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Grigory K. Olkhovikov

Publications and source records attributed to Grigory K. Olkhovikov.

11 recordsLinked to original sources
arXiv2025

On connexivity in modal and conditional contexts

We define and axiomatize three new logics based on the connexive logic $\mathsf{C}$, the modal logic $\mathsf{CnK}$ and the conditional logics $\mathsf{CnCK}$ and $\mathsf{CnCK}_R$. These logics display strong connexivity properties and are connected to one another, since $\mathsf{CnCK}_R$ is the reflexive extension of $\mathsf{CnCK}$ and $\mathsf{CnK}$ is faithfully embeddable into both $\mathsf{CnCK}$ and $\mathsf{CnCK}_R$ in a multitude of natural ways. We argue that all the three logics provide (albeit in different ways) natural expansions of $\mathsf{C}$ to their respective languages that preserve and further develop several core properties of $\mathsf{C}$, especially its connexivity profile.

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arXiv2023

A basic system of paraconsistent Nelsonian logic of conditionals

We define a Kripke semantics for a conditional logic based on the propositional logic $\mathsf{N4}$, the paraconsistent variant of Nelson's logic of strong negation; we axiomatize the minimal system induced by this semantics. The resulting logic, which we call $\mathsf{N4CK}$, shows strong connections both with the basic intuitionistic logic of conditionals $\mathsf{IntCK}$ introduced earlier in arXiv:2306.10402 and with the $\mathsf{N4}$-based modal logic $\mathsf{FSK}^d$ introduced by S. Odintsov and H. Wansing as one of the possible counterparts to the classical modal system $\mathsf{K}$. We map these connections by looking into the embeddings which obtain between the aforementioned systems.

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arXiv2022

Craig interpolation theorem fails in bi-intuitionistic predicate logic

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 (G. Mints, G. K. Olkhovikov and A. Urquhart. Failure of Interpolation in Constant Domain Intuitionistic Logic. Journal of Symbolic Logic, 78: 937--950 (2013)). More precisely, we show that there is a valid implication $ϕ\rightarrow ψ$ with no interpolant (i.e. a formula $θ$ in the intersection of the vocabularies of $ϕ$ and $ψ$ such that both $ϕ\rightarrow θ$ and $θ\rightarrow ψ$ are valid). Importantly, this result does not contradict the unfortunately named `Craig interpolation' theorem established by Rauszer in (Cecylia Rauszer. Craig Interpolation Theorem for an Extention of Intuitionistic Logic. Bull. Ac. Pol. Sc., 25(4), 337--341 (1977)) since that article is about the property more correctly named `deductive interpolation' (see Galatos, Jipsen, Kowalski and Ono's use of this term in N. Galatos, P. Jipsen, T. Kowalski, \& H. Ono. Residuated Lattices: An Algebraic Glimpse at Substructural Logics. Studies in Logic and the Foundations of Mathematics, Vol. 151. Amsterdam: Elsevier B. V. (2007)) 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.

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arXiv2018

Restricted Interpolation and Lack Thereof in Stit Logic

We consider the propositional logic equipped with Chellas stit operators for a finite set of individual agents plus the historical necessity modality. We settle the question of whether such a logic enjoys restricted interpolation property, which requires the existence of an interpolant only in cases where the consequence contains no Chellas stit operators occurring in the premise. We show that if action operators count as logical symbols, then such a logic has restricted interpolation property iff the number of agents does not exceed three. On the other hand, if action operators are considered to be non-logical symbols, the restricted interpolation fails for any number of agents exceeding one. It follows that unrestricted Craig interpolation also fails for almost all versions of stit logic.

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arXiv2017

Explicit justification stit logic: a completeness result

We consider the explicit fragment of the basic justification stit logic 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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arXiv2017

Justification announcements in discrete time. Part II: Frame definability results

In Part I of this paper, we presented a Hilbert-style system $Σ_D$ axiomatizing of stit logic of justification announcements (JA-STIT) interpreted over models with discrete time structure. In this part, we prove three frame definability results for $Σ_D$ using three different definitions of a frame plus a yet another version of completeness result.

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arXiv2017

A completeness result for implicit justification stit logic

We present a completeness result for the implicit fragment of justification stit logic. Although this fragment allows for no strongly complete axiomatization, we show that a restricted form of strong completeness (subsuming weak completeness) is available, as well as deduce a version of restricted compactness property.

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arXiv2012

Model-theoretic characterization of intuitionistic propositional formulas

Notions of k-asimulation and asimulation are introduced as asymmetric counterparts to k-bisimulation and bisimulation, respectively. It is proved that a first-order formula is equivalent to a standard translation of an intuitionistic propositional formula iff it is invariant with respect to k-asimulations for some k, and then that a first-order formula is equivalent to a standard translation of an intuitionistic propositional formula iff it is invariant with respect to asimulations. Finally, it is proved that a first-order formula is intuitionistically equivalent to a standard translation of an intuitionistic propositional formula iff it is invariant with respect to asimulations between intuitionistic models.

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arXiv2012

Model-theoretic characterization of predicate intuitionistic formulas

Notions of asimulation and k-asimulation introduced in [Olkhovikov, 2011] are extended onto the level of predicate logic. We then prove that a first-order formula is equivalent to a standard translation of an intuitionistic predicate formula iff it is invariant with respect to k-asimulations for some k, and then that a first-order formula is equivalent to a standard translation of an intuitionistic predicate formula iff it is invariant with respect to asimulations. Finally, it is proved that a first-order formula is equivalent to a standard translation of an intuitionistic predicate formula over a class of intuitionistic models (intuitionistic models with constant domain) iff it is invariant with respect to asimulations between intuitionistic models (intuitionistic models with constant domain).

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