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Grigory Olkhovikov

Publications and source records attributed to Grigory Olkhovikov.

10 recordsLinked to original sources
arXiv2026

Definable Classes of Models and Frames in Bi-intuitionistic Logic

The question of the expressive power of a given logical language with Kripke relational semantics has at least two dimensions: (1) what the language can say about frames, and (2) what it can say about models. The Goldblatt-Thomason theorem provides a model-theoretic characterisation of modal axiomatisability for elementary classes of frames in terms of closure under taking generated subframes, disjoint unions, bounded morphic images, and reflection of ultrafilter extensions. Goldblatt also provides a similar characterisation for axiomatisability in intuitionistic logic of classes of models rather than frames. In this article we provide analogous results for bi-intuitionistic logic, a natural expressive extension of intuitionistic logic obtained by adding a binary connective dual to the intuitionistic implication, introduced in the 1970s independently by Dieter Klemke and Cecylia Rauszer. Together with previous results, such as a van Benthem bisimulation characterisation theorem and a Lindstrom theorem, this provides a complete picture of the expressive power of propositional bi-intuitionistic logic.

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arXiv2024

Conditional reasoning and the shadows it casts onto the first-order logic: the Nelsonian case

We define a natural notion of standard translation for the formulas of conditional logic which is analogous to the standard translation of modal formulas into the first-order logic. We briefly show that this translation works (modulo a lightweight first-order encoding of the conditional models) for the minimal classical conditional logic $\mathsf{CK}$ introduced by Brian Chellas; however, the main result of the paper is that a classically equivalent reformulation of these notions (i.e. of standard translation plus theory of conditional models) also faithfully embeds the basic Nelsonian conditional logic $\mathsf{N4CK}$, introduced in arXiv:2311.02361 into $\mathsf{QN4}$, the paraconsistent variant of Nelson's first-order logic of strong negation. Thus $\mathsf{N4CK}$ is the logic induced by the Nelsonian reading of the classical Chellas semantics of conditionals and can, therefore, be considered a faithful analogue of $\mathsf{CK}$ on the non-classical basis provided by the propositional fragment of $\mathsf{QN4}$. Moreover, the methods used to prove our main result can be easily adapted to the case of modal logic, which allows to improve an older result by S. Odintsov and H. Wansing about the standard translation embedding of the Nelsonian modal logic $\mathsf{FSK}^d$ into $\mathsf{QN4}$.

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arXiv2023

An intuitionistically complete system of basic intuitionistic conditional logic

We introduce a basic intuitionistic conditional logic $\mathsf{IntCK}$ that we show to be complete both relative to a special type of Kripke models and relative to a standard translation into first-order intuitionistic logic. We show that $\mathsf{IntCK}$ stands in a very natural relation to other similar logics, like the basic classical conditional logic $\mathsf{CK}$ and the basic intuitionistic modal logic $\mathsf{IK}$. As for the basic intuitionistic conditional logic $\mathsf{ICK}$ proposed by Y. Weiss, $\mathsf{IntCK}$ extends its language with a diamond-like conditional modality, but its diamond-conditional-free fragment is also a proper extension of $\mathsf{ICK}$. We briefly discuss the resulting gap between the two candidate systems of basic intuitionistic conditional logic and the possible pros and cons of both candidates.

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arXiv2021

Maximality of bi-intuitionistic propositional logic

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 previous work done by the authors for intuitionistic logic in: G. Badia and G. Olkhovikov. A Lindström theorem for intuitionistic propositional logic. Notre Dame Journal of Formal Logic, 61 (1): 11-30 (2020).

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arXiv2021

A Lindström theorem for intuitionistic first-order logic

We extend the main result of (G. Badia and G. Olkhovikov. A Lindström theorem for intuitionistic propositional logic. Notre Dame Journal of Formal Logic, 61 (1): 11--30 (2020)) to the first-order intuitionistic logic (with and without equality), showing that it is the maximal (with respect to expressive power) abstract logic satisfying a certain form of compactness, the Tarski union property and preservation under asimulations. A similar result is also shown for the intuitionistic logic of constant domains.

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arXiv2018

A Lindström theorem for intuitionistic propositional logic

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.

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arXiv2017

Stit logic of justification announcements: a completeness result

We present a completeness result for a logical system which combines stit logic and justification logic in order to represent proving activity of the agents. This logic is interpreted over the semantics introduced in earlier publications. We define a Hilbert-style axiomatic system for this logic and show that this system is strongly complete relative to the intended semantics.

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arXiv2016

On generalized Van-Benthem-type characterizations

The paper continues the line of model-theoretic characterizations for versions of intuitionistic logic previously achieved by the author, further generalizing them. This results in a model-theoretic characterization of expressive powers of arbitrary finite sets of guarded connectives of degree not exceeding 1 and regular connectives of degree 2 over the language of bounded lattices.

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arXiv2015

Expressive power of basic modal intuitionistic logic as a fragment of classical FOL

The paper treats 4 different fragments of first-order logic induced by their respective versions of Kripke style semantics for modal intuitionistic logic. In order to capture these fragments, the notion of asimulation is modified and extended to yield Van Benthem type of semantic characterization of their respective expressive powers. It is shown, further, that this characterization can be easily carried over to arbitrary first-order definable subclasses of classical first-order models.

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