ORCID Profile
0000-0002-5597-6794
Current Organisation
University of Queensland
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Expanding Knowledge in the Mathematical Sciences | Expanding Knowledge in Philosophy and Religious Studies |
Publisher: Institute of Electrical and Electronics Engineers (IEEE)
Date: 09-2022
Publisher: Institute of Electrical and Electronics Engineers (IEEE)
Date: 06-2021
Publisher: Oxford University Press (OUP)
Date: 03-08-2022
Abstract: 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 this theorem we obtain, as by-products, some results on tableaux (understood as pairs of sets of formulas) and their consistency and satisfiability and a generalization of the Tarski–Vaught theorem on unions of elementary chains. Finally, we provide a structural characterization of $\\kappa $-saturation in terms of the completion of a diagram representing a certain configuration of models and mappings.
Publisher: Elsevier BV
Date: 06-2020
Publisher: Elsevier BV
Date: 08-2018
Publisher: Cambridge University Press (CUP)
Date: 29-06-2020
DOI: 10.1017/S175502032000012X
Abstract: 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 a much weaker arithmetic theory, a version of Robinson’s R (with arithmetic operations also interpreted as mere relations). Our result is based on a structural version of the undecidability argument introduced by Kleene and we show that it goes well beyond the scope of the Boolean, intuitionistic, or fuzzy logic.
Publisher: Cambridge University Press (CUP)
Date: 30-06-2023
DOI: 10.1017/S1755020323000205
Abstract: We generalize the notion of consequence relation standard in abstract treatments of logic to accommodate intuitions of relevance . The guiding idea follows the use criterion , according to which in order for some premises to have some conclusion(s) as consequence(s), the premises must each be used in some way to obtain the conclusion(s). This relevance intuition turns out to require not just a failure of monotonicity, but also a move to considering consequence relations as obtaining between multisets . We motivate and state basic definitions of relevant consequence relations, both in single conclusion (asymmetric) and multiple conclusion (symmetric) settings, as well as derivations and theories, guided by the use intuitions, and prove a number of results indicating that the definitions capture the desired results (at least in many cases).
Publisher: Cambridge University Press (CUP)
Date: 07-06-2023
DOI: 10.1017/S175502032300014X
Abstract: In this note we study a counterpart in predicate logic of the notion of logical friendliness , introduced into propositional logic in [15]. The result is a new consequence relation for predicate languages with equality using first-order models. While compactness, interpolation and axiomatizability fail dramatically, several other properties are preserved from the propositional case. Divergence is diminished when the language does not contain equality with its standard interpretation.
Publisher: Springer Science and Business Media LLC
Date: 27-02-2016
Publisher: Springer Science and Business Media LLC
Date: 22-08-2018
Publisher: Oxford University Press (OUP)
Date: 05-08-2023
Abstract: By limiting the range of the predicate variables in a second-order language, one may obtain restricted versions of second-order logic such as weak second-order logic or definable subset logic. In this note, we provide an infinitary strongly complete axiomatization for several systems of this kind having the range of the predicate variables as a parameter. The completeness argument uses simple techniques from the theory of Boolean algebras. This article is dedicated to our friend John N. Crossley on the occasion of his 86th birthday.
Publisher: Oxford University Press (OUP)
Date: 27-09-2022
Abstract: In the style of Lindström’s theorem for classical first-order logic, this article characterizes propositional bi-intuitionistic logic as the maximal (with respect to expressive power) abstract logic satisfying a certain form of compactness, the Tarski union property and preservation under bi-asimulations. Since bi-intuitionistic logic introduces new complexities in the intuitionistic setting by adding the analogue of a backwards looking modality, the present paper constitutes a non-trivial modification of the previous work done by the authors for intuitionistic logic (Badia and Olkhovikov, 2020, Notre Dame Journal of Formal Logic, 61, 11–30).
Publisher: Cambridge University Press (CUP)
Date: 10-01-2023
DOI: 10.1017/JSL.2023.2
Abstract: 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 validity under certain conditions. In the proofs, we use a form of strong upwards Löwenheim–Skolem theorem not available in the framework with identity.
Publisher: Elsevier BV
Date: 03-2021
Publisher: Victoria University of Wellington Library
Date: 13-07-2023
Abstract: This is an introduction to the special issue of the AJL on Val Plumwood's manuscript "False Laws of Logic" and her other work in logic.
Publisher: Elsevier BV
Date: 07-2023
Publisher: Springer Science and Business Media LLC
Date: 22-02-2019
Publisher: Cambridge University Press (CUP)
Date: 12-08-2022
DOI: 10.1017/S1755020322000296
Abstract: In this article we show that bi-intuitionistic predicate logic lacks the Craig Interpolation Property. We proceed by adapting the counterex le 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 is about the property more correctly named ‘deductive interpolation’ (see Galatos, Jipsen, Kowalski and Ono’s use of this term in [5]) for global consequence. Given that the deduction theorem fails for bi-intuitionistic logic with global consequence, the two formulations of the property are not equivalent.
Publisher: Elsevier BV
Date: 03-2022
Publisher: Springer Science and Business Media LLC
Date: 24-03-2022
DOI: 10.1007/S10992-022-09651-X
Abstract: 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(s) 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-capture’ versions of standard theorems, given suitable restrictions and background assumptions but the shift to non-classical logic may recast the meanings of these apparently ‘absolute’ theorems.
Publisher: Oxford University Press (OUP)
Date: 06-09-2023
Publisher: Elsevier BV
Date: 12-2023
Start Date: 2022
End Date: 2025
Funder: Australian Research Council
View Funded ActivityStart Date: 07-2022
End Date: 07-2025
Amount: $356,000.00
Funder: Australian Research Council
View Funded Activity