•  59
    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
  •  39
    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
  •  38
    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.
  •  19
    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.
  •  19
    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
  •  16
    Infinitary propositional relevant languages with absurdity
    Review of Symbolic Logic 10 (4): 663-681. 2017.
  •  13
    Currying Omnipotence: A Reply to Beall and Cotnoir
    with Andrew Tedder
    Thought: A Journal of Philosophy 7 (2): 119-121. 2018.
  •  11
    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
  •  7
    On elimination of quantifiers in some non-classical mathematical theories
    with Andrew Tedder
    Mathematical Logic Quarterly 64 (3): 140-154. 2018.
  •  6
    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.
  •  6
    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.
  •  4
    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.
  •  3
    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.
  •  2
    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.
  • Fraïssé classes of graded relational structures
    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.
  • 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.
  • A Lindström theorem for intuitionistic propositional logic
    Notre Dame Journal of Formal Logic. forthcoming.
    It is shown that propositional intuitionistic logic is the maximal (with respect to expressive power) abstract logic satisfying a certain topological property reminiscent of compactness, the Tarski union property and preservation under asimulations.