My research focuses on developing fine-grained analyses of concepts and conceptual connections using the tools of philosophical logic.
Within philosophical logic, my main areas of interest are proof theory, paradoxes, relevant logics, and modal logics. I am currently working on a manuscript on relevant logics.
Abstract: This is an intermediate logic text suitable for undergraduate and graduate students. It develops natural deduction proof systems and models for classical and intuitionistic logic, S4 an S5 modal logics, and classical first-order logics. Models for intuitionistic logic, some three-valued loigcs, two-dimensional logics of actuality, and quantiifed modal logic are discussed. The details of important meta-theoretic results are presented, including Normalization for intuitionistic logic as well as Soundness and Completness for many logics. Each chapter includes many excercises, divided into basic and advanced, to provide practice for the interested student.
Abstract: This paper will develop ideas from Savić and Studer. 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 variety of base logics. Finally, we will strengthen Belnap's variable sharing property for the justification logic context, demonstrating that the justifications are properly relevant justifications.
"A substructural approach to explicit modal logic", Journal of Logic, Language and Information. Forthcoming. (2022). doi: 10.1007/s10849-022-09380-z
Abstract: In this paper, we build on earlier work by Standefer 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 axioms for D, 4, and 5, which requires additional justification term operators, following the work of Pacuit and Rubtsova, and present the required modifications to the frame semantics. We present a simplification of the neighborhood frames from the earlier work and we close by investigating the distinctive contribution of the + operator to the logic.
Abstract: 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 complete for the basic positive distributive substructural logic B, that collection frames on multisets are sound and complete for RW (the relevant logic R, without contraction, or equivalently, positive multiplicative and additive linear logic with distribution for the additive connectives), and that collection frames on sets are sound for the positive relevant logic R. The completeness of set frames for R is, currently, an open question.
Abstract: 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 S5-ish extension of R. It further reveals a divide between the universal conception of necessity on the one hand, and the axiomatic conception on the other: The latter is consistent with motivations for rel- evant logics while the former is not. For the committed relevant logician, necessity cannot be the truth in all possible worlds.
Abstract: 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, I explore the limits of what a relevant connective is, showing how some basic crite- ria motivated by the ideology of relevant logicians provide robust limits on potential connectives. These criteria provide some plausible necessary conditions on being a relevant connective.
"Completeness via Metacompleteness", In Katalin Bimbó, editor, Relevance Logics and other Tools for Reasoning: Essays in Honor of J. Michael Dunn p. 394-409. College Publications, (2022). Preprint pdf
Abstract: We show that all logics in a certain class of modal relevant logics are complete with respect to their reduced frames. The proof uses a combination of the canonical frame method and metacompleteness results.
"An incompleteness theorem for modal relevant logics", Notre Dame Journal of Formal Logic 62(4): 669-681 (2021). doi:10.1215/00294527-2021-0035Preprint pdf
Abstract: 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.
"Identity in Mares-Goldblatt models for quantified relevant logic, Journal of Philosophical Logic 50(6): 1389-1415 (2021). doi:10.1007/s10992-021-09603-xPreprint pdf
Abstract: Mares and Goldblatt (2006) provided an alternative frame semantics for two quantified extensions of the relevant logic R. In this paper, I show how to extend the Mares-Goldblatt frames to accommodate identity. Simpler frames are provided for two zero-order 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 Fine-Mares models for relevant logics with identity and a discussion of constant and variable domains.
"Revisiting Semilattice Semantics", Alasdair Urquhart on Nonclassical and Algebraic Logic and Complexity of Proofs, p. 241-258 (2021), edited by Ivo Duntsch and Edwin Mares, in the series Outstanding Contributions to Logic. Springer. doi:10.1007/978-3-030-71430-7_7Preprint pdf
Abstract: The operational semantics of Urquhart [1972a,b] 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.
"Translations between linear and tree natural deduction systems for relevant logics", Review of Symbolic Logic (2021) 14(2): 285-306 doi:10.1017/S1755020319000133
Abstract: Anderson and Belnap presented indexed Fitch-style 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.
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 proof-theoretic 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 [2010. What Are We to Accept, and What Are We to Reject, While Saving Truth from Paradox? Philosophical Studies 147/3: 433--43].
We present an extension of the basic revision theory of circular definitions with a unary operator, []. We present a Fitch-style proof system that is sound and complete with respect to the extended semantics.
Abstract: 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 suggest their pairing will provide good logics for tracking reasons in a fine-grained 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 these logics. For the first kind of frame, I show soundness and highlight a difficulty in proving completeness. This motivates the two alternative kinds of frames, with respect to which completeness results are obtained. Axioms to strengthen the justification logic portions of these logics are considered. I close by developing an analogy between the dot operator of justification logic and theory fusion in relevant logics.
"Translations Between Gentzen–Prawitz and Jaśkowski–Fitch Natural Deduction Proofs", Studia Logica 107(6): 1103--1134 (2019) doi:10.1007/s11225-018-9828-2
Abstract: 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.
Abstract: A tree natural deduction system for Anderson and Belnap's relevant logic E is presented and shown equivalent to a Hilbert-style axiomatization of E. Using an idea from Prawitz, a variant tree system is motivated and shown equivalent to the Hilbert-style system via a detour through Anderson and Belnap's Fitch system for E.
"Natural deduction systems for E," (co-authored with Ross T. Brady) Logique et Analyse, 242: 163–182 2018, Peeters Publishers. doi: 10.2143/LEA.242.0.3284749.
Abstract: Anderson and Belnap [1975] presented a Fitch natural deduction system, FE, for their logic E of entailment as well as Fitch systems for the relevant logics T and R. The system FE is obtained from the system for R through a restriction on the rule of reiteration. Brady [1984] presents Fitch systems for a range of relevant logics, none of which uses a restriction on the rule of reiteration. However, no Fitch system for E was presented. We fill this lacuna by providing two Fitch systems for E, neither of which uses a restriction on the rule of reiteration. We close by discussing their differences and possible connections to other extant systems for E.
"Proof Theory for Functional Modal Logic," Studia Logica 106(1): 49–84 2018, Springer. doi:10.1007/s11225-017-9725-0.
Abstract: We present some proof-theoretic results for the normal modal logic whose characteristic axiom is □~A≡~□A. We present a sequent system for this logic and a hypersequent system for its first-order form and show that these are equivalent to Hilbert-style axiomatizations. We show that the question of validity for these logics reduces to that of classical tautologyhood and first-order logical truth, respectively.
"Inferentialism, Structure, and Conservativeness," (co-authored with Ole Hjortland), (2018) From Rules to Meanings: New Essays on Inferentialism, eds. Ondrej Beran, Vojtech Kolman, and Ladislav Koren, pp. 115–140, Routledge. Preprint pdf
"Intersubstitutivity Principles and the Generalization Function of Truth,” (co-authored with Anil Gupta) Synthese, 195(3): 1065–1075 2018, Springer. doi:
10.1007/s11229-017-1318-y.
Abstract: 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 biconditionals of the form `` `A' is true iff A.''
"The Relevant Logic E and Some Close Neighbours: A Reinterpretation," (co-authored with Ed Mares) The IfColog Journal of Logics and their Applications (4:3) 2017, 695–730. Special Issue: Proceedings of the Third Workshop, 16-17 May 2016, Edmonton, Canada, edited by Katalin Bimbó and J. Michael Dunn. Open access pdf. Published pdf.
Abstract: 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 deduction 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.
"Non-Classical Circular Definitions," Australasian Journal of Logic (14:1) 2017, Article no. 6, 147–180. Special issue: Non-Classicality: Logic, Mathematics, Philosophy, edited by Zach Weber, Maarten McKubre-Jordens, and Patrick Girard. doi: 10.26686/ajl.v14i1.4030. Open access pdf.
Abstract: Circular definitions have primarily been studied in revision theory in the classical scheme. I present systems of circular definitions in the Strong Kleene and supervaluation schemes and provide complete proof systems for them. One class of definitions, the intrinsic definitions, naturally arises in both schemes. I survey some of the features of this class of definitions.
"Conditionals in Theories of Truth," (co-authored with Anil Gupta) Journal of Philosophical Logic 46(1) 2017, pp. 27--63. doi:10.1007/s10992-015-9393-3Preprint pdf.
Abstract: 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.
Abstract: An important question for proponents of non-contractive approaches to paradox is why contraction fails. One non-contractive theorist, Elia Zardini, offers the answer that the 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 non-contractive theorists want to maintain.
"On Artifacts and Truth-Preservation," Australasian Journal of Logic (12:3) 2015, Article no. 1, pp. 135–158. doi: 10.26686/ajl.v12i3.2045Open access pdf.
Abstract: 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 truth-preservation claims. One way of adjusting the theory adequately responds to the truth-preservation 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. I close with some general considerations on the use of revision-theoretic methods in theories of truth.
Abstract: We present an extension of the basic revision theory of circular definitions with a unary operator, □. We present a Fitch-style 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 for GL using arithmetical interpretations. We adapt our proof to a special class of circular definitions as well as to the first-order case.
Conference proceedings
"Non-Triviality Done Proof-Theoretically,"' (co-authored with Rohan French) Logic, Rationality, and Interaction: 6th International Workshop, LORI 2017, Sapporo, Japan, September 11-14, 2017, Proceedings, eds. Alexandru Baltag, Jeremy Seligman and Tomoyuki Yamada, pp. 438--450. Springer: Berlin, Heidelberg. doi:10.1007/978-3-662-55665-8_30
"What Is Wrong with the Tarskian Theory of Truth?" The Logica Yearbook 2010, Michael Peliš (ed), pp. 269-281, College Publications.
"Philosophical Aspects of Display Logic," The Logica Yearbook 2009, Michael Peliš (ed), pp. 283–296, College Publications.
Reviews and introductions
"Guest Editors' Introduction," (co-authored with Riccardo Bruni) Journal of Philosophical Logic, special issue for the 25th anniversary of the publication of Revision Theory of Truth (2019) 48(1): 1–9. doi: 10.1007/s10992-018-9478-xOpen access pdf.
Review of Stewart Shapiro’s Varieties of Logic, Notre Dame Philosophical Reviews. Published html.
Review of Leon Horsten’s The Tarskian Turn, Philosophical Review (2013) 122(1): 144-147; doi:10.1215/00318108-1728795.