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In the Begriffschrift Frege drew no distinction—or anyway signalled no importance to the distinction—between quantifying into positions occupied by what he called eigennamen—singular terms—in a sentence and quantification into predicate position or, more generally, quantification into open sentences—into what remains of a sentence when one or more occurrences of singular terms are removed. He seems to have conceived of both alike as perfectly legitimate forms of generalisation, each properly belonging to logic. More accurately: he seems to have conceived of quantification (...) as such as an operation of pure logic, and in effect to have drawn no distinction between first-order, second-order and higherorder quantification in general. (shrink)
It is a great pleasure to have the opportunity to contribute to this volume dedicated to the critical celebration of Stephen Schiffer’s very considerable philosophical achievements. My focus will be on his recent work on vagueness.1 The broad direction of Schiffer’s researches in this area has been to give priority to what we may call the characterisation problem: the problem of saying what the vagueness of expressions of natural language consists in or, more specifically – since Schiffer takes it as (...) a given that the vagueness he is targeting consist in a propensity of vague expressions to give rise to borderline cases —the problem of saying what being a borderline case of the concept expressed by a vague expression consists in. This has not been a main preoccupation of most of the work in the field since the vagueness “boom” started in the mid 1970s. There has been a tendency to jump straight into devising semantic theories for vague languages, usually aimed at twin desiderata of saving classical logic and dissolving the various paradoxes of vagueness, with a principal focus on the standard sorites, and occasional glances at the Forced March, and others.2 Of course, such work has inevitably implicated commitment to broad conceptions of vagueness, and of borderline cases, of various kinds. The classical epistemicist approach, for example, conceives of borderline cases as instances whose correct classification in terms of the relevant concept is, for reasons it attempts to explain, unknowable. Semantic indeterminist approaches, by contrast, tend (often implicitly) to conceive of borderline cases as items to which the concept in question neither applies nor fails to apply and as coming about because our practice with the concept leaves it, in effect, merely partially defined and so ‘gappy’. A variation on this, still semantic indeterminist, regards vagueness as consisting in a phenomenon akin to divided reference, whereby a predicate, for example, may be associated with a range of extensionally distinct best candidates to be the property it refers to; borderline cases are then items which exemplify some but not all of these properties.. (shrink)
The setting of relativistic ideas about truth in the general style of semantic-theoretic apparatus pioneered by Lewis, Kaplan and others has persuaded many that they should at least be taken seriously as competition in the space of explanatory linguistic theory, a type of view which properly formulated, may offer an at least coherent — and indeed, in the view of some, a superior —account of certain salient linguistic data manifest in, for example, discourse about epistemic modals, about knowledge and about (...) matters of taste and value, and may also offer the prospect of a coherent regimentation of the Aristotelian "Open Future" (along with, perhaps, the Dummettian ’anti-real’ past.) My main purpose here to enter a reminder of certain underlying philosophical issues about relativism — about its metaphysical coherence, its metasemantic obligations, and the apparent limitations of the kind of local linguistic evidence which contemporary proponents have adduced in its favour —of which there is a risk that its apparent rehabilitation in rigorous semantic dress may encourage neglect. (shrink)
This is a short, and therefore necessarily very incomplete discussion of one of the great questions of modern philosophy. I return to a station at which an interpretative train of thought of mine came to a halt in a paper written almost 20 years ago, about Wittgenstein and Chomsky,[1] hoping to advance a little bit further down the track. The rule-following passages in the Investigations and Remarks on the Foundations of Mathematics in fact raise a number of distinct (though connected) (...) issues about rules, meaning, objectivity, and reasons, whose conflation is encouraged by the standard caption, "the Rule-following Considerations".[2] So, let me begin by explaining my focus here. (shrink)
There is now a widespread accord among philosophers that the vagueness of natural language gives rise to some particularly deep and perplexing problems and paradoxes. It was not always so. For most of the first century of analytical philosophy, vagueness was generally regarded as a marginal, slightly irritating phenomenon, —receiving some attention, to be sure, in parts of the Philosophical Investigations and in the amateur linguistics enjoyed by philosophers in Oxford in the 1950s, but best idealised away in any serious (...) theoretical treatment of meaning, understanding and valid inference. Frege, as is well known, had come to be thoroughly mistrustful of vagueness, supposing that a language fit for the purpose of articulating scientific and mathematical knowledge would have to be purified of it. Later trends in philosophical logic and semantics followed his lead, not indeed in setting about the (futile) task of expurgating vagueness from natural language but by largely restricting theoretical attention to artificial languages in whose workings vagueness was assigned no role. (shrink)
It is common among philosophers who take an interest in the phenomenon of vagueness in natural language not merely to acknowledge higher-order vagueness but to take its existence as a basic datum— so that views that lack the resources to account for it, or that put obstacles in the way, are regarded as deficient just on that score. My main purpose in what follows is to loosen the hold of this deeply misconceived idea. Higher-order vagueness is no basic datum but (...) an illusion, fostered by misunderstandings of the nature of (ordinary, if you will ‘first-order’) vagueness itself. To see through the illusion is to take a step that is prerequisite for a correct understanding of vagueness, and for any satisfying dissolution of its attendant paradoxes. (shrink)
In “Double Vision Two Questions about the Neo-Fregean Programme”, John MacFarlane’s raises two main questions: (1) Why is it so important to neo-Fregeans to treat expressions of the form ‘the number of Fs’ as a species of singular term? What would be lost, if anything, if they were analysed instead as a type of quantifier-phrase, as on Russell’s Theory of Definite Descriptions? and (2) Granting—at least for the sake of argument—that Hume’s Principle may be used as a means of implicitly (...) defining the number operator, what advantage, if any, does adopting this course possess over a direct stipulation of the Dedekind-Peano axioms? This paper attempts to answer them. In response to the first, we spell out the links between the recognition of numerical terms as vehicles of singular reference and the conception of numbers as possible objects of singular, or object-directed, thought, and the role of the acknowledgement of numbers as objects in the neo-Fregean attempt to justify the basic laws of arithmetic. In response to the second, we argue that the crucial issue concerns the capacity of either stipulation—of Hume’s Principle, or of the Dedekind-Peano axioms—to found knowledge of the principles involved, and that in this regard there are crucial differences which explain why the former stipulation can, but the latter cannot, play the required foundational role. (shrink)
In the paper we argue that truth-relativism is potentially hostage to a problem of exhibiting witnesses of its own truth. The problem for the relativist stems from acceptance of a trumping principle according to which there is a dependency between ascriptions of truth of an utterance and ascriptions of truth to other ascriptions of truth of that utterance. We argue that such a dependency indeed holds in the case of future contingents and the case of epistemic modals and that, consequently, (...) the relativist about these domains cannot exhibit witnesses to his relativism. In the appendix we provide some results on the relation between trumping and multi-order relativism. (shrink)
What is wrong with abstraction, Michael Potter and Peter Sullivan explain a further objection to the abstractionist programme in the foundations of mathematics which they first presented in their Hale on Caesar and which they believe our discussion in The Reason's Proper Study misunderstood. The aims of the present note are: To get the character of this objection into sharper focus; To explore further certain of the assumptions—primarily, about reference-fixing in mathematics, about certain putative limitations of abstractionist set theory, and (...) about the effects of impredicativity in abstraction principles—which underlie it; and To advance the debate of the issues thereby raised. Thanks for helpful comments to Roy Cook and to an anonymous referee. CiteULike Connotea Del.icio.us What's this? (shrink)
What I am calling New Age Relativism is usually proposed as a thesis about the truth-conditions of utterances, where an utterance is an actual historic voicing or inscription of a sentence of a certain type. Roughly, it is the view that, for certain discourses, whether an utterance is true depends not just on the context of its making—when, where, to whom, by whom, in what language, and so on—and the “circumstances of evaluation”—the state of the world in relevant respects—but also (...) on an additional parameter: a context of assessment. Vary the latter and the truth-value of the utterance can vary, even though the context of its making and the associated state of the world remain fixed. (shrink)
This paper is a reply to George Boolos's three papers (Boolos (1987a, 1987b, 1990a)) concerned with the status of Hume's Principle. Five independent worries of Boolos concerning the status of Hume's Principle as an analytic truth are identified and discussed. Firstly, the ontogical concern about the commitments of Hume's Principle. Secondly, whether Hume's Principle is in fact consistent and whether the commitment to the universal number by adopting Hume's Principle might be problematic. Also the so-called `surplus content' worry is discussed, (...) which points out that the conceptual resources to grasp Hume's Principle vastly outstrip the conceptual resources employed in arithmetical reasoning. And lastly whether Hume's Principle is in bad company with other unsuccessful implicit definitions. In the last section, an account towards our entitlement to Hume&'s Principle is sketched. (shrink)
Lucas and Penrose have contended that, by displaying how any characterisation of arithmetical proof programmable into a machine allows of diagonalisation, generating a humanly recognisable proof which eludes that characterisation, Gödel's incompleteness theorem rules out any purely mechanical model of the human intellect. The main criticisms of this argument have been that the proof generated by diagonalisation (i) will not be humanly recognisable unless humans can grasp the specification of the object-system (Benacerraf); and (ii) counts as a proof only on (...) the (unproven) hypothesis that the object system is consistent (Putnam). The present paper argues that criticism (ii) may be met head-on by an intuitionistic proponent of the anti-mechanist argument; and that criticism (i) is simply mistaken. However the paper concludes by questioning the sufficiency of the situation for an interesting anti-mechanist conclusion. (shrink)
This book is an important collection of new essays on various topics relating to realism and its rivals in metaphysics, logic, metaethics, and epistemology. The contributors include some of the leading authors in these fields and in several cases their essays constitute definitive statements of their views. In some cases authors write in response to the essays of other contributors, in other cases they proceed independently. Although not primarily historical this collection includes discussions of philosophers from the middle ages to (...) the present day, from Aquinas to Wittgenstein. No one seriously interested in questions about realism, whether as a general philosophical outlook or as a particular position within specific debates, can afford to be without this collection. (shrink)
