Throughout the twentieth century, the classical logic of Frege and Russell dominated the field of formal logic. But, as Ross Brady argues, a new type of weak relevant logic may prove to be better equipped to present new solutions to persistent paradoxes. _Universal Logic _begins with an overview of classical and relevant logic and discusses the limitations of both in analyzing certain paradoxes. It is the first text to demonstrate how the main set-theoretic and semantic paradoxes can be solved (...) in a systematic way and as such will be of great interest to both scholars and students of logic. (shrink)

We start by noting that the set-theoretic and semantic paradoxes are framed in terms of a definition or series of definitions. In the process of deriving paradoxes, these definitions are logically represented by a logical equivalence. We will firstly examine the role and usage of definitions in the derivation of paradoxes, both set-theoretic and semantic. We will see that this examination is important in determining how the paradoxes were created in the first place and indeed how they are to be (...) solved in a uniform way. There are three features that are special about these definitions used in the derivation of most of the above paradoxes. The first is the use of self-reference between the definiens and the definiendum, the second is the generality of the definiendum, and the third is the under-determination and over-determination of concepts that usually occur as a result of these definitions. We will examine the impact of these three features on the logical representation of definitions and show how this representation then leads to a uniform paradox solution using an appropriate logic that is both paraconsistent and paracomplete. However, it is the paracompleteness, exhibited through the rejection of the Law of Excluded Middle, together with the rejection of contraction principles, that enables the solution of the paradoxes to go through. We characterize definitions as involving syntactic identity and/or meaning identity. We point out that some paradoxes do not have explicit self-reference or circularity and some may not utilize the generality of the definiendum, but the general characterizations of definitions that we give will still apply. We also look beyond all this to paradoxes that rely on illicit definitions between objects that are essentially different to start with. (shrink)

The main aim is to extend the range of logics which solve the set-theoretic paradoxes, over and above what was achieved by earlier work in the area. In doing this, the paper also provides a link between metacomplete logics and those that solve the paradoxes, by finally establishing that all M1-metacomplete logics can be used as a basis for naive set theory. In doing so, we manage to reach logics that are very close in their axiomatization to that of the (...) logic R of relevant implication. A further aim is the use of metavaluations in a new context, expanding the range of application of this novel technique, already used in the context of negation and arithmetic, thus providing an alternative to traditional model theoretic approaches. (shrink)

This is a general account of metavaluations and their applications, which can be seen as an alternative to standard model-theoretic methodology. They work best for what are called metacomplete logics, which include the contraction-less relevant logics, with possible additions of Conjunctive Syllogism, & →.A→C, and the irrelevant, A→.B→A, these including the logic MC of meaning containment which is arguably a good entailment logic. Indeed, metavaluations focus on the formula-inductive properties of theorems of entailment form A→B, splintering into two types, M1- (...) and M2-, according to key properties of negated entailment theorems. Metavaluations have an inductive presentation and thus have some of the advantages that model theory does, but they represent proof rather than truth and thus represent proof-theoretic properties, such as the priming property, if ├ A $\vee$ B then ├ A or ├ B, and the negated-entailment properties, not-├ ∼ and ├ ∼ iff ├ A and ├ ∼ B. Topics to be covered are their impact on naive set theory and paradox solution, and also Peano arithmetic and Godel’s First and Second Theorems. Interesting to note here is that the familiar M1- and M2-metacomplete logics can be used to solve the set-theoretic paradoxes and, by inference, the Liar Paradox and key semantic paradoxes. For M1-logics, in particular, the final metavaluation that is used to prove the simple consistency is far simpler than its correspondent in the model-theoretic proof in that it consists of a limit point of a single transfinite sequence rather than that of a transfinite sequence of such limit points, as occurs in the model-theoretic approach. Additionally, it can be shown that Peano Arithmetic is simply consistent, using metavaluations that constitute finitary methods. Both of these results use specific metavaluational properties that have no correspondents in standard model theory and thus it would be highly unlikely that such model theory could prove these results in their final forms. (shrink)

Criminal courts make decisions that can remove the liberty and even life of those accused. Civil trials can cause the bankruptcy of companies employing thousands of people, asylum seekers being deported, or children being placed into state care. Selecting the right standards when deciding legal cases is of utmost importance in giving those affected a fair deal. This Element is an introduction to the philosophy of legal proof. It is organised around five questions. First, it introduces the standards of proof (...) and considers what justifies them. Second, it discusses whether we should use different standards in different cases. Third, it asks whether trials should end only in binary outcomes (e.g. guilty or not guilty) or rather use more fine-grained or precise verdicts. Fourth, it considers whether proof is simply about probability, concentrating on the famous 'Proof Paradox'. Finally, it examines who should be trusted with deciding trials, focusing on the jury system. (shrink)

Democracy surrounds us like the air we breath, and is normally taken very much for granted. Across the world democracy has become accepted as an unquestionably good thing. Yet upon further examination the merits of democracy are both paradoxical and problematic, and the treasured values of liberty and equality can be used to argue both for and against it. In the historical section of the book, Ross Harrison clearly traces the history of democracy by examining the works of, amongst (...) others, Plato and Aristotle, Hegel and Marx. Informed by facts and detailed knowledge of these famous thinkers, Harrison provides a clear and cogent justification of democracy. (shrink)

Anthropological inquiry suggests that all societies classify animals and plants in similar ways. Paradoxically, in the same cultures that have seen large advances in biological science, citizenry's practical knowledge of nature has dramatically diminished. Here we describe historical, cross-cultural and developmental research on how people ordinarily conceptualize organic nature, concentrating on cognitive consequences associated with knowledge devolution. We show that results on psychological studies of categorization and reasoning from “standard populations” fail to generalize to humanity at large. Usual populations have (...) impoverished experience with nature, which yields misleading results about knowledge acquisition and the ontogenetic relationship between folkbiology and folkpsychology. We also show that groups living in the same habitat can manifest strikingly distinct behaviors, cognitions and social relations relative to it. This has novel implications for environmental decision making and management, including commons problems. (shrink)

Adam Smith is perceived, through his best-known book, An Inquiry into the Nature and Causes of the Wealth of Nations, as the founder of economics as a science. His thought has shaped modern ideas about the market economy and the role of the state in relation to it. Yet Smith needs to be recognized as more than this, as a man of letters, moralist, historian, and critic, as well as an economist, if we are to get full value for his (...) ideas and perspectives in contemporary applications. Ian Simpson Ross is the biographer of Lord Kames, Smith's patron, and of the Scottish poet William Dunbar, and has edited, with E C Mossner, Smith's correspondence for the Glasgow edition of his works. In this, the first full-scale biography of Adam Smith for a hundred years, Ross brings his subject in to historical light as a thinker and author by examining his family circumstance, education, career, and social and intellectual circle, including David Hume and Francois Quesnay, revealed through his correspondence, archival documents, the reports of contemporaries, and the record of his publications. Readers will meet Smith as a student at a lively Glasgow and sleepy Oxford; freelance lecturer in rhetoric; innovative university teacher; tutor travelling abroad with a Duke; acclaimed political economist; policy advisor to governments during and after the American crisis; and finally, if paradoxically in view of his tenets, a Commissioner of Customs coping with the free traders in the smuggling business. This is the life of a Scottish moral philosopher whose legacy of thought concerns and affects us all. Its lively and informed account will appeal to those interested in the social and intellectual milieu of the eighteenth century, and in scottish history. Economists and philosophers will find much to read about the history of their disciplines, supported by full documentation. (shrink)

Until recently, Albert Einstein's complaints in his later years about the intelligibility of Quantum Mechanics often led philosophers and physicists to dismiss him as, essentially, an old fool in his dotage. Happily, this kind of thing is now coming to an end as philosophers and mathematicians of the caliber of Karl Popper and Roger Penrose conspicuously point out the continuing conceptual difficulties of quantum theory [cf. Penrose's searching discussion in The Emperor's New Mind, chapter 6, "Quantum magic and quantum mystery," (...) Oxford 1990]. The Paradox of Schrödinger's Cat is sometimes now even presented, not as a wonderful exciting implication of the theory, but for what it originally was: a reductio ad absurdum argument against the "new" quantum mechanics of Heisenberg and Bohr. Schrödinger shared the misgivings of Einstein and others. (shrink)

Recent years have seen fresh impetus brought to debates about the proper role of statistical evidence in the law. Recent work largely centres on a set of puzzles known as the ‘proof paradox’. While these puzzles may initially seem academic, they have important ramifications for the law: raising key conceptual questions about legal proof, and practical questions about DNA evidence. This article introduces the proof paradox, why we should care about it, and new work attempting to resolve it.

Democracy surrounds us like the air we breath, and is normally taken very much for granted. Across the world democracy has become accepted as an unquestionably good thing. Yet upon further examination the merits of democracy are both paradoxical and problematic, and the treasured values of liberty and equality can be used to argue both for and against it. In the historical section of the book, Ross Harrison clearly traces the history of democracy by examining the works of, amongst (...) others, Plato and Aristotle, Hegel and Marx. Informed by facts and detailed knowledge of these famous thinkers, Harrison provides a clear and cogent justification of democracy. (shrink)

This paper defends the heretical view that, at least in some cases, we ought to assign legal liability based on purely statistical evidence. The argument draws on prominent civil law litigation concerning pharmaceutical negligence and asbestos-poisoning. The overall aim is to illustrate moral pitfalls that result from supposing that it is never appropriate to rely on bare statistics when settling a legal dispute.

Recently, the practice of deciding legal cases on purely statistical evidence has been widely criticised. Many feel uncomfortable with finding someone guilty on the basis of bare probabilities, even though the chance of error might be stupendously small. This is an important issue: with the rise of DNA profiling, courts are increasingly faced with purely statistical evidence. A prominent line of argument—endorsed by Blome-Tillmann 2017; Smith 2018; and Littlejohn 2018—rejects the use of such evidence by appealing to epistemic norms that (...) apply to individual inquirers. My aim in this paper is to rehabilitate purely statistical evidence by arguing that, given the broader aims of legal systems, there are scenarios in which relying on such evidence is appropriate. Along the way I explain why popular arguments appealing to individual epistemic norms to reject legal reliance on bare statistics are unconvincing, by showing that courts and individuals face different epistemic predicaments (in short, individuals can hedge when confronted with statistical evidence, whilst legal tribunals cannot). I also correct some misconceptions about legal practice that have found their way into the recent literature. (shrink)

A question, long discussed by legal scholars, has recently provoked a considerable amount of philosophical attention: ‘Is it ever appropriate to base a legal verdict on statistical evidence alone?’ Many philosophers who have considered this question reject legal reliance on bare statistics, even when the odds of error are extremely low. This paper develops a puzzle for the dominant theories concerning why we should eschew bare statistics. Namely, there seem to be compelling scenarios in which there are multiple sources of (...) incriminating statistical evidence. As we conjoin together different types of statistical evidence, it becomes increasingly incredible to suppose that a positive verdict would be impermissible. I suggest that none of the dominant views in the literature can easily accommodate such cases, and close by offering a diagnosis of my own. (shrink)

Legal epistemology has been an area of great philosophical growth since the turn of the century. But recently, a number of philosophers have argued the entire project is misguided, claiming that it relies on an illicit transposition of the norms of individual epistemology to the legal arena. This paper uses these objections as a foil to consider the foundations of legal epistemology, particularly as it applies to the criminal law. The aim is to clarify the fundamental commitments of legal epistemology (...) and suggest a way to vindicate it. (shrink)

This volume offers an updated analysis of the use, meaning, and scope of the classical notion of aitía. It clarifies philosophical and philological questions about aitia and offers bold and innovative interpretations of this key concept of ancient philosophy. The numerous meanings and nuances of aitia remain difficult to grasp. Ancient philosophers use aitia to explain the existence and activity of substances, bodies, souls, or gods, Paradoxically, its own definition remains difficult to establish. This book reconstructs some of the most (...) important uses, variants, and scopes of the term aitia within different philosophical perspectives in antiquity, including early Greek philosophy, Plato, Aristotle, Stoicism, and Islamic philosophy. The chapters analyze metaphysical aspects, epistemological issues, and logical implications of aitia. They engage with the most relevant critical literature generated in several modern languages. In doing so, they offer an inclusive and overarching re-evaluation of our assumptions about causation and explanation in ancient philosophy. Cause and Explanation in Ancient Philosophy will be of interest to scholars and advanced students working on pre-Socratic philosophy, Plato, Aristotle, Hellenistic philosophy, late antiquity, and medieval philosophy. (shrink)

Following a suggestion of Feys, we use “rigorous implication” as a translation of Ackermann's strenge Implikation ([1]). Interest in Ackermann's system stems in part from the fact that it formalizes the properties of a strong, natural sort of implication which provably avoids standard implicational paradoxes, and which is consequently a good candidate for a formalization of entailment (considered as a narrower relation than that of strict implication). Our present purpose will not be to defend this suggestion, but rather to present (...) some information about rigorous implication. In particular, we show first that the structure of modalities (in the sense of Parry [4]) in Ackermann's system is identical with the structure of modalities in Lewis's S4, and secondly that (Ackermann's apparent conjecture to the contrary notwithstanding) it is possible to define modalities with the help of rigorous implication. (shrink)

It is argued that claims about personal obligation (of the form "s ought to 0") cannot be reduced to claims about impersonal obligation (of the form "it ought to be the case that p"). The most common attempts at such a reduction are shown to have unacceptable implications in cases involving a plurality of agents. It is then argued that similar problems will face any attempt to reduce personal obligation to impersonal obligation.

Discusses whether legal proof is merely probabilistic, focusing on the famous proof paradox.

That caricature succeeds at all seems paradoxical. That its dictum is “less is more” seems more puzzling still. In this paper I hope to investigate how caricature transforms exaggeration, distortion, and falsification into vehicles for succinct comment and easy identification. I shall examine and discard several views of how caricature functions, and conclude by arguing that correctly identifying a caricature is no more, and no less, paradoxical than correctly identifying any of the everyday objects that clutter our world.

In this paper, I propose that one can have reason to choose a few tickets in a very large lottery and arbitrarily believe of them that they will lose. Such a view fits nicely within portions of Lehrer's theory of rational acceptance. Nonetheless, the reasonability of believing a lottery ticket will lose should not be taken to constitute the kind of justification required in an analysis of knowledge. Moreover, one should not accept what one takes to have a low chance (...) of being true. Accordingly, one should take care not to believe of too many tickets that they are to lose. Finally, while arbitrariness is no absolute barrier to epistemic reasonability, one may not be able to believe that one's lottery ticket will lose if one cannot regard oneself as knowing it will lose. (shrink)

Berkeley and Helmholtz proposed different indirect mechanisms for size perception: Berkeley, that size was conditioned to various cues, independently of perceived distance; Helmholtz, that it was unconsciously calculated from angular size and perceived distance. The geometrical approach cannot explain size-distance paradoxes (e.g., moon illusion). The dorsal/ventral solution is dubious for close displays and untestable for far displays.

This paper examines the use of “pleasure” as the distinguishing mark of aesthetic experience in post-Kantian philosophy. It shows how the distinctive features of aesthetic experience, such as pleasure, qualify this experience as a platform for social criticism. The key argument is that the autonomy of the aesthetic experience is not “false”, rather it is paradoxical in the strong sense that the fact of its communicative efficacy, which follows from distinctive, “autonomous” aesthetic features, necessarily loads it with functions and expectations (...) that are external to the aesthetic moment. Kant takes a complicated path to qualify aesthetic judgment as disinterested in order that it may eloquently testify for morality. He thereby sets up the cogency of the modern pattern of looking to aesthetic experience as a locus of meaningful communication for ideas that are experientially poor or remote. (shrink)

This introduction to the Common Knowledge symposium titled “Comparative Relativism” outlines a variety of intellectual contexts where placing the unlikely companion terms comparison and relativism in conjunction offers analytical purchase. If comparison, in the most general sense, involves the investigation of discrete contexts in order to elucidate their similarities and differences, then relativism, as a tendency, stance, or working method, usually involves the assumption that contexts exhibit, or may exhibit, radically different, incomparable, or incommensurable traits. Comparative studies are required to (...) treat their objects as alike, at least in some crucial respects; relativism indicates the limits of this practice. Jensen argues that this seeming paradox is productive, as he moves across contexts, from Lévi-Strauss's analysis of comparison as an anthropological method to Peter Galison's history of physics, and on to the anthropological, philosophical, and historical examples offered in symposium contributions by Barbara Herrnstein Smith, Eduardo Viveiros de Castro, Marilyn Strathern, and Isabelle Stengers. Comparative relativism is understood by some to imply that relativism comes in various kinds and that these have multiple uses, functions, and effects, varying widely in different personal, historical, and institutional contexts that can be compared and contrasted. Comparative relativism is taken by others to encourage a “comparison of comparisons,” in order to relativize what different peoples—say, Western academics and Amerindian shamans—compare things “for.” Jensen concludes that what is compared and relativized in this symposium are the methods of comparison and relativization themselves. He ventures that the contributors all hope that treating these terms in juxtaposition may allow for new configurations of inquiry. (shrink)

'Post this letter!' does not entail 'Post this letter or drink up my wine!' (the Ross Paradox) because one can be in a state with the content of the former without being in a state with the content of the latter; in turn, because the latter can rationalize drinking up my wine but the former cannot; in turn, because practical rationalization flows toward one's present situation, in contrast with the flow of theoretical rationalization from one's present situation. Formally, (...) this is implemented with a semantics for imperatives involving a partition to set up an issue of significance and a proposition to establish how to settle it. I characterize a series of languages and language-games to implement all this, and appeal to an 'endorsement-theoretic' conception of modals as quantifiers over mental states to explain why 'You must post this letter' does not entail 'You must either post this letter or drink up my wine'. (shrink)

In a number of earlier writings, I have outlined what may perhaps be called a game-theoretical semantics. This semantics can be applied both to formal first-order languages, with or without modalities, and to certain fragments of English. The basic idea of this semantics is to associate with each sentence S under consideration a two-person game G. The players are called Myself and Nature.

It is argued that Ross's Paradox in deontic logic is a problem which should be taken seriously, and which can be given a solution which also solves some other wellknown paradoxes and the traditional problems with conditional obligation.

In a recent paper, Sven Danielsson argued that the ‘original paradoxes' of deontic logic, in particular Ross's paradox and Prior's paradox of derived obligation, can be solved by restricting the modal inheritance rule. I argue that this does not solve the paradoxes.

Martine Nida-Rümelin (1996) argues that color science indicates behaviorally undetectable spectrum inversion is possible and raises this possibility as an objection to functionalist accounts of visual states of color. I show that her argument does not rest solely on color science, but also on a philosophically controversial assumption, namely, that visual states of color supervene on physiological states. However, this assumption, on the part of philosophers or vision scientists, has the effect of simply ruling out certain versions of functionalism. While (...) Nida-Rümelin is quite right to search for empirical tests for claims about the nature of visual states, philosophical issues remain pivotal in determining the correctness of these claims. (shrink)

Original essays by leading philosophers of science explore the question of whether metaphysics can and should be naturalized--conducted as part of natural science.