The paper aims at developing the idea that the standard operator of noncontingency, usually symbolized by Δ, is a special case of a more general operator of dyadic noncontingency Δ(−, −). Such a notion may be modally defined in different ways. The one examined in the paper is __Δ__(B, A) = df ◊B ∧ (A ⥽ B ∨ A ⥽ ¬B), where ⥽ stands for strict implication. The operator of dyadic contingency __∇__(B, A) is defined as the negation of __Δ__(B, (...) A). Possibility (◊A) may be then defined as __Δ__(A, A), necessity (□A) as __∇__(¬A, ¬A) and standard monadic noncontingency (__Δ__A) as __Δ__( \({\textsf{T}}\), A). In the second section it is proved that the deontic system KD is translationally equivalent to an axiomatic system of dyadic noncontingency named KDΔ 2, and that the minimal system KΔ of monadic contingency is a fragment of KDΔ 2. The last section suggests lines for further inquiries. (shrink)

This work concerns the use of operators for past and future contingency in Priorean temporal logic. We will develop a system named C_t, whose language includes a propositional constant and prove that (I) C_t is complete with respect to a certain class of general frames and (II) the usual operators for past and future necessity are definable in such system. Furthermore, we will introduce the extension C_t(lin) that can be interpreted on linear and transitive general frames. The theoretical result of (...) the current work is that contingency can be treated as a primitive notion in reasoning about temporal modalities. (shrink)

We study an extension of first-degree entailment (FDE) by Dunn and Belnap with a non-contingency operator |$\blacktriangle \phi $| which is construed as ‘|$\phi $| has the same value in all accessible states’ or ‘all sources give the same information on the truth value of |$\phi $|’. We equip this logic dubbed |$\textbf {K}^\blacktriangle _{\textbf {FDE}}$| with frame semantics and show how the bi-valued models can be interpreted as interconnected networks of Belnapian databases with the |$\blacktriangle $| operator modelling search (...) for inconsistencies in the provided information. We construct an analytic cut system for the logic and show its soundness and completeness. We prove that |$\blacktriangle $| is not definable via the necessity modality |$\Box $| of |$\textbf {K}_{\textbf{FDE}}$|⁠. Furthermore, we prove that in contrast to the classical non-contingency logic, reflexive, |$\textbf {S4}$| and |$\textbf {S5}$| (among others) frames are definable. (shrink)

It is well known that no consistent normal modal logic contains (as theorems) both ♦A and ♦¬A (for any formula A). Here we observe that this claim can be strengthened to the following: for any formula A, either no consistent normal modal logic contains ♦A, or else no consistent normal modal logic contains ♦¬A.

As shown in Fan, there are some similarities/resemblances between contingency and accident. Given this, one may naturally ask if we can unify the two operators to manifest all of their similarities/resemblances. In this article, instead of looking at the interactions between the two operators like in Fan, we turn our attention to the resemblances between the two operators. We extend the unification method in Fan to the current setting. The main results include some model-theoretical ones, such as expressivity, frame definability, (...) bisimulation, and some axiomatization ones. (shrink)

In this paper, a suitable notion of bisimulation is proposed for the bimodal logic with contingency and accident. We obtain several van Benthem Characterization Theorems, and axiomatize the bimodal logic over the class of Eulidean frames and over some more restricted classes, showing their strong completeness via a novel strategy, thereby answering two open questions raised in the literature. With the new bisimulation notion, we also correct an error in the expressivity results in the literature.

This article proposes the axiomatizations of contingency logics of various natural classes of neighborhood frames. In particular, by defining a suitable canonical neighborhood function, we give sound and complete axiomatizations of monotone contingency logic and regular contingency logic, thereby answering two open questions raised by Bakhtiari, van Ditmarsch, and Hansen. The canonical function is inspired by a function proposed by Kuhn in 1995. We show that Kuhn’s function is actually equal to a related function originally given by Humberstone.

In Fan's 2019 article, “Symmetric Contingency Logic with Unlimitedly Many Modalities”, it is left as an open question in Fan (2019b) how to (completely) axiomatize contingency logic over the class of symmetric and transitive frames, and conjectured that is the desired axiomatization. In the current article, we show that the conjecture is false, and then propose a desired axiomatization, thereby answering the open question. Beyond these results, we also present a family of axiomatizations of contingency logic over Kripke frames.

In this paper, we introduce a notion of ‘disjunctive ignorance’, which is a weak combination of two forms of ignorance in the literature. We propose a logical language with ‘disjunctive ignorance’ as a sole modality, explore the logical properties of this notion and its related notions, and axiomatize it over various frame classes. By finding suitable reduction axioms, we extend the results to the case of public announcements and apply it to Moore-like sentences.