Ian Rumfitt

In ‘Meaning and Truth’, P.F. Strawson staged a ‘Homeric struggle’ between the theorists of communication-intention and the theorists of formal semantics over the question of what meaning consists in. This chapter draws on recent work on the speech act of telling in order to refute Strawson’s argument in favour of the first party. On Strawson’s analysis, a speaker _S_ succeeds in telling an audience _H_ that _p_ only if _H_ comes to believe that _S_ believes that _p_; this chapter argues that belief in the speaker’s sincerity is not necessary for successful telling. It then criticizes John McDowell’s alternative account, whereby telling someone that _p_ is a ‘cognitive stand-in’ for a direct confrontation with the state of affairs that _p_; this neglects the role believing the speaker plays in communication. We do better to follow the Peircean account recently developed by Richard Moran: to tell _H_ something is to offer _H_ one’s guarantee of the truth of what one says. Since truth-conditions now emerge as intrinsic to acts of telling, and since Strawson’s theorists of formal semantics assign truth-conditions to declarative sentences, what they provide is central to an account of telling. However, the notion of truth-conditions thereby vindicated differs from some popular conceptions of them, notably that of Donald Davidson. The chapter concludes with some initial remarks about the shape of a semantic theory cast on Strawsonian lines.

This chapter considers the question: should we employ intuitionistic logic, not classical logic, when reasoning with vague concepts? In his commentary on Michael Dummett’s “Wang’s Paradox,” Crispin Wright presents an apparently powerful argument in favour of an affirmative answer to this question. This chapter advocates a less conclusive answer than Wright’s. It is argued that intuitionistic logic may be the strongest logic we are entitled to use in reasoning with any vague predicate, but there may also be common and central families of vague term where we are entitled to use classical logic.

This chapter considers what form a neo-Fregean account of ordinal numbers might take. It begins by discussing how the natural abstraction principle for ordinals yields a contradiction (the Burali-Forti Paradox) when combined with impredicative second-order logic. It continues by arguing that the fault lies in the use of impredicative logic rather than in the abstraction principle _per se_. As the focus is on a form of predicative logic which reflects a philosophical diagnosis of the source of the paradox, the chapter considers how far Hale and Wright’s neo-logicist programme in cardinal arithmetic can be carried out in that logic.

P. F. Strawson explained truth, as it applies to statements, by saying: ‘one who makes a statement or assertion makes a true statement if and only if things are as, in making the statement, he states them to be’. This explanation differs from others in taking a statement’s having a content (i.e. its saying that things are thus-and-so) to be a presupposition of an attribution of truth to it. This paper shows how this feature opens the way to a distinctive solution to the Liar Paradox and to a foundation for the axiomatic theories of truth now favoured by many logicians.

In _The Things We Mean_, Schiffer notes an asymmetry between ordinary singular terms and that-clauses: while we determine the truth of ‘Henri admires Braque’ by determining whether the referent of ‘Henri’ bears _admires_ to the referent of ‘Braque,’ we cannot proceed in similar fashion with ‘Ralph believes that Eliot was a woman.’ Here, we first determine truth conditions; only then do we determine the that-clause’s reference. But Schiffer is silent about when two that-clauses co-refer. The present chapter tries to fill this lacuna, investigating the possibility that identity conditions for the referents—propositions—can be provided in a manner that mimics Frege’s account of identity conditions for numbers (i.e., by supplying an equivalence relation among beliefs guaranteeing that any two beliefs so related possess the same content). The chapter investigates a promising approach, but concludes that is ultimately too limited in scope to be of use to Schiffer.

Liar sentences say nothing, according to this chapter—which, it claims, we can, in effect, prove. But extending the proof as the chapter does appears to result in revenge. The solution to this problem is to restrict the laws of logic by distinguishing _expressing a falsehood_ from _failing to express a truth_. But the question that presses is how we can signify that a given sentence—a liar sentence, for example—fails to express a truth without being mired in paradox. To this end, the chapter revisits the sort of bilateral system that Rumfitt (2000) has discussed. The chapter shows that there is a way of developing Aristotle’s conception of truth into a definition of truth that does not yield a contradiction, even when applied to a semantically closed language. If successful, the proposal will enable us to reject a Strengthened Liar as untrue without asserting its negation.

This chapter develops F. P. Ramsey's theory of truth. Everyone knows that Ramsey had a theory of truth. Rather fewer people know that the theory came in two significantly different versions, and that the later and less famous version is more fertile than its predecessor. The more famous version is, of course, Ramsey's redundancy theory of truth, which he expounded in his great paper of 1927, ‘Facts and Propositions’. The Ramsey–Prior theory, _contra_ Ramsey's original approach, confirms that the word ‘true’ has a determinate sense. It does so by specifying the contribution that the word makes to what is expressed by a statement that contains it. That specification vindicates the common-sense view that ‘true’ is grammatically and semantically a predicate.

This chapter explores the case for referentialism about plurals. It presents a version of referentialism characterized by the following theses: (i) semantic predicates, such as ‘designate’ and ‘satisfy’, remain constant in sense whether they attach to a plural subject or a singular subject; (ii) in the principles that state the designations of plural terms the expressions that follow the verb ‘designates’ (or ‘refers to’) are themselves plural expressions; and (iii) the designation relation does not distribute, so there is no entailment from ‘α refers to _x_ & _y_ & _z_’ to ‘α refers to _x_’. The chapter deploys Boolos's second-order account of the logic of plurals to explore how plurals behave in modal contexts.

The paper defends the intelligibility of unrestricted quantification. For any natural number n, 'There are at least n individuals' is logically true, when the quantifier is unrestricted. In response to the objection that such sentences should not count as logically true because existence is contingent, it is argued by consideration of cross-world counting principles that in the relevant sense of 'exist' existence is not contingent. A tentative extension of the upward L?wenheim-Skolem theorem to proper classes is used to argue that a sound and complete axiomatization of the logic of unrestricted universal quantification results from adding all sentences of the form 'There are at least n individuals' as axioms to a standard axiomatization of the first-order predicate calculus.

This chapter concerns that harmony is a particular relationship between the introduction rule and the elimination rule for a given connective. The Harmony Thesis says that a connective is defective unless its associated introduction and elimination rules are in harmony. It also says that a connective is defective if the logical principles which regulate its use go beyond a pair of harmonious introduction and elimination rules. The chapter scrutinizes the most influential arguments which have been put forward for the Harmony Thesis. It discusses Dummett‐Prawitz's argument for Harmony, which revealed the huge difficulties that confront the project of trying to explicate the notions of consequence and validity directly in terms of the rules which, for the Inferential Role Semantics (IRS) theorist, constitute the meanings of the connectives. The chapter concludes by discussing briefly how the failure of the Harmony Thesis affects the prospects for IRS.

ABSTRACTIn reply to Linnebo, I defend my analysis of Tait's argument against the use of classical logic in set theory, and make some preliminary comments on Linnebo's new argument for the same conclusion. I then turn to Shapiro's discussion of intuitionistic analysis and of Smooth Infinitesimal Analysis. I contend that we can make sense of intuitionistic analysis, but only by attaching deviant meanings to the connectives. Whether anyone can make sense of SIA is open to doubt: doing so would involve making sense of mathematical quantities whose relationship to zero and to one another is inherently indeterminate.

In his book The Things We Mean, Stephen Schiffer advances a subtle defence of what he calls the ‘face-value’ analysis of attributions of belief and reports of speech. Under this analysis, ‘Harold believes that there is life on Venus’ expresses a relation between Harold and a certain abstract object, the proposition that there is life on Venus. The present essay first proposes an improvement to Schiffer’s ‘pleonastic’ theory of propositions. It then challenges the face-value analysis. There will be such things as propositions only if they possess conditions of identity and distinctness. By analyzing Frege’s theory of propositions (Gedanken), I argue that such conditions may be found for the special case of beliefs and sayings advanced as premises and conclusions of deductive arguments. These conditions, however, are not applicable to most ordinary beliefs and sayings. Ordinary attributions and reports, then, do not place thinkers and speakers in relations to propositions. A bonus is exposure of the fallacy in the Putnam-Taschek objection to Frege’s theory of sense and reference.

I compare three sorts of case in which philosophers have argued that we cannot assert the Law of Excluded Middle for statements of identity. Adherents of Smooth Infinitesimal Analysis deny that Excluded Middle holds for statements saying that an infinitesimal is identical with zero. Derek Parfit contended that, in certain sci-fi scenarios, the Law does not hold for some statements of personal identity. He also claimed that it fails for the statement ‘England in 1065 was the same nation as England in 1067’. I argue that none of these cases poses a serious threat to Excluded Middle. My analysis of the last example casts doubt on the principle of the Determinacy of Distinctness. While David Wiggins's ‘conceptualist realism’ provides a metaphysics which can dispense with that principle, it leaves no house-room for infinitesimals.

This essay assesses the account of truth presented in Wiggins's 2002 paper ‘An indefinibilist cum normative view of truth and the marks of truth'. I agree with Wiggins that we should seek, not to define truth, but to elucidate it by unfolding its connections with other basic notions. However, I give reasons for preferring an elucidation based on Ramsey's account of truth to Wiggins's Tarski-inspired approach. I also cast doubt on Wiggins's thesis that convergence is a mark of truth, arguing instead that a claim which is up for assessment as true or false must be one to which different speakers/hearers can attach, and know that they are attaching, the same sense. I use this principle to rule out an account of indicative conditionals, and bring some considerations to bear on the question of whether those conditionals have truth values. An appendix revisits a debate about the determinateness of distinctness.

Philip Jourdain put this question to Frege in a letter of 28 January 1909. Frege had, indeed, next to nothing to say about ordinals, and in this respect Bob Hale has followed the master. As I hope this chapter will show, though, the topic is worth addressing. The natural abstraction principle for ordinals combines with full, impredicative second-order logic to engender a contradiction, the so-called Burali-Forti Paradox. I shall contend that the best solution involves a retreat to a predicative logic. Such a retreat has implications for other neo-Fregean theories, including the cardinal arithmetic on which Hale has focused. The discussion will touch on a topic which has been at the centre of Hale’s more recent work—namely, the interpretation of plural and higher-order quantifiers.

Intuitionistic logic provides an elegant solution to the Sorites Paradox. Its acceptance has been hampered by two factors. First, the lack of an accepted semantics for languages containing vague terms has led even philosophers sympathetic to intuitionism to complain that no explanation has been given of why intuitionistic logic is the correct logic for such languages. Second, switching from classical to intuitionistic logic, while it may help with the Sorites, does not appear to offer any advantages when dealing with the so-called paradoxes of higher-order vagueness. We offer a proposal that makes strides on both issues. We argue that the intuitionist’s characteristic rejection of any third alethic value alongside true and false is best elaborated by taking the normal modal system S4M to be the sentential logic of the operator ‘it is clearly the case that’. S4M opens the way to an account of higher-order vagueness which avoids the paradoxes that have been thought to infect the notion. S4M is one of the modal counterparts of the intuitionistic sentential calculus and we use this fact to explain why IPC is the correct sentential logic to use when reasoning with vague statements. We also show that our key results go through in an intuitionistic version of S4M. Finally, we deploy our analysis to reply to Timothy Williamson’s objections to intuitionistic treatments of vagueness.