In this work we define contingency logic with arbitrary announcement. In contingency logic, the primitive modality contingency formalises that a proposition may be true but also may be false, so that if it is non-contingent then it is necessarily true or necessarily false. To this logic one can add dynamic operators to describe change of contingency. Our logic has operators for public announcement and operators for arbitrary public announcement, as in the dynamic epistemic logic called arbitrary public announcement logic. However, (...) our language primitive is the more suitable notion of public announcement whether, instead of the public announcement that. We compare the expressive power of our logic and its various fragments to related dynamic epistemic logics. We further present an axiomatisation and show its completeness by adapting a method to demonstrate completeness of arbitrary public announcement logic. Various extensions are also shown to be complete with respect to the corresponding frame classes. (shrink)

Simple finite axiomatizations are given for versions of the modal logics K and K4 with non-contingency (or contingency) as the sole modal primitive. This answers two questions of I. L. Humberstone.

We show the Boxdot Conjecture holds for a limited but familiar range of Lemmon-Scott axioms. We re-introduce the language of essence and accident, first introduced by J. Marcos, and show how it aids our strategy.

We introduce an operator to represent the simple notion of being wrong. Read Wp to mean: the agent is wrong about p . Being wrong about p means believing p though p is false. We add this operator to the language of propositional logic and study it. We introduce a canonical model for logics of being wrong, show completeness for the minimal logic of being wrong and various other systems. En route we examine the expressiveness of the language. In conclusion, (...) we discuss an open question regarding K4. (shrink)

A proposition is noncontingent, if it is necessarily true or it is necessarily false. In an epistemic context, ‘a proposition is noncontingent’ means that you know whether the proposition is true. In this paper, we study contingency logic with the noncontingency operator? but without the necessity operator 2. This logic is not a normal modal logic, because?→ is not valid. Contingency logic cannot define many usual frame properties, and its expressive power is weaker than that of basic modal logic over (...) classes of models without reflexivity. These features make axiomatizing contingency logics nontrivial, especially for the axiomatization over symmetric frames. In this paper, we axiomatize contingency logics over various frame classes using a novel method other than the methods provided in the literature, based on the ‘almost-definability’ schema AD proposed in our previous work. We also present extensions of contingency logic with dynamic operators. Finally, we compare our work to the related work in the fields of contingency logic and ignorance logic, where the two research communities have similar results but are apparently unaware of each other’s work. One goal of our paper is to bridge this gap. (shrink)

Contingency and accident are two important notions in philosophy and philosophical logic. Their meanings are so close that they are mixed up sometimes, in both daily life and academic research. This indicates that it is necessary to study them in a unified framework. However, there has been no logical research on them together. In this paper, we propose a language of a bimodal logic with these two concepts, investigate its model-theoretical properties such as expressivity and frame definability. We axiomatize this (...) logic over various classes of frames, whose completeness proofs are shown with the help of a crucial schema. The interactions between contingency and accident can sharpen our understanding of both notions. Then we extend the logic to a dynamic case: public announcements. By finding the required reduction axioms, we obtain a complete axiomatization, which gives us a good application to Moore sentences. (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.

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.

We propose a logic of temporal contingency, which has operators of past and future contingency as primitive modalities. This logic is less expressive than standard temporal logic over the class of bidirectional frames, and cannot define some basic frame properties such as bidirectionality and transitivity. We present a minimal system based on two key ‘bridge axioms’ and a bimodal version of a so-called ‘almost definability’ schema in the literature. The completeness proof is highly nontrivial due to the requirement that the (...) canonical model be bidirectional. We then extend the results to the simplest dynamic case: public announcements. (shrink)