University of Otago
Department of Philosophy
PhD, 2017
Brisbane, Queensland, Australia
Areas of Specialization
Areas of Interest
•  3
•  10
##### Maximality of Logic Without Identity with Xavier Caicedo and Carles Noguera Journal of Symbolic Logic 1-16. forthcoming.
Lindström’s theorem obviously fails as a characterization of first-order logic without identity ( $\mathcal {L}_{\omega \omega }^{-}$ ). In this note, we provide a fix: we show that $\mathcal {L}_{\omega \omega }^{-}$ is a maximal abstract logic satisfying a weak form of the isomorphism property (suitable for identity-free languages and studied in [11]), the Löwenheim–Skolem property, and compactness. Furthermore, we show that compactness can be replaced by being recursively enumerable for val…Read more
•  2
##### Craig Interpolation Theorem Fails in Bi-Intuitionistic Predicate Logic with Grigory K. Olkhovikov Review of Symbolic Logic 1-23. forthcoming.
In this article we show that bi-intuitionistic predicate logic lacks the Craig Interpolation Property. We proceed by adapting the counterexample given by Mints, Olkhovikov and Urquhart for intuitionistic predicate logic with constant domains [13]. More precisely, we show that there is a valid implication $\phi \rightarrow \psi$ with no interpolant. Importantly, this result does not contradict the unfortunately named ‘Craig interpolation’ theorem established by Rauszer in [24] since that article…Read more
•  22
##### Paraconsistent Metatheory: New Proofs with Old Tools with Zach Weber and Patrick Girard Journal of Philosophical Logic 51 (4): 825-856. 2022.
This paper is a step toward showing what is achievable using non-classical metatheory—particularly, a substructural paraconsistent framework. What standard results, or analogues thereof, from the classical metatheory of first order logic can be obtained? We reconstruct some of the originals proofs for Completeness, Löwenheim-Skolem and Compactness theorems in the context of a substructural logic with the naive comprehension schema. The main result is that paraconsistent metatheory can ‘re-captur…Read more
• ##### How Much Propositional Logic Suffices for Rosser's Essential Undecidability Theorem? with Petr Cintula, Petr Hajek, and Andrew Tedder 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
•  15
##### Lindström theorems in graded model theory with Carles Noguera Annals of Pure and Applied Logic 172 (3): 102916. 2021.
Stemming from the works of Petr Hájek on mathematical fuzzy logic, graded model theory has been developed by several authors in the last two decades as an extension of classical model theory that studies the semantics of many-valued predicate logics. In this paper we take the first steps towards an abstract formulation of this model theory. We give a general notion of abstract logic based on many-valued models and prove six Lindström-style characterizations of maximality of first-order logics in…Read more
•  27
##### How Much Propositional Logic Suffices for Rosser’s Essential Undecidability Theorem? with Petr Cintula, Petr Hajek, and Andrew Tedder 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
•  4
##### Saturated models of first-order many-valued logics with Carles Noguera Logic Journal of the IGPL 30 (1): 1-20. 2022.
This paper is devoted to the problem of existence of saturated models for first-order many-valued logics. We consider a general notion of type as pairs of sets of formulas in one free variable that express properties that an element of a model should, respectively, satisfy and falsify. By means of an elementary chains construction, we prove that each model can be elementarily extended to a $\kappa$-saturated model, i.e. a model where as many types as possible are realized. In order to prove thi…Read more
•  12
##### Incompactness of the A1 Fragment of Basic Second Order Propositional Relevant Logic Australasian Journal of Logic 16 (1): 1-8. 2019.
In this note we provide a simple proof of the incompactness over Routley-Meyer B-frames of the A1 fragment of the second order propositional relevant language.
•  22
##### A Lindström Theorem in Many-Valued Modal Logic over a Finite MTL-chain with Grigory Olkhovikov Fuzzy Sets and Systems. forthcoming.
We consider a modal language over crisp frames and formulas evaluated on a finite MTL-chain (a linearly ordered commutative integral residuated lattice). We first show that the basic modal abstract logic with constants for the values of the MTL-chain is the maximal abstract logic satisfying Compactness, the Tarski Union Property and strong invariance for bisimulations. Finally, we improve this result by replacing the Tarski Union Property by a relativization property.
•  115
##### Syntactic characterizations of ﬁrst-order structures in mathematical fuzzy logic with Pilar Dellunde, Vicent Costa, and Carles Noguera Soft Computing. forthcoming.
This paper is a contribution to graded model theory, in the context of mathematical fuzzy logic. We study characterizations of classes of graded structures in terms of the syntactic form of their first-order axiomatization. We focus on classes given by universal and universal-existential sentences. In particular, we prove two amalgamation results using the technique of diagrams in the setting of structures valued on a finite MTL-algebra, from which analogues of the Łoś–Tarski and the Chang–Łoś–S…Read more
• ##### Model definability in relevant logic IfCoLog Journal of Logics and Their Applications 3 (4): 623-646. 2017.
It is shown that the classes of Routley-Meyer models which are axiomatizable by a theory in a propositional relevant language with fusion and the Ackermann constant can be characterized by their closure under certain model-theoretic operations involving prime filter extensions, relevant directed bisimulations and disjoint unions.
•  9
##### Variable Sharing in Substructural Logics: An Algebraic Characterization Bulletin of the Section of Logic 47 (2): 107-115. 2018.
We characterize the non-trivial substructural logics having the variable sharing property as well as its strong version. To this end, we find the algebraic counterparts over varieties of these logical properties.
• ##### Fraïssé classes of graded relational structures with Carles Noguera Theoretical Computer Science 737. 2018.
We study classes of graded structures satisfying the properties of amalgamation, joint embedding and hereditariness. Given appropriate conditions, we can build a graded analogue of the Fraïssé limit. Some examples such as the class of all finite weighted graphs or the class of all finite fuzzy orders (evaluated on a particular countable algebra) will be examined.
•  10
##### A Lindström Theorem for Intuitionistic Propositional Logic Notre Dame Journal of Formal Logic 61 (1): 11-30. 2020.
We show that propositional intuitionistic logic is the maximal abstract logic satisfying a certain form of compactness, the Tarski union property, and preservation under asimulations.
•  18
##### On elimination of quantifiers in some non-classical mathematical theories with Andrew Tedder 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.
•  36
##### Currying Omnipotence: A Reply to Beall and Cotnoir with Andrew Tedder 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
•  51
##### The relevant fragment of first order logic Review of Symbolic Logic 9 (1): 143-166. 2016.
Under a proper translation, the languages of propositional (and quantified relevant logic) with an absurdity constant are characterized as the fragments of first order logic preserved under (world-object) relevant directed bisimulations. Furthermore, the properties of pointed models axiomatizable by sets of propositional relevant formulas have a purely algebraic characterization. Finally, a form of the interpolation property holds for the relevant fragment of first order logic.
•  20
##### Infinitary propositional relevant languages with absurdity Review of Symbolic Logic 10 (4): 663-681. 2017.
Analogues of Scott's isomorphism theorem, Karp's theorem as well as results on lack of compactness and strong completeness are established for infinitary propositional relevant logics. An "interpolation theorem" for the infinitary quantificational boolean logic L-infinity omega. holds. This yields a preservation result characterizing the expressive power of infinitary relevant languages with absurdity using the model-theoretic relation of relevant directed bisimulation as well as a Beth definabi…Read more
•  13
##### On Sahlqvist Formulas in Relevant Logic Journal of Philosophical Logic 47 (4): 673-691. 2018.
This paper defines a Sahlqvist fragment for relevant logic and establishes that each class of frames in the Routley-Meyer semantics which is definable by a Sahlqvist formula is also elementary, that is, it coincides with the class of structures satisfying a given first order property calculable by a Sahlqvist-van Benthem algorithm. Furthermore, we show that some classes of Routley-Meyer frames definable by a relevant formula are not elementary.
•  62
##### Bi-Simulating in Bi-Intuitionistic Logic Studia Logica 104 (5): 1037-1050. 2016.
Bi-intuitionistic logic is the result of adding the dual of intuitionistic implication to intuitionistic logic. In this note, we characterize the expressive power of this logic by showing that the ﬁrst order formulas equivalent to translations of bi-intuitionistic propositional formulas are exactly those preserved under bi-intuitionistic directed bisimulations. The proof technique is originally due to Lindstrom and, in contrast to the most common proofs of this kind of result, it does not use th…Read more
•  33
##### A Lindström-style theorem for finitary propositional weak entailment languages with absurdity Logic Journal of the IGPL 24 (2): 115-137. 2016.
Following a result by De Rijke for modal logic, it is shown that the basic weak entailment model-theoretic language with absurdity is the maximal model-theoretic language having the finite occurrence property, preservation under relevant directed bisimulations and the finite depth property. This can be seen as a generalized preservation theorem characterizing propositional weak entailment formulas among formulas of other model-theoretic languages.
•  83
##### What Is an Inconsistent Truth Table? with Zach Weber and Patrick Girard Australasian Journal of Philosophy 94 (3): 533-548. 2016.
ABSTRACTDo truth tables—the ordinary sort that we use in teaching and explaining basic propositional logic—require an assumption of consistency for their construction? In this essay we show that truth tables can be built in a consistency-independent paraconsistent setting, without any appeal to classical logic. This is evidence for a more general claim—that when we write down the orthodox semantic clauses for a logic, whatever logic we presuppose in the background will be the logic that appears …Read more
•  16
##### The languages of relevant logic: a model-theoretic perspective
A traditional aspect of model theory has been the interplay between formal languages and mathematical structures. This dissertation is concerned, in particular, with the relationship between the languages of relevant logic and Routley-Meyer models. One fundamental question is treated: what is the expressive power of relevant languages in the Routley-Meyer framework? In the case of finitary relevant propositional languages, two answers are provided. The first is that finitary propositional releva…Read more
•  9
##### A Remark on Maksimova's Variable Separation Property in Super-Bi-Intuitionistic Logics Australasian Journal of Logic 14 (1). 2017.
We provide a sucient frame-theoretic condition for a super bi-intuitionistic logic to have Maksimova's variable separation property. We conclude that bi-intuitionistic logic enjoys the property. Furthermore, we offer an algebraic characterization of the super-bi-intuitionistic logics with Maksimova's property.