Alex Oliver and Timothy Smiley provide a new account of plural logic. They argue that there is such a thing as genuinely plural denotation in logic, and expound a framework of ideas that includes the distinction between distributive and collective predicates, the theory of plural descriptions, multivalued functions, and lists.

Multiple -conclusion logic extends formal logic by allowing arguments to have a set of conclusions instead of a single one, the truth lying somewhere among the conclusions if all the premises are true. The extension opens up interesting possibilities based on the symmetry between premises and conclusions, and can also be used to throw fresh light on the conventional logic and its limitations. This is a sustained study of the subject and is certain to stimulate further research. Part I reworks (...) the fundamental ideas of logic to take account of multiple conclusions, and investigates the connections between multiple - and single - conclusion calculi. Part II draws on graph theory to discuss the form and validity of arguments independently of particular logical systems. Part III contrasts the multiple - and the single - conclusion treatment of one and the same subject, using many-valued logic as the example; and Part IV shows how the methods of 'natural deduction' can be matched by direct proofs using multiple conclusions. (shrink)

To prepare for potential human infection challenge studies involving SARS-CoV-2, we convened a multidisciplinary working group to address ethical questions regarding whether and how much SAR...

This essay discusses the nature of collective responsibility and explores various controversies associated with its possibility and normative value.

The history of the idea of predicate is the history of its emancipation. The lesson of this paper is that there are two more steps to take. The first is to recognize that predicates need not have a fixed degree, the second that they can combine with plural terms. We begin by articulating the notion of a multigrade predicate: one that takes variably many arguments. We counter objections to the very idea posed by Peirce, Dummett's Frege, and Strawson. We show (...) that the arguments of a multigrade predicate must be grouped into places, with perhaps several arguments occupying positions at a place. Variability may relate to places or positions. Russell's multiple judgement predicate turns out to be just one example of a family-'is necessarily true of', 'is said of', 'is instantiated by' and so on-of predicates with variably many places. Our main concern, however, is lists. Any adequate account of lists must include plural as well as singular terms. On one account, lists are mere strings of separate arguments, which occupy variably many positions within a place of a multigrade predicate. A quite different account takes the list itself to be a compound plural term. We compare these rival conceptions, and reach some surprising conclusions. As a coda, we deploy the conceptual apparatus developed in the paper to assess Morton's pioneer system of multigrade logic. (shrink)

T. J. Smiley; XII.—Entailment and Deducibility, Proceedings of the Aristotelian Society, Volume 59, Issue 1, 1 June 1959, Pages 233–254, https://doi.org/10.1093.

Alex Oliver and Timothy Smiley provide a new account of plural logic. They argue that there is such a thing as genuinely plural denotation in logic, and expound a framework of ideas that includes the distinction between distributive and collective predicates, the theory of plural descriptions, multivalued functions, and lists.

Russell had two theories of definite descriptions: one for singular descriptions, another for plural descriptions. We chart its development, in which ‘On Denoting’ plays a part but not the part one might expect, before explaining why it eventually fails. We go on to consider many-valued functions, since they too bring in plural terms—terms such as ‘4’ or the descriptive ‘the inhabitants of London’ which, like plain plural descriptions, stand for more than one thing. Logicians need to take plural reference seriously (...) if only because mathematicians take many-valued functions seriously. We assess the objection (by Russell, Frege and others) that many-valued functions are illegitimate because the corresponding terms are ambiguous. We also assess the various methods proposed for getting rid of them. Finding the objection ill-founded and the methods ineffective, we introduce a logical framework that admits plural reference, and use it to answer some earlier questions and to raise some more. (shrink)

We present a plural logic that is as expressively strong as it can be without sacrificing axiomatisability, axiomatise it, and use it to chart the expressive limits set by axiomatisability. To the standard apparatus of quantification using singular variables our object-language adds plural variables, a predicate expressing inclusion (is/are/is one of/are among), and a plural definite description operator. Axiomatisability demands that plural variables only occur free, but they have a surprisingly important role. Plural description is not eliminable in favour of (...) quantification; on the contrary, quantification is definable in terms of it. Predicates and functors (function signs) can take plural as well as singular terms as arguments, and both many-valued and single-valued functions are expressible. The system accommodates collective as well as distributive predicates, and the condition for a predicate to be distributive is definable within it; similarly for functors. An essential part of the project is to demonstrate the soundness and completeness of the calculus with respect to a semantics that does without set-theoretic domains and in which the use of settheoretic extensions of predicates and functors is replaced by the sui generis relations and functions for which the extensions were at best artificial surrogates. Our metalanguage is designed to solve the difficulties involved in talking plurally about individuals and about the semantic values of plural items. (shrink)

We all learn about the mistake of treating ‘nothing’ as if it were a term standing for something; but is it a mistake to treat it as an empty term, denoting nothing? We argue not, and we introduce ‘zilch’, defined as ‘the non-self-identical thing’, as a term which is empty as a matter of logical necessity. We contrast its behaviour with that of the quantifier ‘nothing’, and illustrate its uses. We use the same idea to vindicate Locke’s, Descartes’ and Hume’s (...) handling of ‘nothing’, and we show how Heidegger’s ‘das Nichts nichtet’ comes out as logically true. (shrink)

This book has three goals. The first is to demonstrate that the modern, distinctly Kantian, notion of moral responsibility is incoherent by virtue of the way it fuses free will and blameworthiness. The second is to develop an alternative notion of moral responsibility that separates causal responsibility from blameworthiness and views both as relative to the boundaries of our moral community. The third is to establish a framework for arguing openly about our moral responsibility for particular kinds of harm.

How can we make sense of future-looking collective responsibility? What is its moral basis and how -- under what conditions -- can we ascribe it to particular groups? I address these questions below and, in doing so, argue that in ascribing future-looking collective responsibility we need to bring claims of backward-looking (causal) responsibility together with judgments of fairness, practicality, and group identity.

Explains why Bob Hale's proposed notion of weak sense cannot explain the analyticity of Hume's principle as he claims. Argues that no other notion of the sort Hale wants could do the job either.

This essay argues that while the notion of collective responsibiility is incoherent if it is taken to be an application of the Kantian model of moral responsibility to groups, it is coherent -- and important -- if formulated in terms of the moral reactions that we can have to groups that cause harm in the world. I formulate collective responsibility as such and in doing so refocus attention from intentionality to the production of harm.

The question of responsibility plays a critical role not only in our attempts to resolve social and political problems, but in our very conceptions of what those problems are. Who, for example, is to blame for apartheid in South Africa? Is the South African government responsible? What about multinational corporations that do business there? Will uncovering the "true facts of the matter" lead us to the right answer? In an argument both compelling and provocative, Marion Smiley demonstrates how attributions of (...) blame—far from being based on an objective process of factual discovery—are instead judgments that we ourselves make on the basis of our own political and social points of view. She argues that our conception of responsibility is a singularly modern one that locates the source of blameworthiness in an individual's free will. After exploring the flaws inherent in this conception, she shows how our judgments of blame evolve out of our configuration of social roles, our conception of communal boundaries, and the distribution of power upon which both are based. The great strength of Smiley's study lies in the way in which it brings together both rigorous philosophical analysis and an appreciation of the dynamics of social and political practice. By developing a pragmatic conception of moral responsibility, this work illustrates both how moral philosophy can enhance our understanding of social and political practices and why reflection on these practices is necessary to the reconstruction of our moral concepts. (shrink)

A follow-up, showing why Bob Hale's revision of his notion of weak sense is still inadequate.

Almost all set theorists pay at least lip service to Cantor’s definition of a set as a collection of many things into one whole; but empty and singleton sets do not fit with it. Adapting Dana Scott’s axiomatization of the cumulative theory of types, we present a ‘Cantorian’ system which excludes these anomalous sets. We investigate the consequences of their omission, examining their claim to a place on grounds of convenience, and asking whether their absence is an obstacle to the (...) theory’s ability to represent ordered pairs or to support the arithmetization of analysis or the development of the theory of cardinals and ordinals. (shrink)

Questions about knowledge, and about the relation between logic and language, are at the heart of philosophy. Eleven distinguished philosophers from Britain and America contribute papers on such questions. All the contributions are examples of recent philosophy at its best. The first half of the book constitutes a running debate about knowledge, evidence and doubt. The second half tackles questions about logic and its relation to language.

Tyler Cowen's ?Why Keynesianism Triumphed? proposed that only Keynesian economists have presented a successful explanation for the Great Depression of 1929?1933 and the continuing slow and intermittent recovery of the rest of the 1930s. This paper examines recent scholarship on the 1930s and finds that there is increasing doubt about the validity of Keynesian explanations, lending credence to both older and recent scholarship that vindicates free?market views of why the Depression happened and why the recovery was so slow and uneven.

Timothy Smiley; I*—The Presidential Address: The Schematic Fallacy, Proceedings of the Aristotelian Society, Volume 83, Issue 1, 1 June 1983, Pages 1–18, https.

Timothy Smiley; I*—The Presidential Address: The Schematic Fallacy, Proceedings of the Aristotelian Society, Volume 83, Issue 1, 1 June 1983, Pages 1–18, https.