English has plural terms as well as singular terms. But our standard formal languages, e.g., the predicate calculus, feature only singular terms. How can the plural idiom be formalized?‘Changing the subject’ is by far the most common plurals strategy among both philosophers and linguists: a plural term is replaced by a singular term standing for some complex object that ‘contains’ the individuals to which the plural term alludes. For example, one might simply replace ‘A, B imply C’ with ‘{A, B} (...) implies [singular] C’. We uncover a surprising variety of ways to change the subject, of ever‐increasing complexity and ingenuity. Our question is whether any can made to work. (shrink)

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.

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.

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)

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)

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)

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.

1. ExpositionRichard Sharvy's ‘A more general theory of definite descriptions’ was published in 1980. Its aim was to replace Russell's paradigm by " a general theory of definite descriptions, of which definite mass descriptions, definite plural descriptions, and Russellian definite singular count descriptions are species. … We have an account of the generic ‘the’ along these same lines. " By now his theory has attained the status of a new paradigm. Even a casual trawl of the literature throws up over (...) a score of citations and endorsements, and no rebuttals of its central contention that the primary use of ‘the’ is to indicate totality . We shall argue that it is wrong for mass description, wrong for singular description, wrong for plural description and wrong for generic ‘the’. Sharvy would have approved: his obituary says that his car displayed the slogan ‘Subvert the Dominant Paradigm’.He follows Russell in treating descriptions as incomplete symbols, but nothing in his paper turns on this aspect of it, and we can bypass the issue by considering them in the context of complete sentences. Where Russell translates F into the predicate calculus as Formula Sharvy translates it as Formula Here ≤ is supposed to stand for some or other species of part of relation. His other governing idea is of a predicate's being cumulative. G is cumulative if any number of things that satisfy G have a sum, i.e. a least upper bound with respect to ≤ which also satisfies G. The meanings of ≤, sum and cumulativity vary according to the different kinds of description, which we take in the order in which Sharvy presents them. Meanwhile, we note that he originally defined ‘sum’ to mean simply the l.u.b. , but subsequently redefined …. (shrink)

Timothy Smiley has made ground-breaking contributions to modal logic, free logic, multiple-conclusion logic, and plural logic. He has illuminated Aristotle’s syllogistic, the ideas of logical form and consequence, and the distinction between assertion and rejection, and has worked to debunk the theory of descriptions. This volume brings together new articles by an international roster of leading logicians and philosophers in order to honour Smiley’s work. Their essays will be of significant interest to those working across the logical (...) spectrum—in philosophy of language, philosophical and mathematical logic, and philosophy of mathematics. (shrink)

Two objections are raised against Oliver and Smiley’s analysis of the collective–distributive opposition in their 2016 book: They take it as a basic premise that the collective reading of ‘baked a cake’ corresponds to a predicate different from its distributive reading, and the same applies to all predicate expressions that admit both a collective and a distributive interpretation. At the same time, however, they argue that inflectional forms of the same lexeme reveal a univocity that should be preserved (...) in a formal representation of English. These two assumptions sit uneasily. In developing their analysis, Oliver and Smiley come to the conclusion that even a singular predication such as ‘Tom baked a cake’ must be regarded as ambiguous between a collective and a distributive reading. This is so artificial that it hardly makes sense, and yet there seems to be no way out of the difficulty unless we are prepared to give up the basic premise just mentioned. (shrink)

Views of fire in the contemporary physical sciences arguably accord with Heraclitus’ proposal that ‘all things are an exchange for fire, and fire for all things, as goods for gold and gold for goods.’ Fire is a media, as John Durham Peters has stated, a species of transformative biochemical reactions between the flammable gases found in air, such as oxygen, and those found in fuels, such as plants. Inspired by an ignition source, these materials react and transform themselves and their (...) surrounds into light and heat energy, carbon dioxide, water vapour, char and much else besides. Fire is conjunctural, durational and transformative. Fire is a dialectician, at once consuming living and dead organic matter and providing both the space and ingredients for new and renewed organic life. In this article, we draw upon our experience of combustible contexts—Australia, Canada and the Philippines—to consider the diverse ways in which fire is today framed as a social problem, an ecological process, an ancient tool, a natural disaster, a source of economic wealth and much more. In this way, we seek to explore the value and limits of ‘elemental thinking’ in relation to the planetary predicaments described by ‘the Anthropocene’. (shrink)

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.

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)

Can we ever truly answer the question, “Who am I?” Moderated by Alex Voorhoeve (London School of Economics), neuro-philosopher Elie During (University of Paris, Ouest Nanterre), cognitive scientist David Jopling (York University, Canada), social psychologist Timothy Wilson (University of Virginia),and ethicist Frances Kamm (Harvard University) examine the difficulty of achieving genuine self-knowledge and how the pursuit of self-knowledge plays a role in shaping the self.

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.