Reasoning under uncertainty, that is, making judgements with only partial knowledge, is a major theme in artificial intelligence. Professor Paris provides here an introduction to the mathematical foundations of the subject. It is suited for readers with some knowledge of undergraduate mathematics but is otherwise self-contained, collecting together the key results on the subject, and formalising within a unified framework the main contemporary approaches and assumptions. The author has concentrated on giving clear mathematical formulations, analyses, justifications and consequences of the (...) main theories about uncertain reasoning, so the book can serve as a textbook for beginners or as a starting point for further basic research into the subject. It will be welcomed by graduate students and research workers in logic, philosophy, and computer science as a textbook for beginners, a starting point for further basic research into the subject, and not least, an account of how mathematics and artificial intelligence can complement and enrich each other. (shrink)

ABSTRACTAlthough formative of modern value theory, the moral beauty view—which states that moral virtue is beautiful and moral vice is ugly—is now mostly neglected by philosophers. The two contemporary defences of the view mostly capitalize on its intuitive attractiveness, but to little avail: such considerations hardly convince sceptics of what is nowadays a rather unpopular view. Historically, the view was supported by thought experiments; and although these greatly increase its plausibility, they also raise empirical questions, which they leave unanswered. Here, (...) I offer a novel defence of the moral beauty view, capitalizing on empirical evidence and arguing via an inference to the best explanation. (shrink)

I offer a set of sufficient conditions for beauty, drawing on Parsons and Carlson’s account of ‘functional beauty’. First, I argue that their account is flawed, whilst falling short of...

There is a tendency in contemporary (analytic) aesthetics to consider- ably restrict the scope of things that can be beautiful or ugly. This peculiarly modern tendency is holding back progress in aesthetics and robbing it of its potential contribution to other domains of inquiry. One view that has suffered neglect as a result of this tendency is the moral beauty view, whereby the moral virtues are beautiful and the moral vices are ugly. This neglect stems from an assumption to the (...) effect that virtues and vices simply cannot be beautiful or ugly. The aim of this paper is twofold. First, it develops an account of form, under which, it argues, possession of form suffices for an object’s candidature for beauty and ugliness. Second, it argues that, under the foregoing proposal, the moral beauty view turns out to be a coherent position, and so should be taken seriously in both aesthetics and ethics. (shrink)

Can one extend crisp Peano arithmetic PA by a possibly many-valued predicate Tr(x) saying "x is true" and satisfying the "dequotation schema" $\varphi \equiv \text{Tr}(\bar{\varphi})$ for all sentences φ? This problem is investigated in the frame of Lukasiewicz infinitely valued logic.

Ugliness is a neglected topic in contemporary analytic aesthetics. This is regrettable given that this topic is not just genuinely fascinating, but could also illuminate other areas in the field, seeing as ugliness, albeit unexplored, does feature rather prominently in several debates in aesthetics. This paper articulates a ‘deformity-related’ conception of ugliness. Ultimately, I argue that deformity, understood in a certain way, and displeasure, jointly suffice for ugliness. First, I motivate my proposal, by locating a ‘deformity-related’ conception of ugliness in (...) aesthetic tradition, offering examples in support, and rejecting related alternative suggestions. Second, I argue that the proposal boasts considerable merits. Not only does it capture much of what we ordinarily think of as ugly, but it also comprises an objective criterion for ugliness, offers unity and comprehensiveness, and is informative and explanatorily potent. Third, I discuss a number of objections, thereby demonstrating that the proposal withstands reflective scrutiny. (shrink)

According to immoralists, some artworks are better aesthetically in virtue of their immorality. A. W. Eaton recently offered a novel defence of this view, seeking to overcome shortcomings in previous accounts, thereby occasioning a reconsideration of immoralism. Yet, as I argue in this paper, Eaton’s attempt is unsuccessful, insofar as it consists partly of inadequately supported claims, and partly—and more interestingly, albeit paradoxically––of covert moralist assumptions that are, eo ipso, incompatible with immoralism. I then turn to a parallel debate in (...) ethics concerning the possibility of admirable immorality, suggesting that a consideration of the state of that debate further supports my argument against immoralism. I close by suggesting some strategies by which immoralists may offer a rejoinder, although I note that their prospects are rather dim, not least because available defences of immoralism and similar positions all seem to share the same flawed pattern. (shrink)

This paper is a sequel to an earlier result of the authors that in making inferences from certain probabilistic knowledge bases the maximum entropy inference process, ME, is the only inference process respecting “common sense.” This result was criticized on the grounds that the probabilistic knowledge bases considered are unnatural and that ignorance of dependence should not be identified with statistical independence. We argue against these criticisms and also against the more general criticism that ME is representation dependent. In a (...) final section, however, we provide a criticism of our own of ME, and of inference processes in general, namely that they fail to satisfy compactness. (shrink)

Pure Inductive Logic is the study of rational probability treated as a branch of mathematical logic. This monograph, the first devoted to this approach, brings together the key results from the past seventy years, plus the main contributions of the authors and their collaborators over the last decade, to present a comprehensive account of the discipline within a single unified context.

This paper concerns the question of how to draw inferences common sensically from uncertain knowledge. Since the early work of Shore and Johnson (1980), Paris and Vencovská (1990), and Csiszár (1989), it has been known that the Maximum Entropy Inference Process is the only inference process which obeys certain common sense principles of uncertain reasoning. In this paper we consider the present status of this result and argue that within the rather narrow context in which we work this complete and (...) consistent mode of uncertain reasoning is actually characterised by the observance of just a single common sense principle (or slogan). (shrink)

We present a comprehensive study of the axiom schemas IΣ - n , BΣ - n (induction and collection schemas for parameter free Σ n formulas) and some closely related schemas.

Frank Plumpton Ramsey (1903–30) made seminal contributions to philosophy, mathematics and economics. Whilst he was acknowledged as a genius by his contemporaries, some of his most important ideas were not appreciated until decades later; now better appreciated, they continue to bear an influence upon contemporary philosophy. His historic significance was to usher in a new phase of analytic philosophy, which initially built upon the logical atomist doctrines of Bertrand Russell and Ludwig Wittgenstein, raising their ideas to a new level of (...) sophistication, but ultimately he became their successor rather than remain a mere acolyte. (shrink)

We examine the closure conditions of the probabilistic consequence relation of Hawthorne and Makinson, specifically the outstanding question of completeness in terms of Horn rules, of their proposed (finite) set of rules O. We show that on the contrary no such finite set of Horn rules exists, though we are able to specify an infinite set which is complete.

We consider the desirability, or otherwise, of various forms of induction in the light of certain principles and inductive methods within predicate uncertain reasoning. Our general conclusion is that there remain conflicts within the area whose resolution will require a deeper understanding of the fundamental relationship between individuals and properties.

We give an account of some relationships between the principles of Constant and Atom Exchangeability and various generalizations of the Principle of Instantial Relevance within the framework of Inductive Logic. In particular we demonstrate some surprising and somewhat counterintuitive dependencies of these relationships on ostensibly unimportant parameters, such as the number of predicates in the overlying language.

In this paper we consider a natural generalization of the Principle of Instantial Relevance and give a complete characterization of the probabilistic belief functions satisfying this principle as a family of discrete probability functions parameterized by a single real δ ∊ [0, 1).

Can one extend crisp Peano arithmetic PA by a possibly many-valued predicate Tr(x) saying “xis true” and satisfying the “dequotation schema”for all sentences φ? This problem is investigated in the frame of Łukasiewicz infinitely valued logic.

Leading experts and rising stars in the field explore whether cosmopolitanism becomes impossible in the theoretical framework that assumed the absence of a final ground. The questions that the volume addresses refer exactly to the foundational predicament that characterizes cosmopolitanism: How is it possible to think cosmopolitanism after the critique of foundations? Can cosmopolitanism be conceived without an ‘ultimate’ ground? Can we construct theories of cosmopolitanism without some certainties about the entire world or about the cosmos? Should we continue to (...) look for foundations of cosmopolitan rights, norms and values? Alternatively, should we aim towards cosmopolitanism without foundations or towards cosmopolitanism with ‘contingent foundations’? Could cosmopolitanism be the very attempt to come to terms with the failure of ultimate grounds? Written accessibly and contributing to key debates on political philosophy, and social and political thought, this volume advances the concept of post-foundational cosmopolitanism by bridging the polarised approaches to the concept. (shrink)

The opening sentence of Christopher Meyers’ Target Article is “Lying to one’s patient is wrong”. The author continues, “This truism is one that bioethicists have heartedly endorsed fo...

The opinion of Mr. Justice Francis of the English High Court which denied the parents of Charlie Gard, who had been born with an extremely rare mutation of a genetic disease, the right to take their child to the United States for a proposed experimental treatment occasioned world wide attention including that of the Pope, President Trump, and the US Congress. The case raise anew a debate as old as the foundation of Western medicine on who should decide and on (...) what standard when there is a conflict between a family and the treating physicians over a possible treatment. This paper will explore the different approaches of the British and American courts on the issue and the various proposals from that of John Rawls in his A Theory of Justice to a processed-based approach for resolving such disputes. As carefully crafted as the opinion of Mr Justice Nicholas Francis in the Gard case proved to be, it left commentators unsatisfied. A widespread criticism, captured in an article by Michael Dougherty in The National Review was for the state to ‘get out of the way of the parents trying to act in the best interests of the child’.1 Although he conceded the parents could be adding to the suffering of the child by taking Charlie to America for an experimental therapy and agreed that such a choice ‘may be the wrong decision,’ in Dougherty’s view, it should still be ‘their decision’. Dougherty’s stand was the popular response to the question of ‘Who should decide?’ It fails, however, to propose any rationale for the decision. It provided no norms, no standards and no guidelines for the parents. Their motive could equally well be indifference to the suffering of the child as concern for his well-being. Furthermore, even good, loving parents may make …. (shrink)

Rational Pavelka logic extends Lukasiewicz infinitely valued logic by adding truth constants r̄ for rationals in [0, 1]. We show that this is a conservative extension. We note that this shows that provability degree can be defined in Lukasiewicz logic. We also give a counterexample to a soundness theorem of Belluce and Chang published in 1963.

We consider the problem of induction over languages containing binary relations and outline a way of interpreting and constructing a class of probability functions on the sentences of such a language. Some principles of inductive reasoning satisfied by these probability functions are discussed, leading in turn to a representation theorem for a more general class of probability functions satisfying these principles.

We propose an Analogy Principle in the context of Unary Inductive Logic and characterize the probability functions which satisfy it. In particular in the case of a language with just two predicates the probability functions satisfying this principle correspond to solutions of Skyrmsʼ ‘Wheel of Fortune’.

We have all heard people say ‘Beauty is only skin-deep’, or ‘Beauty is in the eye of the beholder’: our culture promulgates a conception of beauty as subjective, superficial, and independent of other values like moral goodness or knowledge and understanding. Yet our taste in beauty affects many aspects of our lives, sometimes playing a decisive – and often detrimental – role in areas as wide-ranging as our identity and self-esteem, our morally salient decisions, and our relationship to the environment. (...) This presents us with a choice: we can either ignore the facts – leaving our conception of beauty unchanged and allowing our taste to influence much in our lives while either not acknowledging such influence, or perhaps seeking to reprimand it; or we can take the power of beauty seriously and seek to harmonise our taste with our values. I argue for the latter option and propose a way of bringing beauty and taste in line with what matters to us using the notion of functional beauty. Adopting this strategy, I suggest, can have a powerful – and positive – impact on our self-esteem and wellbeing, our relationship to others, as well as our attitudes towards the environment. (shrink)

We investigate uncertain reasoning with quantified sentences of the predicate calculus treated as the limiting case of maximum entropy inference applied to finite domains.

Volume 20, Issue 7, July 2020, Page 147-150.

A necessary and sufficient condition in terms of a de Finetti style representation is given for a probability function in Polyadic Inductive Logic to satisfy being part of a Language Invariant family satisfying Spectrum Exchangeability. This theorem is then considered in relation to the unary Carnap and Nix–Paris Continua.

We give a complete characterization of Priest's Finite Inconsistent Arithmetics observing that his original putative characterization included arithmetics which cannot in fact be realized.

Linda Zagzebski recently put forward a new theory, moral exemplarism, that is meant to provide an alternative to theories like consequentialism and deontology, and which proposes to define key moral terms by direct reference to exemplars. The theory’s basic structure is straightforward. A virtuous person is defined as a person like that, where that points to individuals like Leopold Socha, Confucius, Jesus Christ, and so on. A key component of this theory is the function played by the emotions, specifically the (...) emotion of admiration, which helps us identify moral exemplars, inspires our emulation of them, and grounds moral motivation. In other words, admiration tracks persons like that. The aim of this article is to show that unless moral exemplarism recognises and incorporates an aesthetic dimension to morality, as did, for instance, eighteenth-century sentimentalists by recognising the categories of moral beauty and ugliness, the theory suffers important theoretical difficulties, whilst failing to yield some of the theoretical merits that it boasts. To this end, I will cast doubt on the prominent role accorded to the emotion of admiration for morality, arguing that it has to either be qualified as a specific kind of admiration––namely that which takes the beautiful as its object––or be replaced altogether with the affective response to the beautiful, a mark of which is pleasure or delight in the contemplation of an object. In short, an aesthetics of character is necessary in order to properly theorise the affective and motivational components of morality under an exemplarist framework. (shrink)

This paper draws from critical data studies and related fields to investigate police officer-involved homicide data for Los Angeles County. We frame police officer-involved homicide data as a rhetorical tool that can reify certain assumptions about the world and extend regimes of power. We highlight the possibility that this type of sensitive civic data can be investigated and employed within local communities through creative practice. Community involvement with data can create a countervailing force to powerful dominant narratives and supplement activist (...) projects that hold local officials accountable for their actions. Our analysis examines four Los Angeles County police officer-involved homicide data sets. First, we provide accounts of the semantics, granularity, scale and transparency of this local data. Then, we describe a “counter data action,” an event that invited members of the community to identify the limits and challenges present in police officer-involved homicide data and to propose new methods for deriving meaning from these indicators and statistics. (shrink)

Considerable progress in personality and social psychology has been largely ignored by philosophers, many of whom still remain sceptical concerning whether the conception of character presupposed by virtue theory is descriptively adequate. Here, I employ the five-factor model of personality, currently the consensus view in personality psychology, to respond to a strong reading of the situationist challenge, whereby most people lack dispositions that are both cross-situationally consistent and temporally stable. I show that situationists rely on a false dichotomy between character (...) traits and situations, and that evidence supports the empirical adequacy of the sorts of character traits presupposed by virtue ethics. Additionally, I suggest that the personality traits of the five-factor model are relevant to virtue theory, in so far as they are malleable, morally salient, and seem to structurally parallel Aristotelian virtues and vices. Thus, contra situationism, the five-factor model supports the descriptive adequacy of a virtue-theoretical framework. (shrink)

We answer some problems set by Priest in [11] and [12], in particular refuting Priest's Conjecture that all LP-models of Th(N) essentially arise via congruence relations on classical models of Th(N). We also show that the analogue of Priest's Conjecture for I δ₀ + Exp implies the existence of truth definitions for intervals [0,a] ⊂ₑ M ⊨ I δ₀ + Exp in any cut [0,a] ⊂e K ⊆ M closed under successor and multiplication.

The notion of the rational closure of a positive knowledge base K of conditional assertions θ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$i$$ \end{document} |∼ φ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$i$$ \end{document} (standing for if θ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$i$$ \end{document} then normally φ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$i$$ \end{document}) was first introduced by Lehmann (1989) and developed by Lehmann and Magidor (...) (1992). Following those authors we would also argue that the rational closure is, in a strong sense, the minimal information, or simplest, rational consequence relation satisfying K. In practice, however, one might expect a knowledge base to consist not just of positive conditional assertions, θ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$i$$ \end{document} |∼ φ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$i$$ \end{document}, but also negative conditional assertions, θi φ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$i$$ \end{document} (standing for not if θ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$i$$ \end{document} then normally φ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$i$$ \end{document}. Restricting ourselves to a finite language we show that the rational closure still exists for satisfiable knowledge bases containing both positive and negative conditional assertions and has similar properties to those exhibited in Lehmann and Magidor (1992). In particular an algorithm in Lehmann and Magidor (1992) which constructs the rational closure can be adapted to this case and yields, in turn, completeness theorems for the conditional assertions entailed by such a mixed knowledge base. (shrink)

We consider the version of Pure Inductive Logic which obtains for the language with equality and a single unary function symbol giving a complete characterization of the probability functions on this language which satisfy Constant Exchangeability.

In What Is Literature? Sartre claims, "One does not write for slaves."1 This takes place in the context of an argument Sartre makes in claiming literature is an appeal to the freedom of others.2 Furthermore, the acts of reading and writing are collaborative occasions that invent and re-invent the world by disclosing it and creating it.3 It is important to be precise about what Sartre believes must be presupposed in order for literature to function. The force of committed literature is (...) not propaganda that attempts to control the reader; it only comes about from the a priori existence of free subjects. Thus, Sartre notes, "The book does not serve my freedom; it requires it. Indeed, one cannot address oneself to... (shrink)

We give a unified account of some results in the development of Polyadic Inductive Logic in the last decade with particular reference to the Principle of Spectrum Exchangeability, its consequences for Instantial Relevance, Language Invariance and Johnson's Sufficientness Principle, and the corresponding de Finetti style representation theorems.

We give a brief account of some de Finetti style representation theorems for probability functions satisfying Spectrum Exchangeability in Polyadic Inductive Logic, together with applications to Non-splitting, Language Invariance, extensions with Equality and Instantial Relevance.