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)

The communication process is analysed on the basis of the notions of trust and agency. The aim of the paper is to clarify the role played by causality on the one hand and the role played by logical consequences of assumptions about trust in information sources on the other hand. The first part is informal and the second part refers to the logical framework of modal logic though it requires a quite limited background in this area.

There are several three-valued logical systems that form a scattered landscape, even if all reasonable connectives in three-valued logics can be derived from a few of them. Most papers on this subject neglect the issue of the relevance of such logics in relation with the intended meaning of the third truth-value. Here, we focus on the case where the third truth-value means unknown, as suggested by Kleene. Under such an understanding, we show that any truth-qualified formula in a large range (...) of three-valued logics can be translated into KD as a modal formula of depth 1, with modalities in front of literals only, while preserving all tautologies and inference rules of the original three-valued logic. This simple information logic is a two-tiered classical propositional logic with simple semantics in terms of epistemic states understood as subsets of classical interpretations. We study in particular the translations of Kleene, Gödel, ᴌukasiewicz and Nelson logics. We show that Priest’s logic of paradox, closely connected to Kleene’s, can also be translated into our modal setting, simply by exchanging the modalities possible and necessary. Our work enables the precise expressive power of three-valued logics to be laid bare for the purpose of uncertainty management. (shrink)

According to Vázquez and Liz (Found Sci 16(4): 383–391, 2011), Points of View (PoV) can be considered in two different ways. On the one hand, they can be explained following the model of propositional attitudes. This model assumes that the internal structure of a PoV is constituted by a subject, a set of contents, and a set of relations between the subject and those contents. On the other hand, we can analyze points of view taking as a model the notions (...) of location and access. If we choose to follow the second approach, instead of the first one, the internal structure of a PoV is not directly addressed, and the emphasized features of PoV are related to the function that PoV are intended to have. That is, PoV are directly identified by their role and they can solely be understood as ways of accessing the world that bring some kind of perspective about it. Having this in mind, we would like to propose a notation that explains how to understand such access as a sort of models (that can allow the creation of concepts), independently of whether the precise PoV under consideration is impersonal or non-impersonal, its kind of content, and its subjective or objective character. First, we will present an account of some previous approaches to the study of points of view. Then, we will analyze what kind of structure the world is assumed to posses and how the access to it is possible. Third, we will develop a notation that explains PoV as qualitative dimensions by means of which it is possible to valuate objects and states of the world. (shrink)

We present sequent calculi for normal modal logics where modal and propositional behaviours are separated, and we prove a cut elimination theorem for the basic system K, so as completeness theorems both for K itself and for its most popular enrichments. MSC: 03B45, 03F05.

We develop cut-free calculi of sequents for normal modal logics by using treesequents, which are trees of sequences of formulas. We introduce modal operators corresponding to the ways we move formulas along the branches of such trees, only considering fixed distance movements. Finally, we exhibit syntactic cut-elimination theorems for all the main normal modal logics.

The basic bimodal systemK/K can be interpreted as an analysis of the logic of ability developed in [1]. Where in [1] we would express the claimI can bring it about that P using the formula, with its non-normal operator, we will now use the formula. Here is a normal alethic possibilitation operator.is a normal necessitation operator, but it is independent of, and not subject to an alethic interpretation. Rather, is interpreted to meanI bring it about that P. The result is (...) a simplification and clarification of a combined logic of ability and action like that in [2], but employing only normal operators.A number of extensions of the basic systemK/K are constructed, first by strengthening the two normal sublogics independently and then by linking the two sublogics via axiom schemata involving both operators. The result is a series of increasingly strong systems which more and more adequately fulfill our expectations for a satisfactory logic of action and ability. (shrink)

We extend stit logic by adding a spatial dimension. This enables us to distinguish between powers and opportunities of agents. Powers are agent-specific and do not depend on an agent’s location. Opportunities do depend on locations, and are the same for every agent. The central idea is to define the real possibility to see to the truth of a condition in space and time as the combination of the power and the opportunity to do so. The focus on agent-relative powers (...) and space-relative opportunities firmly roots effectivity of an autonomous choice making agent in a space–time picture. Our space–time view will be classically Newtonian, since we will assume relativistic phenomena do not play a role in agentive effectivity. We show how our semantics naturally distinguishes between different kinds of histories; histories that reflect real possibilities and histories that reflect counterfactual possibilities. Furthermore, we discuss how the spatial picture sheds light on conceptual problems plaguing the central stit property of ‘independence of agency’. At several points in the article we will emphasise and defend the differences with Belnap’s theory of agency in relativistic branching space–times. (shrink)

Normal systems of modal logic, interpreted as deontic logics, are unsuitable for a logic of conflicting obligations. By using modal operators based on a more complex semantics, however, we can provide for conflicting obligations, as in [9], which is formally similar to a fragment of the logic of ability later given in [2], Having gone that far, we may find it desirable to be able to express and consider claims about the comparative strengths, or degrees of urgency, of the conflicting (...) obligations under which we stand. This paper, building on the formalism of the logic of ability in [2], provides a complete and decidable system for such a language. (shrink)

Abstract: In the course of seeking an answer to the question "How do you know you are not a zombie?" Floridi (2005) issues an ingenious, philosophically rich challenge to artificial intelligence (AI) in the form of an extremely demanding version of the so-called knowledge game (or "wise-man puzzle," or "muddy-children puzzle")—one that purportedly ensures that those who pass it are self-conscious. In this article, on behalf of (at least the logic-based variety of) AI, I take up the challenge—which is to (...) say, I try to show that this challenge can in fact be met by AI in the foreseeable future. (shrink)

I articulate a novel modal argument for P=NP.

Kripke-style models with two accessibility relations, one intuitionistic and the other modal, are given for analogues of the modal systemK based on Heyting's prepositional logic. It is shown that these two relations can combine with each other in various ways. Soundness and completeness are proved for systems with only the necessity operator, or only the possibility operator, or both. Embeddings in modal systems with several modal operators, based on classical propositional logic, are also considered. This paper lays the ground for (...) an investigation of intuitionistic analogues of systems stronger thanK. A brief survey is given of the existing literature on intuitionistic modal logic. (shrink)

Important decisions are often based on a distributed process of information processing, from a knowledge base that is itself distributed among agents. The simplest such situation is that where a decision-maker seeks the recommendations of experts. Because experts may have vested interests in the consequences of their recommendations, decision-makers usually seek the advice of experts they trust. Trust, however, is a commodity that is usually built through repeated face time and social interaction and thus cannot easily be built in a (...) global world where we have immediate internet access to a vast pool of experts. In this article, we integrate findings from experimental psychology and formal tools from Artificial Intelligence to offer a preliminary roadmap for solving the problem of trust in this computer-mediated environment. We conclude the article by considering a diverse array of extended applications of such a solution. (shrink)

According to a simple form of consequentialism, we should base decisions on our judgments about their consequences for achieving our goals. Our goals give us reason to endorse consequentialism as a standard of decision making. Alternative standards invariably lead to consequences that are less good in this sense. Yet some people knowingly follow decision rules that violate consequentialism. For example, they prefer harmful omissions to less harmful acts, they favor the status quo over alternatives they would otherwise judge to be (...) belter, they provide third-party compensation on the basis of the cause of an injury rather than the benefit from the compensation, they ignore deterrent effects in decisions about punishment, and they resist coercive reforms they judge to be beneficial. I suggest that nonconsequentialist principles arise from overgeneralizing rules that are consistent with consequentialism in a limited set of cases. Commitment to such rules is detached from their original purposes. The existence of such nonconsequentialist decision biases has implications for philosophical and experimental methodology, the relation between psychology and public policy, and education. (shrink)

According to a simple form of consequentialism, we should base decision on our judgments about their consequences for achieving out goals. Our goals give us reason to endorse consequentialism as a standard of decision making. Alternative standards invariably lead to consequences that are less good in this sense. Yet some people knowingly follow decision rules that violate consequentialism. For example, they prefer harmful omissions to less harmful acts, they favor the status quo over alternatives they would otherwise judge to be (...) better, they provide third-party compensation on the basis of the cause of an injury rather than the benefit from the compensation, they ignore deterrent effects in decisions about punishment, and they resist coercive reforms they judge to be beneficial. I suggest that nonconsequentialist principles arise from overgeneralizing rules that are consistent with consequentialism in a limited set of cases. Commitment to such rules is detached from their original purposes. The existence of such nonconsequentialist decision biases has implications for philosophical and experimental methodology, the relation between psychology and public policy, and education. (shrink)

In this paper I present a more refined analysis of the principles of deductive closure and positive introspection. This analysis uses the expressive resources of logics for different types of group knowledge, and discriminates between aspects of closure and computation that are often conflated. The resulting model also yields a more fine-grained distinction between implicit and explicit knowledge, and places Hintikka’s original argument for positive introspection in a new perspective.

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)