend Horwich’s move by isolating the grade of intersubstitutivity required by the generalization function and by providing a new reading of the…Read more

586On Artifacts and TruthPreservationAustralasian Journal of Logic 12 (3): 135158. 2015.In Saving Truth from Paradox, Hartry Field presents and defends a theory of truth with a new conditional. In this paper, I present two criticisms of this theory, one concerning its assessments of validity and one concerning its treatment of truthpreservation claims. One way of adjusting the theory adequately responds to the truthpreservation criticism, at the cost of making the validity criticism worse. I show that in a restricted setting, Field has a way to respond to the validity criticism. …Read more
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Revision Theory of Truth 
Proof Theory 