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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 $\mathsf{QN4}$., Comment: 53 pages, 1 figure

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., Comment: 35 pages, 3 diagrams. arXiv admin note: text overlap with arXiv:2306.10402

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., Comment: 36 pages, 0 figures

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 $\phi \rightarrow \psi$ with no interpolant (i.e. a formula $\theta$ in the intersection of the vocabularies of $\phi$ and $\psi$ such that both $\phi \rightarrow \theta$ and $\theta \rightarrow \psi$ 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., Comment: 23 pages, 0 figures

In the style of Lindstr\"om'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 Lindstr\"om theorem for intuitionistic propositional logic. Notre Dame Journal of Formal Logic, 61 (1): 11-30 (2020)., Comment: 27 pages, 0 figures. arXiv admin note: text overlap with arXiv:2103.17024

We extend the main result of (G. Badia and G. Olkhovikov. A Lindstr\"om 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., Comment: 50 pages, 0 figures

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.

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., Comment: 25 pages, 0 figures

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., Comment: 32 pages, 0 figures. arXiv admin note: substantial text overlap with arXiv:1707.03292

In Part I of this paper, we presented a Hilbert-style system $\Sigma_D$ 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 $\Sigma_D$ using three different definitions of a frame plus a yet another version of completeness result., Comment: 21 pages, 0 figures

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., Comment: 33 pages, 0 figures. arXiv admin note: text overlap with arXiv:1705.09119

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., Comment: 38 pages, 0 figures, 1 table

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., Comment: 44 pages, 0 figures

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., Comment: 29 pages, 0 figures. arXiv admin note: text overlap with arXiv:1207.4414

We formulate a Hilbert-style axiomatic system for STIT logic of imagination recently proposed by H. Wansing and prove its completeness by the method of canonical models., Comment: 8 pages, 0 figures

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., Comment: 16 pages, 0 figures. arXiv admin note: substantial text overlap with arXiv:1202.1195

Drawing on the previous work on interpolation failure, we show that Beth's definability theorem does not hold for intuitionistic predicate logic of constant domains without identity., Comment: 9 pages, 0 figures. Several misprints corrected

This paper shows that the interpolation theorem fails in the intuitionistic logic of constant domains. This result refutes two previously published claims that the interpolation property holds., Comment: 13 pages, 0 figures. Overlaps with arXiv 1202.1195 removed, the text thouroughly reworked in terms of notation and style, historical notes as well as some other minor details added

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).