The paper surveys the currently available axiomatizations of common belief (CB) and common knowledge (CK) by means of modal propositional logics. (Throughout, knowledge- whether individual or common- is defined as true belief.) Section 1 introduces the formal method of axiomatization followed by epistemic logicians, especially the syntax-semantics distinction, and the notion of a soundness and completeness theorem. Section 2 explains the syntactical concepts, while briefly discussing their motivations. Two standard semantic constructions, Kripke structures and neighbourhood structures, are introduced in Sections (...) 3 and 4, respectively. It is recalled that Aumann's partitional model of CK is a particular case of a definition in terms of Kripke structures. The paper also restates the well-known fact that Kripke structures can be regarded as particular cases of neighbourhood structures. Section 3 reviews the soundness and completeness theorems proved w.r.t. the former structures by Fagin, Halpern, Moses and Vardi, as well as related results by Lismont. Section 4 reviews the corresponding theorems derived w.r.t. the latter structures by Lismont and Mongin. A general conclusion of the paper is that the axiomatization of CB does not require as strong systems of individual belief as was originally thought- only monotonicity has thusfar proved indispensable. Section 5 explains another consequence of general relevance: despite the "infinitary" nature of CB, the axiom systems of this paper admit of effective decision procedures, i.e., they are decidable in the logician's sense. (shrink)

The problem of Common Knowledge will be considered in two classes of models: a class K.* of Kripke models and a class S of Scott models. Two modal logic systems will be defined. Those systems, KC and MC, include an axiomatisation of Common Knowledge. We prove determination of each system by the corresponding class of models. MSC: 03B45, 68T25.

The coordinated attack scenario and the electronic mail game are two paradoxes of common knowledge. In simple mathematical models of these scenarios, the agents represented by the models can coordinate only if they have common knowledge that they will. As a result, the models predict that the agents will not coordinate in situations where it would be rational to coordinate. I argue that we should resolve this conflict between the models and facts about what it would be rational to do (...) by rejecting common knowledge assumptions implicit in the models. I focus on the assumption that the agents have common knowledge that they are rational, and provide models to show that denying this assumption suffices for a resolution of the paradoxes. I describe how my resolution of the paradoxes fits into a general story about the relationship between rationality in situations involving a single agent and rationality in situations involving many agents. (shrink)

We study two logics of knowledge and belief stemming from the work of Stalnaker, omitting positive introspection for knowledge. The two systems are equivalent with positive introspection, but not without. We show that while the logic of beliefs remains unaffected by omitting introspection for knowledge in one system, it brings significant changes to the other. The resulting logic of belief is non-normal, and its complete axiomatization uses an infinite hierarchy of coherence constraints. We conclude by returning to the philosophical interpretation (...) underlying both models of belief, showing that neither is strong enough to support a probabilistic interpretation, nor an interpretation in terms of certainty or the “mental component” of knowledge. (shrink)

We study a multiple-succedent sequent calculus with both of the structural rules Left Weakening and Left Contraction but neither of their counterparts on the right, for possible application to the treatment of multiplicative disjunction against the background of intuitionistic logic. We find that, as Hirokawa dramatically showed in a 1996 paper with respect to the rules for implication, the rules for this connective render derivable some new structural rules, even though, unlike the rules for implication, these rules are what we (...) call ipsilateral: applying such a rule does not make any formula change sides—from the left to the right of the sequent separator or vice versa. Some possibilities for a semantic characterization of the resulting logic are also explored. The paper concludes with three open questions. (shrink)

Let us say that a normative conflict is a situation in which an agent ought to perform an action A, and also ought to perform an action B, but in which it is impossible for the agent to perform both A and B. Not all normative conflicts are moral conflicts, of course. It may be that the agent ought to perform the action A for reasons of personal generosity, but ought to perform the action B for reasons of prudence: perhaps (...) A involves buying a lavish gift for a friend, while B involves depositing a certain amount of money in the bank. In general, our practical deliberation is shaped by a concern with a variety of morally neutral goods—not just generosity and prudence, but any number of others, such as etiquette, aesthetics, fun—many of which are capable of providing conflicting reasons for action. I mention these ancillary values in the present setting, however, only to put them aside. We will be concerned here, not with normative conflicts more generally, but precisely with moral conflicts—situations in which, even when our attention is restricted entirely to moral reasons for action, it is nevertheless true that an agent ought to do A and ought to do B, where it is impossible to do both. (shrink)

It is known that a theory in S5‐epistemic logic with several agents may have numerous models. This is because each such model specifies also what an agent knows about infinite intersections of events, while the expressive power of the logic is limited to finite conjunctions of formulas. We show that this asymmetry between syntax and semantics persists also when infinite conjunctions (up to some given cardinality) are permitted in the language. We develop a strengthened S5‐axiomatic system for such infinitary logics, (...) and prove a strong completeness theorem for them. Then we show that in every such logic there is always a theory with more than one model. (shrink)

This article argues that it is possible to hold autonomous agents themselves, and not only their makers, users or owners, responsible for the acts of these agents. In this connection autonomous systems are computer programs that interact with the outside world without human interference. They include such systems as ‘intelligent’ weapons and self-driving cars. The argument is based on an analogy between human beings and autonomous agents and its main element asserts that if humans can be held responsible, so can, (...) in principle, autonomous agents, as humans are more like autonomous agents than is often assumed. (shrink)

On the face of it, normative conflicts are commonplace. Yet standard deontic logic declares them to be logically impossible. That prompts the question, What are the proper principles of normative reasoning if such conflicts are possible? This paper examines several alternatives that have been proposed for a logic of 'ought' that can accommodate normative conflicts, and finds all of them unsatisfactory as measured against three criteria of adequacy. It then introduces a new logic that does meet all three criteria, and (...) so allows for the possibility of genuine normative conflicts. (shrink)

We provide a syntactic model of unawareness. By introducing multiple knowledge modalities, one for each sub-language, we specifically model agents whose only mistake in reasoning (other than their unawareness) is to underestimate the knowledge of more aware agents. We show that the model is a complete and sound axiomatization of the set-theoretic model of Galanis (University of Southampton Discussion paper 709, 2007) and compare it with other unawareness models in the literature.

We study a modal system $\overline{T}$, that extends the classical modal system T and whose language is provided with modal operators $M_{n}$ to be interpreted, in the usual kripkean semantics, as "there are more than n accessible worlds such that...". We find reasonable axioms for $\overline{T}$ and we prove for it completeness, compactness and decidability theorems.

We prove a completeness theorem for Kmath image, the infinitary extension of the graded version K0 of the minimal normal logic K, allowing conjunctions and disjunctions of countable sets of formulas. This goal is achieved using both the usual tools of the normal logics with graded modalities and the machinery of the predicate infinitary logics in a version adapted to modal logic.

We study modal logics based on neighbourhood semantics using methods and theorems having their origin in topological model theory. We thus obtain general results concerning completeness of modal logics based on neighbourhood semantics as well as the relationship between neighbourhood and Kripke semantics. We also give a new proof for a known interpolation result of modal logic using an interpolation theorem of topological model theory.

This paper arose out of the 2017 international conference on AI and law doctoral consortium. There were five students who presented their Ph.D. work, and each of them has contributed a section to this paper. The paper offers a view of what topics are currently engaging students, and shows the diversity of their interests and influences.

Many philosophers are convinced that rationality dictates that one’s overall set of intentions be consistent. The starting point and inspiration for our study is Bratman’s planning theory of intentions. According to this theory, one needs to appeal to the fulfilment of characteristic planning roles to justify norms that apply to our intentions. Our main objective is to demonstrate that one can be rational despite having mutually inconsistent intentions. Conversely, it is also shown that one can be irrational despite having a (...) consistent overall set of intentions. To overcome this paradox, we argue that it is essential for a successful planning system that one’s intentions are practically consistent rather than being consistent or applying an aggregation procedure. Our arguments suggest that a new type of norm is needed: whereas the consistency requirement focuses on rendering the contents of one’s intentions consistent, our new practical consistency requirement demands that one’s intentions be able to simultaneously and unconditionally guide one’s action. We observe that for intentions that conform to the ‘own-action condition’, the practical consistency requirement is equivalent to the traditional consistency requirement. This implies that the consistency requirement only needs to be amended in scenarios of choice under uncertainty. (shrink)

This paper, a sequel to Models for normal intuitionistic modal logics by M. Boi and the author, which dealt with intuitionistic analogues of the modal system K, deals similarly with intuitionistic analogues of systems stronger than K, and, in particular, analogues of S4 and S5. For these prepositional logics Kripke-style models with two accessibility relations, one intuitionistic and the other modal, are given, and soundness and completeness are proved with respect to these models. It is shown how the holding of (...) formulae characteristic for particular logics is equivalent to conditions for the relations of the models. Modalities in these logics are also investigated. (shrink)

در نگاه نخست به نظر می‌رسد می‌توان شباهت‌هایی بین جهان‌های ممکن مورد نظر لوئیس و نظریۀ جهان‌های موازی فیزیک یافت. هر دو این نظریات به امکان وجود جهان‌هایی بیش از جهانی که در آن زندگی می‌کنیم، اشاره دارند. پس از رسالۀ اورت، توجه فیزیک‌دانان به چندجهانی و جهان‌های موازی جلب شد. اکنون جهان‌های موازی یکی از نظریه‌های مورد توجه دانشمندان و پژوهشگران حوزۀ فیزیک است. دیوید لوئیس با تکیه بر فیزیکالیسم و رئالیسم موجهاتی از جمله طرفداران نظریۀ جهان‌های ممکن است. (...) در این مقاله کوشیده‌ایم با استفاده از روش تحلیلی-توصیفی ابتدا مبانی دو نظریه را تبیین کرده و سپس نظریۀ لوئیس را در مورد جهان‌های ممکن با توجه به نظریۀ جهان‌های موازی مورد بررسی قرار دهیم. هر دو نظریه وجود جهان‌های دیگر را ممکن می‌دانند. نظریه لوئیس شش ویژگی اصلی دارد: ۱- وجود و واقعیت جهان‌های ممکن، ۲- عدم وجود رابطه علی میان‌ جهان‌های ممکن، ۳- شباهت جهان‌های ممکن با جهان ما و تفاوت در محتوای جهان‌ها، ۴- عدم امکان تقلیل جهان‌های ممکن، ۵- نمایه‌ای بودن واقعیت و ۶- اتحاد مکان- زمان اجزای هر جهان و مجزا بودن روابط زمان-مکان جهان‌ها. نتایج پژوهش نشان می‌دهد در صورتی که جهان‌های موازی، زمانی به عنوان یک حقیقت علمی مورد تأیید قرار بگیرند، آنگاه ممکن است ویژگی‌های دوم، سوم و ششم نظریۀ لوئیس با واقعیت فیزیکی در تضاد قرار بگیرند. در صورت وجود تضاد -به جهت بنیادی بودن این ویژگی‌ها- نظریۀ جهان‌های ممکن لوئیس دچار تزلزل فراوان خواهد شد. (shrink)