The completeness of the axiomatization of contingency logic over symmetric frames has been thought of as a nontrivial job, the unimodal case of which cannot be generalized to the finitely multimodal case, which in turn cannot be generalized to the infinitely multimodal case. This paper deals with the completeness of symmetric contingency logic with unlimitedly many modalities, no matter whether the set of modalities is finite or infinite.

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. (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.

A formula is a contingent logical truth when it is true in every model M but, for some model M , false at some world of M . We argue that there are such truths, given the logic of actuality. Our argument turns on defending Tarski’s definition of truth and logical truth, extended so as to apply to modal languages with an actuality operator. We argue that this extension is the philosophically proper account of validity. We counter recent arguments (...) to the contrary presented in Hanson’s ‘Actuality, Necessity, and Logical Truth’ (Philos Stud 130:437–459, 2006 ). (shrink)

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. (shrink)

At least since Aristotle’s famous 'sea-battle' passages in On Interpretation 9, some substantial minority of philosophers has been attracted to the doctrine of the open future--the doctrine that future contingent statements are not true. But, prima facie, such views seem inconsistent with the following intuition: if something has happened, then (looking back) it was the case that it would happen. How can it be that, looking forwards, it isn’t true that there will be a sea battle, while also being true (...) that, looking backwards, it was the case that there would be a sea battle? This tension forms, in large part, what might be called the problem of future contingents. A dominant trend in temporal logic and semantic theorizing about future contingents seeks to validate both intuitions. Theorists in this tradition--including some interpretations of Aristotle, but paradigmatically, Thomason (1970), as well as more recent developments in Belnap, et. al (2001) and MacFarlane (2003, 2014)--have argued that the apparent tension between the intuitions is in fact merely apparent. In short, such theorists seek to maintain both of the following two theses: (i) the open future: Future contingents are not true, and (ii) retro-closure: From the fact that something is true, it follows that it was the case that it would be true. It is well-known that reflection on the problem of future contingents has in many ways been inspired by importantly parallel issues regarding divine foreknowledge and indeterminism. In this paper, we take up this perspective, and ask what accepting both the open future and retro-closure predicts about omniscience. When we theorize about a perfect knower, we are theorizing about what an ideal agent ought to believe. Our contention is that there isn’t an acceptable view of ideally rational belief given the assumptions of the open future and retro-closure, and thus this casts doubt on the conjunction of those assumptions. (shrink)

Contingent conditionals, at the comparative logical works, has yielded different, and often inconsistent, analyses. At contemporaries’ works, there are two approaches to the contingent conditionals: modal and truth-table methods. At the former, there has been used modal connectives of necessity and possibility to analyze the contingent conditionals; but at the latter, some truth-tables has been proposed. At the paper, besides presenting and criticizing the extant theories, we gain new analyses and formulations of the disputed subject matters.

The philosophical, logical, and terminological junctions between Description Logics (DLs) and Modal Logic (ML) are important because they can support the formal analysis of modal notions of ‘possibility’ and ‘necessity’ through the lens of DLs. This paper introduces functional contingents in order to (i) structurally and terminologically analyse ‘functional possibility’ and ‘functional necessity’ in DL world descriptions and (ii) logically and terminologically annotate DL world descriptions based on functional contingents. The most significant contributions of this research are the logical (...) characterisation and terminological analysis of functional contingents in DL world descriptions. The ultimate goal is to investigate how modal operators can – logically and terminologically – be expressed within DL world descriptions. (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)

I show that intuitive and logical considerations do not justify introducing Leibniz’s Law of the Indiscernibility of Identicals in more than a limited form, as applying to atomic formulas. Once this is accepted, it follows that Leibniz’s Law generalises to all formulas of the first-order Predicate Calculus but not to modal formulas. Among other things, identity turns out to be logically contingent.

We consider the modal logic of non-contingency in a general setting, without making special assumptions about the accessibility relation. The basic logic in this setting is axiomatized, and some of its extensions are discussed, with special attention to the expressive weakness of the language whose sole modal primitive is non-contingency , by comparison with the usual language based on necessity.

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)

Current orthodoxy in modal logic and metaphysics has it that actuality is non-contingent in the following sense: for all p, if actually, p, then necessarily, actually, p. Call this thesis (Actuality) Necessitism and its negation (Actuality) Contingentism. Thus, according to Contingentism, there is at least one proposition p which is actually true but which could have been actually false. In another paper, one of us (Glazier 2023) has recently defended Contingentism. The present paper explores the logic of actuality (...) under Contingentism. After informally introducing Contingentism and outlining what we take to be the main motivations for accepting it, we consider how contingent actuality should be taken to be. We argue that the nature, or essence, of actuality imposes certain limits on the extent to which actuality could have been different, and these then inform our view of the logic of actuality. Turning to the formal study of the logic of contingent actuality, we specify a formal language and a possible world semantics for that language and distinguish a number of natural notions of validity and consequence to which it gives rise. We present axiomatizations of the different resulting logics and examine the variety of iterated modalities in our systems. We establish soundness and completeness results. (shrink)

It is argued here that Prior's non-standard modal system Q, and the Parry-Dunn system of analytic implication, though entirely independent and independently motivated systems, together provide a rationale for explicating the concept of validity in a non-standard way; their implications are explored for the theory of natural deduction as well as for modal logic and the concept of entailment. I give an account of formal logic from this non-standard viewpoint, together with an informal presentation of the system that (...) unites the insights of Prior (drawing on Russell) and, Parry (drawing on Kant), and the motivations for both in the concept of the contingent existence - as opposed to the contingent truth or falsehood - of a proposition. (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 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)

There are two general approaches to handling contingencies in decision-theoretic planning. State-space planners reason globally, building a map of the parts of the world relevant to the planning problem, and then attempt to distill a plan out of the map. POCL planners reason locally, attempting to build the plan up from local relationships. A planning problem is constructed that humans find trivial, but no state-space planner can solve. This motivates an investigation of decision-theoretic POCL contingency planners. Existing POCL (...) class='Hi'>contingency planners attempt to generalize the results of classical POCL contingency planning. However, this paper argues that the nature of contingency planning changes dramatically in decision-theoretic contexts, and results from classical contingency planning are of little relevance. In particular, in classical planning contingencies can only be attached to conditional forks, but in most uses of contingencies in decision-theoretic planning they are attached to single branches of the plan rather than to conditional forks. A criterion of adequacy for contingency planners is formulated, following from ordinary completeness, and it is shown that existing decision-theoretic POCL contingency planners do not satisfy it. Some tentative suggestions are made regarding how to construct a planner that does satisfy the adequacy condition. (shrink)

Timothy Williamson gives an original and provocative treatment of deep metaphysical questions about existence, contingency, and change, using the latest resources of quantified modal logic. Contrary to the widespread assumption that logic and metaphysics are disjoint, he argues that modal logic provides a structural core for metaphysics.

Speech given by Markus Gabriel within the international meeting “Presente del idealismo alemán” organized by the Philosophy Department of the Universidad Nacional de Colombia. This conference was held on October 9, 2009.

In constructing the three-valued logic, Jan Łukasiewicz was highly inspirited by the Aristotelian idea of logical contingency. Nevertheless, we can construct a four-valued logic for explicating the Stoic idea of logical determinacy. In this system, we have the following truth values: 0 (‘possibly false), 1 (‘necessarily false’), 2 (‘possibly true’), 3 (‘necessarily true’), where the designated truth value is represented by the two values: 2 and 3.

In this essay I propose to examine Hegel’s account of necessity and contingency in the Science of Logic. Anyone who dares to take Hegel’s Logic seriously in public risks being accused by legions of formal logicians of “elementary logical fallacies”. Nevertheless, John Burbidge, Dieter Henrich, and others have demonstrated that it is possible to discuss the Logic with clarity and intelligibility, and I shall endeavor to emulate their example as best as I can. One should take (...) heed, however; even Hegel admits that “the concept of necessity is very difficult,” and rendering the concept of necessity intelligible will not eliminate that difficulty. All I can hope to do, therefore, is take us from the point at which we are not even sure what the difficulty is supposed to be to the point at which the nature of the difficulty itself becomes clear. (shrink)

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)

Critical (necessary or sufficient) features in categorisation have a long history, but the empirical evidence makes their existence questionable. Nevertheless, there are some cases that suggest critical feature effects. The purpose of the present work is to offer some insight into why classification decisions might misleadingly appear as if they involve critical features. Utilising Tversky's (1977) contrast model of similarity, we suggest that when an object has a sparser representation, changing any of its features is more likely to lead to (...) a change in identity than it would in objects that have richer representations. Experiment 1 provides a basic test of this suggestion with artificial stimuli, whereby objects with a rich or a sparse representation were transformed by changing one of their features. As expected, we observed more identity judgements in the former case. Experiment 2 further confirms our hypothesis, with realistic stimuli, by assuming that superordinate categories have sparser representations than subordinate ones. These results offer some insight into the way feature changes may or may not lead to identity changes in classification decisions. (shrink)

Recently, Bourne constructed a system of three-valued logic that he supposed to replace Łukasiewicz’s three-valued logic in view of the problems of future contingents. In this paper, I will show first that Bourne’s system makes no improvement to Łukasiewicz’s system. However, finding some good motivations and lessons in his attempt, next I will suggest a better way of achieving his original goal in some sense. The crucial part of my way lies in reconsidering the significance of the intermediate (...) truth-value so as to reconstruct Łukasiewicz’s three-valued logic as a kind of extensional modal logic based on partial logic. (shrink)

Prior's three-valued modal logic Q was developed as a philosophically interesting modal logic. Thus, we should be able to modify Q as a temporal logic. Although a temporal version of Q was suggested by Prior, the subject has not been fully explored in the literature. In this paper, we develop a three-valued temporal logic $Q_t $ and give its axiomatization and semantics. We also argue that $Q_t $ provides a smooth solution to the problem of future (...) contingents. (shrink)

Inspired by G W F Hegel's system, Johan Ludvig Heiberg authored a series of essays and monographs on different philosophical issues in both Danish and German; ...

In this paper, I argue against the view there are contingent a priori truths, and against the related view that there are contingent logical truths. I will suggest that in general, predicates ›a priori‹ and ›contingent‹ are implicitly relativized to circumstances, and argue that apriority entails necessity, whenever the two are relativized to the same circumstance. I will then criticize the idea, inspired by David Kaplan's framework, of contingent contents "knowable under a priori characters." I will also argue, against Kaplan, (...) that sentences of the form »The actual F is F« do not deserve the status of logical truths, since what they express is neither necessary nor a priori. (shrink)

Echte Freiheit läßt sich nach Leibniz nicht auf die Möglichkeit reduzieren, so zu handeln, wie man handeln will. Vielmehr muß echte Freiheit mit dem moralischen Verdienst in Verbindung gebracht werden. In diesem Zusammenhang erweist sich die Idee der logischen Kontingenz von besonderer Wichtigkeit. Eine Person hat kein moralisches Verdienst, wenn sie blindlings durch das Gute geleitet wird. Sie hat allein dann ein moralisches Verdienst, wenn sie das Schlechte zurückweist und bewußt das Gute wählt, das sich vom Schlechten abhebt. Insofern ist (...) derjenige, der moralisch richtig handelt, sich anderer Möglichkeiten bewußt. Leibniz trägt dieser Erfahrungstatsache durch den Gedanken Rechnung, daß eine freie Handlung logisch kontingent ist. Diesem Gedanken verleiht Leibniz einen metaphysischen Status: die möglichen Welten haben einen objektiven Gehalt. (shrink)

To answer the question, we need first to consider the notion of necessity and the related notion of contingency. These are so-called "modal" notions. Other modal notions include those of possibility, impossibility, non-necessity, and noncontingency. All play a crucial role in philosophical thinking about matters to do with logic, metaphysics, morality, law, etc. This is because none of these modal notions is univocal in meaning. There are, so to speak, different "species" of the generic notions of necessity, (...) class='Hi'>contingency, possibility, and the rest. (shrink)

For reasons that have more to do with the historiographical traditions of modern Jewish history and the history of critical thought than history itself, Georg Simmel – of Jewish descent – is rarely discussed within the frame of modern Jewish history. Bringing the two together as a theoretical contribution to Simmel studies and modern Jewish history alike, this article explores Simmel’s logic of contingency in the context of modern Jewish history. Which forms and types could Jews realistically seek (...) to fulfill from the perspective of Simmel’s thought? Which could they hope to escape and what could they expect from the future? The author suggests that in answering these questions we disclose a peculiar notion of ‘bounded contingency’ embedded in Simmel’s positing of a non-binary, ‘gray area’ between the necessary and the impossible. This hypothesis is tested in several distinct contexts: a letter on Zionism from around 1900; an odd passage on love in the excursus on ‘The Stranger’ ; and a letter on individuals of Jewish background in German academia. Simmel’s coherency lies in the ‘realist’ approach he adopts to reality as the domain of the contingently possible. When applied to Jewish history in general and Zionism in particular, this notion not only explains why Zionism is unfeasible but also demystifies its emergence as an attempt to escape the status of stranger in Europe. It also evinces why this Jewish national enterprise inevitably carried within it the seeds of the problem it sought to solve. (shrink)