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The provability logic of a theory $T$ captures the structural behavior of formalized provability in $T$ as provable in $T$ itself. Like provability, one can formalize the notion of relative interpretability giving rise to interpretability…
This paper continues the author's previous study \cite{Kura20}, showing that several weak principles inspired by non-normal modal logic suffice to derive various refined forms of the second incompleteness theorem. Among the main results of…
Metric Temporal Logic, $\mtlfull$ is amongst the most studied real-time logics. It exhibits considerable diversity in expressiveness and decidability properties based on the permitted set of modalities and the nature of time interval…
We establish completeness for intuitionistic first-order logic, iFOL, showing that a formula is provable if and only if its embedding into minimal logic, mFOL, is uniformly valid under the Brouwer Heyting Kolmogorov (BHK) semantics, the…
We introduce syntactic modal operator $\BOX$ for \textit{being a thesis} into first-order logic. This logic is a modern realization of R. Carnap's old ideas on modality, as logical necessity (J. Symb. Logic, 1946) \cite{Ca46}. We place it…
Thomason \cite{Thomason74} showed that a certain modal logic $\mathbf{L}\subset \mathbf{S4}$ is incomplete with respect to Kripke semantics. Later Gerson \cite{Gerson75} showed that $\mathbf{L}$ is also incomplete with respect to…
Canonical models are of central importance in modal logic, in particular as they witness strong completeness and hence compactness. While the canonical model construction is well understood for Kripke semantics, non-normal modal logics…
We determine the ZF-provable modal logic of the modality $\Box_{\mathrm{sym}}$, where $\Box_{\mathrm{sym}}\varphi$ means '$\varphi$ holds in every finite symmetry-preserving iteration' of the symmetric method. We prove that the exact logic…
Vaught's Conjecture states that if $T$ is a complete first order theory in a countable language that has more than $\aleph_0$ pairwise non-isomorphic countably infinite models, then $T$ has $2^{\aleph_0}$ such models. Morley showed that if…
We determine the modal logic of fixed-point models of truth and their axiomatizations by Solomon Feferman via Solovay-style completeness results. Given a fixed-point model $\mathcal{M}$, or an axiomatization $S$ thereof, we find a modal…
We investigate algebraic and topological semantics of the modal logic S4CI and obtain strong completeness of the given system in the case of local semantic consequence relations. In addition, we consider an extension of the logic S4CI with…
For an ordinal $\lambda>0$, we use the Erd\H{o}s--Rado partition theorem to prove the failure of strong completeness of $\mathsf{GL}$ for modal languages of cardinality $(2^{|\lambda|+\aleph_0})^{+}$ with respect to models on ordinals…
By Solovay's celebrated completeness result on formal provability we know that the provability logic $\mathrm GL$ describes exactly all provable structural properties for any sound and strong enough arithmetical theory with a decidable…
In this paper we investigate the complexity-theoretical aspects of cyclic and non-wellfounded proofs in the context of parsimonious logic, a variant of linear logic where the exponential modality ! is interpreted as a constructor for…
We study elementary modal logics, i.e. modal logic considered over first-order definable classes of frames. The classical semantics of modal logic allows infinite structures, but often practical applications require to restrict our…
In a previous paper, we showed that profinite $L$-algebras (where $L$ is a variety of modal algebras generated by its finite members) are monadic over $\mathbf{Set}$. This monadicity result suggests that profinite $L$-algebras could be…
Modal dependence logic was introduced recently by V\"a\"an\"anen. It enhances the basic modal language by an operator =(). For propositional variables p_1,...,p_n, =(p_1,...,p_(n-1);p_n) intuitively states that the value of p_n is…
Polymodal provability logic GLP is incomplete w.r.t. Kripke frames. It is known to be complete w.r.t. topological semantics, where the diamond modalities correspond to topological derivative operations. However, the topologies needed for…
This paper presents the logic QRC$_1$, which is a strictly positive fragment of quantified modal logic. The intended reading of the diamond modality is that of consistency of a formal theory. Predicate symbols are interpreted as…
We deal with the fragment of modal logic consisting of implications of formulas built up from the variables and the constant `true' by conjunction and diamonds only. The weaker language allows one to interpret the diamonds as the uniform…