Eli Hirsch recently suggested the metaontological doctrine of so-called "quantifier variance", according to which ontological disputes—e.g. concerning the question whether arbitrary, possibly scattered, mereological fusions exist, in the sense that these are recognised as objects proper in our ontology—can be defused as insubstantial. His proposal is that the meaning of the quanti er `there exists' varies in such debates: according to one opponent in this dispute, some existential statement claiming the existence of, e.g., a scattered object is true, according to (...) the other it is not. This paper argues that Hirsch's proposal leads into inconsistency. (shrink)
George Boolos has suggested a plural interpretation of second-order logic for two purposes: (i) to escape W.V. Quine’s allegation that second-order logic is set theory in disguise, and (ii) to avoid the paradoxes arising if the second-order variable are given a set-theoretic interpretation in second-order set theory. Since the plural interpretation accounts only for monadic second-order logic, Agustín Rayo and Stephen Yablo suggest an new interpretation for polyadic second-order logic in the Boolosian spirit. The present paper argues that Rayo and (...) Yablo’s interpretation does not achieve the goal. (shrink)
This paper investigates feasible ways of destroying artworks, assuming they are abstract objects, or works of a particular art-form, where the works of at least this art-form are assumed to be abstracta. If artworks are eternal, mind-independent abstracta, and hence discovered, rather than created, then they cannot be destroyed, but merely forgotten. For more moderate conceptions of artworks as abstract objects, however, there might be logical space for artwork destruction. Artworks as abstracta have been likened to impure sets (i.e., sets (...) of concrete things, as opposed to pure sets, i.e., sets of nothing but other sets) that have a beginning in time, namely when their members come into being, and an end in time, namely when their members cease to exist. Alternatively, artworks as abstracta have been thought of as types that are created with their first token. Artwork destruction is harder on this account: merely destroying every token might not yet destroy the type. To what extent such similes can be spelt out and made plausible as an ontology of artworks, and what options there are on the different accounts for artwork destruction, is explored in this paper. (shrink)
Paraconsistent and dialetheist approaches to a theory of truth are faced with a problem: the expressive resources of the logic do not suffice to express that a sentence is just true—i.e., true and not also false—or to express that a sentence is consistent. In his recent book, Spandrels of Truth, Jc Beall proposes a ‘just true’-operator to identify sentences that are true and not also false. Beall suggests seven principles that a ‘just true’-operator must fulfill, and proves that his operator (...) indeed fulfills all of them. He concludes that just true has been expressed in the language. I argue that, while the seven conditions may be necessary for an operator to express just true, they are not jointly sufficient. Specifically, first, I prove that a further plausible desideratum for necessary conditions on ‘just true’ is not fulfilled by Beall's proposal, namely that ‘just true’ ascriptions should themselves be just true, and not also false (or equivalently, that the ‘just true’-operator iterates). Second, I show that Beall's operator does not adequately express just true, but that it merely captures an arbitrary proper subset of the just true sentences. Further, there is no prospect of extending the proposal in order to encompass a more reasonable subset of the just true sentences without presupposing that we have antecedent means to characterize the class of just true sentences. (shrink)
Second-order axiomatizations of certain important mathematical theories—such as arithmetic and real analysis—can be shown to be categorical. Categoricity implies semantic completeness, and semantic completeness in turn implies determinacy of truth-value. Second-order axiomatizations are thus appealing to realists as they sometimes seem to offer support for the realist thesis that mathematical statements have determinate truth-values. The status of second-order logic is a controversial issue, however. Worries about ontological commitment have been influential in the debate. Recently, Vann McGee has argued that one (...) can get some of the technical advantages of second-order axiomatizations—categoricity, in particular—while walking free of worries about ontological commitment. In so arguing he appeals to the notion of an open-ended schema—a schema that holds no matter how the language of the relevant theory is extended. Contra McGee, we argue that second-order quantification and open-ended schemas are on a par when it comes to ontological commitment. (shrink)
In 1885, Georg Cantor published his review of Gottlob Frege's Grundlagen der Arithmetik . In this essay, we provide its first English translation together with an introductory note. We also provide a translation of a note by Ernst Zermelo on Cantor's review, and a new translation of Frege's brief response to Cantor. In recent years, it has become philosophical folklore that Cantor's 1885 review of Frege's Grundlagen already contained a warning to Frege. This warning is said to concern the defectiveness (...) of Frege's notion of extension. The exact scope of such speculations varies and sometimes extends as far as crediting Cantor with an early hunch of the paradoxical nature of Frege's notion of extension. William Tait goes even further and deems Frege 'reckless' for having missed Cantor's explicit warning regarding the notion of extension. As such, Cantor's purported inkling would have predated the discovery of the Russell-Zermelo paradox by almost two decades. In our introductory essay, we discuss this alleged implicit (or even explicit) warning, separating two issues: first, whether the most natural reading of Cantor's criticism provides an indication that the notion of extension is defective; second, whether there are other ways of understanding Cantor that support such an interpretation and can serve as a precisification of Cantor's presumed warning. (shrink)
In this short letter to Ed Zalta we raise a number of issues with regards to his version of Neo-Logicism. The letter is, in parts, based on a longer manuscript entitled “What Neo-Logicism could not be” which is in preparation. A response by Ed Zalta to our letter can be found on his website: http://mally.stanford.edu/publications.html (entry C3).
This paper investigates the relation of the Calculus of Individuals presented by Henry S. Leonard and Nelson Goodman in their joint paper, and an earlier version of it, the so-called Calculus of Singular Terms, introduced by Leonard in his Ph.D. dissertation thesis Singular Terms. The latter calculus is shown to be a proper subsystem of the former. Further, Leonard’s projected extension of his system is described, and the definition of an intensional part-relation in his system is proposed. The final section (...) discusses to what extend Goodman might have contributed to the formulation of the Calculus of Individuals. (shrink)
It is often claimed that nominalistic programmes to reconstruct mathematics fail, since they will at some point involve the notion of logical consequence which is unavailable to the nominalist. In this paper we use an idea of Goodman and Quine to develop a nominalistically acceptable explication of logical consequence.
This paper introduces and evaluates two contemporary approaches of neo-logicism. Our aim is to highlight the diﬀerences between these two neo-logicist programmes and clarify what each projects attempts to achieve. To this end, we ﬁrst introduce the programme of the Scottish school – as defended by Bob Hale and Crispin Wright1 which we believe to be a..
(R. Schwartz, “In Memoriam Nelson Goodman (August 7, 1906—November 25, 1998)", Erkenntnis 50 (1999), 7410, esp. 8) . With Goodman's car, Quine took almost all of his students from his seminar on Carnap's Logische Syntax to Baltimore.
This paper investigates the claim that the second-order consequence relation is intractable because of the incompleteness result for SOL. The opponents’ claim is that SOL cannot be proper logic since it does not have a complete deductive system. I argue that the lack of a completeness theorem, despite being an interesting result, cannot be held against the status of SOL as a proper logic.