
3Omitting types theorem in hybrid dynamic firstorder logic with rigid symbolsAnnals of Pure and Applied Logic 174 (3): 103212. 2023.

10Maximality of Logic Without IdentityJournal of Symbolic Logic 116. forthcoming.Lindström’s theorem obviously fails as a characterization of firstorder 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 identityfree 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

2Craig Interpolation Theorem Fails in BiIntuitionistic Predicate LogicReview of Symbolic Logic 123. forthcoming.In this article we show that biintuitionistic 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

22Paraconsistent Metatheory: New Proofs with Old ToolsJournal of Philosophical Logic 51 (4): 825856. 2022.This paper is a step toward showing what is achievable using nonclassical 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, LöwenheimSkolem and Compactness theorems in the context of a substructural logic with the naive comprehension schema. The main result is that paraconsistent metatheory can ‘recaptur…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

15Lindström theorems in graded model theoryAnnals 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 manyvalued 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 manyvalued models and prove six Lindströmstyle characterizations of maximality of firstorder logics in…Read more

27How 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

4Saturated models of firstorder manyvalued logicsLogic Journal of the IGPL 30 (1): 120. 2022.This paper is devoted to the problem of existence of saturated models for firstorder manyvalued 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

12Incompactness of the A1 Fragment of Basic Second Order Propositional Relevant LogicAustralasian Journal of Logic 16 (1): 18. 2019.In this note we provide a simple proof of the incompactness over RoutleyMeyer Bframes of the A1 fragment of the second order propositional relevant language.

22A Lindström Theorem in ManyValued Modal Logic over a Finite MTLchainFuzzy Sets and Systems. forthcoming.We consider a modal language over crisp frames and formulas evaluated on a finite MTLchain (a linearly ordered commutative integral residuated lattice). We first show that the basic modal abstract logic with constants for the values of the MTLchain 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.

115Syntactic characterizations of ﬁrstorder structures in mathematical fuzzy logicSoft 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 firstorder axiomatization. We focus on classes given by universal and universalexistential sentences. In particular, we prove two amalgamation results using the technique of diagrams in the setting of structures valued on a finite MTLalgebra, from which analogues of the Łoś–Tarski and the Chang–Łoś–S…Read more

Model definability in relevant logicIfCoLog Journal of Logics and Their Applications 3 (4): 623646. 2017.It is shown that the classes of RoutleyMeyer 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 modeltheoretic operations involving prime filter extensions, relevant directed bisimulations and disjoint unions.

9Variable Sharing in Substructural Logics: An Algebraic CharacterizationBulletin of the Section of Logic 47 (2): 107115. 2018.We characterize the nontrivial 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 structuresTheoretical 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.

10A Lindström Theorem for Intuitionistic Propositional LogicNotre Dame Journal of Formal Logic 61 (1): 1130. 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.

18On 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.

36Currying 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

51The relevant fragment of first order logicReview of Symbolic Logic 9 (1): 143166. 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 (worldobject) 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.

20Infinitary propositional relevant languages with absurdityReview of Symbolic Logic 10 (4): 663681. 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 Linfinity omega. holds. This yields a preservation result characterizing the expressive power of infinitary relevant languages with absurdity using the modeltheoretic relation of relevant directed bisimulation as well as a Beth definabi…Read more

13On Sahlqvist Formulas in Relevant LogicJournal of Philosophical Logic 47 (4): 673691. 2018.This paper defines a Sahlqvist fragment for relevant logic and establishes that each class of frames in the RoutleyMeyer 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 Sahlqvistvan Benthem algorithm. Furthermore, we show that some classes of RoutleyMeyer frames definable by a relevant formula are not elementary.

62BiSimulating in BiIntuitionistic LogicStudia Logica 104 (5): 10371050. 2016.Biintuitionistic 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 biintuitionistic propositional formulas are exactly those preserved under biintuitionistic 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

33A Lindströmstyle theorem for finitary propositional weak entailment languages with absurdityLogic Journal of the IGPL 24 (2): 115137. 2016.Following a result by De Rijke for modal logic, it is shown that the basic weak entailment modeltheoretic language with absurdity is the maximal modeltheoretic 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 modeltheoretic languages.

83What Is an Inconsistent Truth Table?Australasian Journal of Philosophy 94 (3): 533548. 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 consistencyindependent 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

16A 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 RoutleyMeyer models. One fundamental question is treated: what is the expressive power of relevant languages in the RoutleyMeyer framework? In the case of finitary relevant propositional languages, two answers are provided. The first is that finitary propositional releva…Read more

9A Remark on Maksimova's Variable Separation Property in SuperBiIntuitionistic LogicsAustralasian Journal of Logic 14 (1). 2017.We provide a sucient frametheoretic condition for a super biintuitionistic logic to have Maksimova's variable separation property. We conclude that biintuitionistic logic enjoys the property. Furthermore, we offer an algebraic characterization of the superbiintuitionistic logics with Maksimova's property.
Brisbane, Queensland, Australia
Areas of Specialization
Logic and Philosophy of Logic 
Philosophy of Mathematics 
Areas of Interest
Logic and Philosophy of Logic 
Philosophy of Mathematics 