Stewart Shapiro

Most philosophers are familiar with the metaphysical puzzle of the statue and the clay. A sculptor begins with some clay, eventually sculpting a statue from it. Are the clay and the statue one and the same thing? Apparently not, since they have different properties. For example, the clay could survive being squashed, but the statue could not. The statue is recently formed, though the clay is not, etc. Godehart Link 1983’s highly influential analysis of the count/mass distinction recommends that English draws a distinction between uncountable “stuff” and countable “things”. There are two mereological relations, related in specific ways. Our primary question here is whether an empirically adequate account of the mass/count distinction really does require distinguishing “things” from “stuff”, and thus postulating two corresponding mereological relations, or if instead positing only one sort of entity and corresponding mereological relation is sufficient, as other semantic theories would have it. This question is meant to be one of what we call natural language mereology. We are asking about the mereological commitments of English, or perhaps competent speakers of English, and not about ultimate reality as such. There is no pretense that we will definitively solve the metaphysical puzzle of the statue and clay.

SummaryAccording to Dummett's understanding of Frege, the sense of a denoting expression is a procedure for determining its denotation. The purpose of this article is to pursue this suggestion and develop a semi‐formal interpretation of Fregean sense for the special case of a first‐order language of arithmetic. In particular, we define the sense of each arithmetic expression to be a hypothetical process to determine the denoted number or truth value. The sense‐process is “hypothetical” in that the senses of some expressions cannot be completed in a finite number of steps. The motivation for this work is to explicate an intensional notion of effectiveness in terms of the senses of function names. It is proposed, in particular, that an expression is effective if and only if its sense can be completed in a finite number of steps. It is shown that a function is computable if and only if it has an effective name.RésuméSelon la lecture que Dummet donne de Frege, le sens ?on;une expression consiste en une procédure pour déterminer sa dénotation. Le but de cet article est de développer cette suggestion par une interprétation semi‐formelle du sens frégéen dans le cas particulier ?on;un langage de premier ordre en arithmétique. En particulier, nous définissons comme sens de toute expression mathématique un procédé hypothétique pour déterminer le nombre dénoté ou la valeur de vérité. Le procédé créateur de sens est hypothetique en ce sens que les sens de certaines expressions ne peuvent pas être donnés en un nombre fini de pas. Le but de cette étude est ?on;expliciter une notion intensionnelle ?on;effectuabilite en termes de sens des noms de fonction. II est en particulier propose qu'une expression soit dite effectuable si et seulement si son sens peut être épuisé en un nombre fini de pas. On montre qu'une fonction est calculable si et seulement si elle a un nom effectuable.ZusammenfassungDummets Auffassung von Frege gemäss besteht der Sinn eines denotierenden Ausdruckes in einem Verfahren, das dessen Denotation bestimmt. Es ist der Zweck der vorliegenden Arbeit, die‐sen Vorschlag weiterzufuUhren und eine halbformale Interpretation des Fregeschen Sinnes fiir den besonderen Fall der Sprache erster Stufe der Arithmetik zu entwickeln. Es wird im besonderen der Sinn eines jeden arithmetischen Ausdruckes als hypothetisches Verfahren zur Bestimmung der denotierten Zahl oder des Wahrheitswertes definiert. Die Sinngebung ist insofern hypothe‐tisch, als der Sinn gewisser Ausdrucke nicht in einer endlichen Zahl von Schritten gegeben wer‐den kann. Das Anliegen der Arbeit grundet in der Absicht, einen intensionalen Begriff der Effek‐tivitat aufgrund des Sinnes von Funktionsnamen zu erlautern. Es wird im besonderen festgelegt, dass ein Ausdruck dann und nur dann effektiv sei, wenn sein Sinn in einer endlichen Zahl von Schritten erreicht werden kann. Es wird gezeigt, dass eine Funktion genau dann berechenbar ist, wenn sie einen effektiven Namen hat

To get to grips with what Shapiro does and can say about higher-order vagueness, it is first necessary to thoroughly review and evaluate his conception of (first-order) vagueness, a conception which is both rich and suggestive but, as it turns out, not so easy to stabilise. In Sections I–IV, his basic position on vagueness (see Shapiro [2003]) is outlined and assessed. As we go along, I offer some suggestions for improvement. In Sections V–VI, I review two key paradoxes of higher-order vagueness, while in Section VII, I explore a possible line of response to such paradoxes given by Keefe [2000]. In Section VIII, I assess whether which Shapiro might adapt Keefe's response to combat both paradoxes.

The primary argument against mereological singularism—the view that definite plural noun phrases like ‘the students’ refer to “set-like entities”—is that it is ultimately incoherent. The most forceful form of this charge is due to Barry Schein, who argues that singularists must accept a certain comprehension principle which entails the existence of things having the contradictory property of being both atomic and non-atomic. The purpose of this paper is to defuse Schein’s argument, by noting three necessary and independently motivated restrictions on the metalinguistic predicates ‘atom’ and ‘non-atom’: both are sort, property, and context-relative. With these restrictions in place, Schein’s problematic assumption becomes evident: his presumed singularist analysis of ‘non-atom’ conflates the metalanguage with the meta-metalanguage, i.e. the language used to talk about the metalanguage.

To get to grips with what Shapiro does and can say about higher-order vagueness, it is first necessary to thoroughly review and evaluate his conception of (first-order) vagueness, a conception which is both rich and suggestive but, as it turns out, not so easy to stabilise. In Sections I–IV, his basic position on vagueness (see Shapiro [2003]) is outlined and assessed. As we go along, I offer some suggestions for improvement. In Sections V–VI, I review two key paradoxes of higher-order vagueness, while in Section VII, I explore a possible line of response to such paradoxes given by Keefe [2000]. In Section VIII, I assess whether which Shapiro might adapt Keefe's response to combat both paradoxes.