Monotone modal logics have emerged in several application areas such as computer science and social choice theory. Since many of the most studied selfextensional logics have a conjunction, in this paper we study some distributive extensions obtained from a semilattice based deductive system with monotonic modal operators, and we give them neighborhood and algebraic semantics. For each logic defined our main objective is to prove completeness with respect to its characteristic class of monotonic frames.

This is a thorough treatment of first-order modal logic. The book covers such issues as quantification, equality (including a treatment of Frege's morning star/evening star puzzle), the notion of existence, non-rigid constants and function symbols, predicate abstraction, the distinction between nonexistence and nondesignation, and definite descriptions, borrowing from both Fregean and Russellian paradigms.

. I give a systematic presentation of a fairly large family of multiple-conclusion modal logics that are paraconsistent and/or paracomplete. After providing motivation for studying such systems, I present semantics and tableau-style proof theories for them. The proof theories are shown to be sound and complete with respect to the semantics. I then show how the “standard” systems of classical, single-conclusion modal logics fit into the framework constructed.

This paper presents ConR (Conditional R), a logic of conditionals based on Anderson and Belnap''s system R. A Routley-Meyer-style semantics for ConR is given for the system (the completeness of ConR over this semantics is proved in E. Mares and A. Fuhrmann, A Relevant Theory of Conditionals (unpublished MS)). Moreover, it is argued that adopting a relevant theory of conditionals will improve certain theories that utilize conditionals, i.e. Lewis'' theory of causation, Lewis'' dyadic deontic logic, and Chellas'' dyadic deontic logic.

In "Doing Well Enough: Toward a Logic for Common Sense Morality", Paul McNamara sets out a semantics for a deontic logic which contains the operator It is supererogatory that. As well as having a binary accessibility relation on worlds, that semantics contains a relative ordering relation, . For worlds u, v and w, we say that u w v when v is at least as good as u according to the standards of w. In this paper we axiomatize logics complete (...) over three versions of the semantics. We call the strongest of these logics DWE for Doing Well Enough. (shrink)

This paper investigates the semantic treatment of conditional obligation, explicit permission (often called positive permission), and prohibition based on models with agents and branched time. In such models branches (rather than moments) are taken as basic, and the branching provides a way to represent the indeterminism which is normally presupposed by talk of free will, responsibility, action and ability. Careful treatment of the relation between ability and responsibility avoids many common problems with accounts of conditional obligation. Recognition of the generality (...) often involved in conditional obligations makes possible a sensitive way of expressing some kinds of general prohibitions, which in turn makes it possible to account for the special role of explicit permission. (shrink)

In this paper we set out a semantics for relevant (counterfactual) conditionals. We combine the Routley-Meyer semantics for relevant logic with a semantics for conditionals based on selection functions. The resulting models characterize a family of conditional logics free from fallacies of relevance, in particular counternecessities and conditionals with necessary consequents receive a non-trivial treatment.

We present a dynamic approach to Peirce’s original construal of abductive logic as a logic of conjecture making, and provide a new decidable, contraction-free and cut-free proof system for the dynamic logic of abductive inferences with neighborhood semantics. Our formulation of the dynamic logic of abduction follows the philosophical and scientific track that led Peirce to his late, post-1903 characterization of abductive conclusions as investigands, namely invitations to investigate propositions conjectured at the level of pre-beliefs.

I offer you some theories of intellectual obligations and rights (virtue Ethics): initially, RBT (a Right to Believe Truth, if something is true it follows one has a right to believe it), and, NDSM (one has no right to believe a contradiction, i.e., No right to commit Doxastic Self-Mutilation). Evidence for both below. Anthropology, Psychology, computer software, Sociology, and the neurosciences prove things about human beliefs, and History, Economics, and comparative law can provide evidence of value about theories of rights. (...) However, insofar as we have methods within Philosophy to help us formulate precise concepts of 'belief' and 'rights', methods that also help us to prove links (or absence thereof) amongst families of concepts of rights and belief, our discipline is in and of itself capable of sound reasoning about issues as puzzling as the following. Suppose a Jane who does not believe in God yet who believes she ought to so believe: Jane is undergoing doxastic moral regret (moral regret for lack of faith). We have all known such Janes and perhaps at one time or another even been one. Paradox: given RBT and NDSM, Jane as described not only does not exist, Jane cannot exist. Thus, to enrich the ways in which Philosophy need not get all its evidence from other academic disciplines, I present a brief introduction to what I call Neutral Universal Frames (NUFs). NUFs solve hard puzzles about interactions among modal concepts of belief and rights, concepts that occur in RBT, NDSM and the description of our Janes. NUFs for theories precisely articulated via any two or more modal concepts are a powerful and immensely general set of tools enabling us to define rich theories of truth ("models on frames") to test philosophical theories for internal consistency and to prove the existence of connections (or absence thereof) amongst alternative articulations of philosophical theories. NUFs thereby add to the constructive knowledge producing way Logics intersect with Philosophy of Religion. And we will soon see why Jane, be she named 'Jane' or known simply as you: cannot exist. Read on at your own risk. (shrink)

We show that several logics of common belief and common knowledge are not only complete, but also strongly complete, hence compact. These logics involve a weakened monotonicity axiom, and no other restriction on individual belief. The semantics is of the ordinary fixed-point type.

I propose a modal logic that enables to reason about self-regarding and otherregarding motivations in strategic games. This logic integrates the concepts of joint action, belief, individual and group payoff. The first part of the article is focused on self-regarding agents. A self-regarding agent decides to perform a certain action only if he believes that this action maximizes his own personal benefit. The second part of the article explores different kinds of other-regarding motivations such as fairness and reciprocity. Differently from (...) self-regarding agents, other-regarding agents also consider the benefits of their choices for the group. Moreover, their decisions can be affected by their beliefs about other agents' willingness to act for the well-being of the group. The analysis also considers team-directed reasoning, i.e. the mode of reasoning that people use when they take themselves to be acting as members of a group or a team. (shrink)

We investigate an extension of the formalism of interpreted systems by Halpern and colleagues to model the correct behaviour of agents. The semantical model allows for the representation and reasoning about states of correct and incorrect functioning behaviour of the agents, and of the system as a whole. We axiomatise this semantic class by mapping it into a suitable class of Kripke models. The resulting logic, KD45ni-j, is a stronger version of KD, the system often referred to as Standard Deontic (...) Logic. We extend this formal framework to include the standard epistemic notions defined on interpreted systems, and introduce a new doubly-indexed operator representing the knowledge that an agent would have if it operates under the assumption that a group of agents is functioning correctly. We discuss these issues both theoretically and in terms of applications, and present further directions of work. (shrink)

The paper presents a formal explication of the early Wittgenstein's views on ontology, the syntax and semantics of an ideal logical language, and the propositional attitudes. It will be shown that Wittgenstein gave a language of thought analysis of propositional attitude ascriptions, and that his ontological views imply that such ascriptions are truth-functions of (and supervenient upon) elementary sentences. Finally, an axiomatization of a quantified doxastic modal logic corresponding to Tractarian semantics will be given.

Weakly Aggregative Modal Logic ) is a collection of disguised polyadic modal logics with n-ary modalities whose arguments are all the same. \ has interesting applications on epistemic logic, deontic logic, and the logic of belief. In this paper, we study some basic model theoretical aspects of \. Specifically, we first give a van Benthem–Rosen characterization theorem of \ based on an intuitive notion of bisimulation. Then, in contrast to many well known normal or non-normal modal logics, we show that (...) each basic \ system \ lacks Craig interpolation. Finally, by model theoretical techniques, we show that an extension of \ does have Craig interpolation, as an example of amending the interpolation problem of \. (shrink)

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.

Two approaches for defining common knowledge coexist in the literature: the infinite iteration definition and the circular or fixed point one. In particular, an original modelization of the fixed point definition was proposed by Barwise in the context of a non-well-founded set theory and the infinite iteration approach has been technically analyzed within multi-modal epistemic logic using neighbourhood semantics by Lismont. This paper exhibits a relation between these two ways of modelling common knowledge which seem at first quite different.

In this paper, we give an alternative semantics to the non-normal logic of knowing how proposed by Fervari et al., based on a class of Kripke neighborhood models with both the epistemic relations and neighborhood structures. This alternative semantics is inspired by the same quantifier alternation pattern of ∃∀\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\exists \forall $$\end{document} in the semantics of the know-how modality and the neighborhood semantics for the standard modality. We show that this new semantics (...) is equivalent to the original Kripke semantics in terms of the validities. A key result is a representation theorem showing that the more abstract Kripke neighborhood models can be represented by the concrete Kripke models with action transitions modulo the valid formulas. We prove the completeness of the logic for the neighborhood semantics. The neighborhood semantics can be adapted to other variants of logics of knowing how. It provides us a powerful technical tool to study these logics while preserving the basic semantic intuition. (shrink)

All standard epistemic logics legitimate something akin to the principle of closure, according to which knowledge is closed under competent deductive inference. And yet the principle of closure, particularly in its multiple premise guise, has a somewhat ambivalent status within epistemology. One might think that serious concerns about closure point us away from epistemic logic altogether—away from the very idea that the knowledge relation could be fruitfully treated as a kind of modal operator. This, however, need not be so. The (...) abandonment of closure may yet leave in place plenty of formal structure amenable to systematic logical treatment. In this paper we describe a family of weak epistemic logics in which closure fails, and describe two alternative semantic frameworks in which these logics can be modelled. One of these—which we term plurality semantics—is relatively unfamiliar. We explore under what conditions plurality frames validate certain much-discussed principles of epistemic logic. It turns out that plurality frames can be interpreted in a very natural way in light of one motivation for rejecting closure, adding to the significance of our technical work. The second framework that we employ—neighbourhood semantics—is much better known. But we show that it too can be interpreted in a way that comports with a certain motivation for rejecting closure. (shrink)

Over the course of more than two millennia the philosophical school of Mīmāṃsā has thoroughly analyzed normative statements. In this paper we approach a formalization of the deontic system which is applied but never explicitly discussed in Mīmāṃsā to resolve conflicts between deontic statements by giving preference to the more specific ones. We first extend with prohibitions and recommendations the non-normal deontic logic extracted in Ciabattoni et al. from Mīmāṃsā texts, obtaining a multimodal dyadic version of the deontic logic \. (...) Sequent calculus is then used to close a set of prima-facie injunctions under a restricted form of monotonicity, using specificity to avoid conflicts. We establish decidability and complexity results, and investigate the potential use of the resulting system for Mīmāṃsā philosophy and, more generally, for the formal interpretation of normative statements. (shrink)

Some people commonly know a proposition just in case they all know it, they all know that they all know it, they all know that they all know that they all know it, and so on. They commonly believe a proposition just in case they all believe it, they all believe that they all believe it, they all believe that they all believe that they all believe it, and so on. A long tradition in economic theory, theoretical computer science, linguistics (...) and philosophy has held that people have some approximation of common knowledge or common belief in a range of circumstances, for example, when they are looking at an object together, or when they have just discussed something explicitly in conversation. In this paper, I argue that people do not have any approximation of common knowledge or common belief in these circumstances. The argument suggests that people never have any approximation of common knowledge or common belief. (shrink)

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)

In this article, it was first presented Moore’s paradox per se and after the author focused on the logical perspective — at first he analyzed these considerations in the field of so-called standard epistemic logic and after on the formal theory of belief change.

This paper presents a formal account of how to determine the discourse relations between propositions introduced in a text, and the relations between the events they describe. The distinct natural interpretations of texts with similar syntax are explained in terms of defeasible rules. These characterise the effects of causal knowledge and knowledge of language use on interpretation. Patterns of defeasible entailment that are supported by the logic in which the theory is expressed are shown to underly temporal interpretation.

This papers gives a survey of recent results about simulations of one class of modal logics by another class and of the transfer of properties of modal logics under extensions of the underlying modal language. We discuss: the transfer from normal polymodal logics to their fusions, the transfer from normal modal logics to their extensions by adding the universal modality, and the transfer from normal monomodal logics to minimal tense extensions. Likewise, we discuss simulations of normal polymodal logics by normal (...) monomodal logics, of nominals and the difference operator by normal operators, of monotonic monomodal logics by normal bimodal logics, of polyadic normal modal logics by polymodal normal modal logics, and of intuitionistic modal logics by normal bimodal logics. (shrink)

Modeling a complex phenomena such as the mind presents tremendous computational complexity challenges. Modeling field theory (MFT) addresses these challenges in a non-traditional way. The main idea behind MFT is to match levels of uncertainty of the model (also, a problem or some theory) with levels of uncertainty of the evaluation criterion used to identify that model. When a model becomes more certain, then the evaluation criterion is adjusted dynamically to match that change to the model. This process is called (...) the Dynamic Logic of Phenomena (DLP) for model construction and it mimics processes of the mind and natural evolution. This paper provides a formal description of DLP by specifying its syntax, semantics, and reasoning system. We also outline links between DLP and other logical approaches. Computational complexity issues that motivate this work are presented using an example of polynomial models. (shrink)

Modelling a complex phenomenon such as the mind presents tremendous computational complexity challenges. Modelling field theory addresses these challenges in a non-traditional way. The main idea behind MFT is to match levels of uncertainty of the model with levels of uncertainty of the evaluation criterion used to identify that model. When a model becomes more certain, then the evaluation criterion is adjusted dynamically to match that change to the model. This process is called the Dynamic Logic of Phenomena for model (...) construction and it mimics processes of the mind and natural evolution. This paper provides a formal description of DLP by specifying its syntax, semantics, and reasoning system. We also outline links between DLP and other logical approaches. Computational complexity issues that motivate this work are presented using an example of polynomial models. (shrink)

Defeasible conditionals are statements of the form ‘if A then normally B’. One plausible interpretation introduced in nonmonotonic reasoning dictates that ) is true iff B is true in ‘most’ A-worlds. In this paper, we investigate defeasible conditionals constructed upon a notion of ‘overwhelming majority’, defined as ‘truth in a cofinite subset of \’, the first infinite ordinal. One approach employs the modal logic of the frame \\), used in the temporal logic of discrete linear time. We introduce and investigate (...) conditionals, defined modally over \\); several modal definitions of the conditional connective are examined, with an emphasis on the nonmonotonic ones. An alternative interpretation of ‘majority’ as sets cofinal ) rather than cofinite ) is examined. For these modal approaches over \\), a decision procedure readily emerges, as the modal logic \ of this frame is well-known and a translation of the conditional sentences can be mechanically checked for validity; this allows also for a quick proof of \-completeness of the satisfiability problem for these logics. A second approach employs the conditional version of Scott-Montague semantics, in the form of \-many possible worlds, endowed with neighborhoods populated by collections of cofinite subsets of \. This approach gives rise to weak conditional logics, as expected. The relative strength of the conditionals introduced is compared to KLM logics and other conditional logics in the literature. (shrink)

Every truth-functional three-valued propositional logic can be conservatively translated into the modal logic S5. We prove this claim constructively in two steps. First, we define a Translation Manual that converts any propositional formula of any three-valued logic into a modal formula. Second, we show that for every S5-model there is an equivalent three-valued valuation and vice versa. In general, our Translation Manual gives rise to translations that are exponentially longer than their originals. This fact raises the question whether there are (...) three-valued logics for which there is a shorter translation into S5. The answer is affirmative: we present an elegant linear translation of the Logic of Paradox and of Strong Three-valued Logic into S5. (shrink)

We are concerned with formal models of reasoning under uncertainty. Many approaches to this problem are known in the literature e.g. Dempster-Shafer theory [29], [42], bayesian-based reasoning [21], [29], belief networks [29], many-valued logics and fuzzy logics [6], non-monotonic logics [29], neural network logics [14]. We propose rough mereology developed by the last two authors [22-25] as a foundation for approximate reasoning about complex objects. Our notion of a complex object includes, among others, proofs understood as schemes constructed in order (...) to support within our knowledge assertions/hypotheses about reality described by our knowledge incompletely. (shrink)

We are concerned with formal models of reasoning under uncertainty. Many approaches to this problem are known in the literature e.g. Dempster-Shafer theory [29], [42], bayesian-based reasoning [21], [29], belief networks [29], many-valued logics and fuzzy logics [6], non-monotonic logics [29], neural network logics [14]. We propose rough mereology developed by the last two authors [22-25] as a foundation for approximate reasoning about complex objects. Our notion of a complex object includes, among others, proofs understood as schemes constructed in order (...) to support within our knowledge assertions/hypotheses about reality described by our knowledge incompletely. (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 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)

The theory of rough sets starts with the notion of an approximation space , which is a pair ( U , R ), U being the domain of discourse, and R an equivalence relation on U . R is taken to represent the knowledge base of an agent, and the induced partition reflects a granularity of U that is the result of a lack of complete information about the objects in U . The focus then is on approximations of concepts (...) on the domain, in the context of the granularity. The present article studies the theory in the situation where information is obtained from different sources. The notion of approximation space is extended to define a multiple-source approximation system with distributed knowledge base , which is a tuple , where N is a set of sources and P ranges over all finite subsets of N . Each R P is an equivalence relation on U satisfying some additional conditions, representing the knowledge base of the group P of sources. Thus each finite group of sources and hence individual source perceives the same domain differently (depending on what information the group/individual source has about the domain), and the same concept may then have approximations that differ with the groups. In order to express the notions and properties related with rough set theory in this multiple-source situation, a quantified modal logic LMSAS D is proposed. In LMSAS D , quantification ranges over modalities, making it different from modal predicate logic and modal logic with propositional quantifiers. Some fragments of LMSAS D are discussed and it is shown that the modal system KTB is embedded in LMSAS D . The epistemic logic is also embedded in LMSAS D , and cannot replace the latter to serve our purpose. The relationship of LMSAS D with first and second-order logics is presented. Issues of expressibility, axiomatization and decidability are addressed. (shrink)

In order to capture the concept of common knowledge, various extensions of multi-modal epistemic logics, such as fixed-point ones and infinitary ones, have been proposed. Although we have now a good list of such proposed extensions, the relationships among them are still unclear. The purpose of this paper is to draw a map showing the relationships among them. In the propositional case, these extensions turn out to be all Kripke complete and can be comparable in a meaningful manner. F. Wolter (...) showed that the predicate extension of the Halpern-Moses fixed-point type common knowledge logic is Kripke incomplete. However, if we go further to an infinitary extension, Kripke completeness would be recovered. Thus there is some gap in the predicate case. In drawing the map, we focus on what is happening around the gap in the predicate case. The map enables us to better understand the common knowledge logics as a whole. (shrink)

We give a reading of binary necessity statements of the form “ϕ is necessary for ψ” in terms of proofs. This reading is based on the idea of interpreting such statements as “Every proof of ψ uses ϕ”.

Abstract.The paper outlines the approach to the analysis of deontic conditionals taken in earlier work by Jones and Pörn, compares it very briefly with two main trends within dyadic deontic logic, and then discusses problems associated with the augmentation principle and the factual detachment principle. The author then modifies Jones and Pörn's previous system, using a classical but not normal (in the sense of Chellas) deontic modality to provide the basis for an alternative analysis of deontic conditionals. This new analysis (...) validates neither the factual detachment nor the augmentation principles. However, influenced by the approach of James Delgrande to default reasoning, the author tries to show how a restricted form of factual detachment may be accommodated within the revised system. (shrink)

This paper is a preliminary investigation into the application of the formal-logical theory of normative positions to the characterisation of normative-informational positions, pertaining to rules that are meant to regulate the supply of information. First, we present the proposed framework. Next, we identify the kinds of nuances and distinctions that can be articulated in such a logical framework. Finally, we show how such nuances can arise in specific regulations. Reference is made to Data Protection Law and Contract Law, among others. (...) The proposed approach is articulated around two essential steps. The first involves identifying the set of possible interpretations that can be given to a particular norm. This is done by using formal methods. The second involves picking out one of these interpretations as the most likely one. This second step can be resolved only by using further information (e.g., the context or other parts of the regulation). (shrink)

There seems to be no clear consensus in the existing literature about the role of deontic logic in legal knowledge representation — in large part, we argue, because of an apparent misunderstanding of what deontic logic is, and a misplaced preoccupation with the surface formulation of legislative texts. Our aim in this paper is to indicate, first, which aspects of legal reasoning are addressed by deontic logic, and then to sketch out the beginnings of a methodology for its use in (...) the analysis and representation of law.The essential point for which we argue is that deontic logic — in some form or other —needs to be taken seriously whenever it is necessary to make explicit, and then reason about, the distinction between what ought to be the case and what is the case, or as we also say, between the ideal and the actual. We take the library regulations at Imperial College as the main illustration, and small examples from genuinely legal domains to introduce specific points. In conclusion, we touch on the role of deontic logic in the development of the theory of normative positions. (shrink)

Partly motivated by description logics, poor man's logics have been proposed as an interesting fragment of modal logics. A poor man's logic is a propositional modal logic where only literals and the connectives ∧, □, and ◊ are allowed. It is known that the complexity of the satisfiability problem may drop dramatically when going from a full modal logic to the corresponding poor man's logic, e.g., in the case of modal logic K one goes from PSPACE-complete to coNP-complete. We prove (...) that it is sometimes possible to extend poor man's logics with restricted disjunctions (i.e., clauses) without increasing the computational complexity. For Horn and Krom clauses, we provide necessary and sufficient conditions for when the resulting logic is polynomial-time. Such extensions correspond to allowing a restricted use of the union operator in description logics. (shrink)

In his recent book Willensfreiheit Geert Keil defends a version of libertarianism. Yet he criticizes a flagship argument for incompatibilism. Van Inwagen's consequence argument, Keil thinks, relies on an irrelevant premise when it claims that agents have no choice about the remote past. I argue that Keil's charge rests on a misunderstanding. I then sketch why discussions of the consequence argument should focus on the question whether or not a certain version of rule Beta is valid.

Standard deontic logic (SDL) is defined on the basis of possible world semantics and is a logic of alethic-deontic modalities rather than deontic modalities alone. The interpretation of the concepts of obligation and permission comes down exclusively to the logical value that a sentence adopts for the accessible deontic alternatives. Here, we set forth a different approach, this being a logic which additionally takes into consideration whether sentences stand in relation to the normative system or to the system of values (...) under which we predicate the deontic qualifications. By taking this aspect into account, we arrive at a logical system which preserves laws proper to a deontic logic but where the standard paradoxes of deontic logic do not arise. It is a logic of strictly-deontic modalities DR. (shrink)

– We offer a new motivation for imprecise probabilities. We argue that there are propositions to which precise probability cannot be assigned, but to which imprecise probability can be assigned. In such cases the alternative to imprecise probability is not precise probability, but no probability at all. And an imprecise probability is substantially better than no probability at all. Our argument is based on the mathematical phenomenon of non-measurable sets. Non-measurable propositions cannot receive precise probabilities, but there is a natural (...) way for them to receive imprecise probabilities. The mathematics of non-measurable sets is arcane, but its epistemological import is far-reaching; even apparently mundane propositions are liable to be affected by non-measurability. The phenomenon of non-measurability dramatically reshapes the dialectic between critics and proponents of imprecise credence. Non-measurability offers natural rejoinders to prominent critics of imprecise credence. Non-measurability even reverses some of the critics’ arguments—by the very lights that have been used to argue against imprecise credences, imprecise credences are better than precise credences. (shrink)