Philosophy of Logic and Mathematics

Proceedings of the 41st International Ludwig Wittgenstein Symposium

Gabriele M. Mras, Paul Weingartner, Bernhard Ritter

• Publisher: De Gruyter
• Year: 2019
• Print ISBN: 9783110654301
• eBook ISBN: 9783110654547
• Edition: 1
• Topic: Philosophy
• Subtopic: History & Philosophy of Mathematics

This volume presents different conceptions of logic and mathematics and discuss their philosophical foundations and consequences. This concerns first of all topics of Wittgenstein's ideas on logic and mathematics; questions about the structural complexity of propositions; the more recent debate about Neo-Logicism and Neo-Fregeanism; the comparison and translatability of different logics; the foundations of mathematics: intuitionism, mathematical realism, and formalism.

The contributing authors are Matthias Baaz, Francesco Berto, Jean-Yves Beziau, Elena Dragalina-Chernya, Günther Eder, Susan Edwards-McKie, Oliver Feldmann, Juliet Floyd, Norbert Gratzl, Richard Heinrich, Janusz Kaczmarek, Wolfgang Kienzler, Timm Lampert, Itala Maria Loffredo D'Ottaviano, Paolo Mancosu, Matthieu Marion, Felix Mühlhölzer, Charles Parsons, Edi Pavlovic, Christoph Pfisterer, Michael Potter, Richard Raatzsch, Esther Ramharter, Stefan Riegelnik, Gabriel Sandu, Georg Schiemer, Gerhard Schurz, Dana Scott, Stewart Shapiro, Karl Sigmund, William W. Tait, Mark van Atten, Maria van der Schaar, Vladimir Vasyukov, Jan von Plato, Jan Wole?ski and Richard Zach.

Part I:Philosophy of Logic

Eric Snyder and Stewart Shapiro

Link’s Revenge: A Case Study in Natural
Language Mereology

Eric Snyder, Smith College, Northampton (MA), USA

Stewart Shapiro, The Ohio State University, Columbus (OH), USA,[email protected]

Abstract: 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.

1Introduction

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.

LEIBNIZ’S LAW is the thesis that identical things have the same properties. It follows from this that the statue and the clay are distinct, since they have different properties: one can survive being squashed, the other cannot; one is recently formed, the other is not, etc.

Many metaphysical puzzles then follow: When exactly does the clay become the statue? How can two co-located objects exist concurrently, etc.

Given the persistence of this puzzle, it would be rather surprising if the relatively nascent science of natural language semantics required taking a stance on the issue. Yet that is precisely what Godehart Link 1983’s highly influential analysis of THE COUNT/MASS DISTINCTION recommends. In particular, Link argues on the basis of examples like (1) that English draws a distinction between uncountable “stuff” and countable “things”.

(1)	This ring is new, but the gold in the ring is old.

Link’s argument is straightforward: The ring and the gold constituting it have different properties, even though the ring is completely constituted by the gold. It follows, from Leibniz’s Law, that the two are distinct. To quote Link 1983: 128 directly:1

Our guide in ontological matters has to be language itself, it seems to me. So if we have, for instance, two expressions a and b that refer to entities occupying the same place at the same time but have different sets of predicates applying to them, then the entities referred to are simply not the same. From this it follows that my ring and the gold making up my ring are different entities.

If providing an empirically adequate semantics for nouns requires postulating such a distinction, and if “our guide to ontological matters has to be language itself”, then it would appear that we have little choice but to accept that the ring and the gold are in fact different, as are the statue and the clay.

More to the point, since ‘ring’ is a count noun and ‘gold’ is a mass noun, Link infers that the denotations of the two nouns must be different sorts of things. Very roughly, rings are “things”, while gold is “stuff”.

More technically, Link distinguishes between ATOMIC INDIVIDUALS and MASS QUANTITIES. Atomic individuals are countable entities serving as the denotations of singular count nouns like ‘ring’. Plural nouns like ‘rings’ then denote PLURALITIES, or mereological sums of atoms. Atoms are related to pluralities, and pluralities to other pluralities, via the INDIVIDUAL PARTHOOD relation, represented as ‘⊑i’, and defined as (2), where ‘x ⊔i y’ is the individual sum of x and y.

(2)	∀x, y. x ⊑i y ↔ x ⊔i y = y

Ordering the atoms via individual parthood results in an atomic join semilatice structure like the following, where arrows represent ⊑i.

Call this THE COUNT DOMAIN.

Mass quantities, in contrast, are related via MATERIAL PARTHOOD, represented as ‘⊑m’, and defined similarly to (2). Ordering mass quantities via material parthood results in a different, but similarly structured, semilattice. Call this THE MASS DOMAIN.

The count and mass domains are related via a homomorphism, a function h, mapping pluralities to mass quantities. If a is an individual (or a sum of individuals), then h(a) is the material stuff it is (or they are) made of. So if a is the notorious ring, then h(a) is the gold constituting it. Since h is a homomorphism, it preserves the material constitution of atomic individuals: if a ⊑i b, then h(a) ⊑m h(b). For example, if Link’s ring a is an individual part of his collection of rings b, then the gold constituting Link’s ring is a material part of that collection.

If Link’s ring and the gold constituting are located in different, though related, domains, then it is hardly surprising that they have different properties. After all, they would be different entities. As Link acknowledges, the resulting ontology is anything but parsimonious, from a purely metaphysical perspective.

[L]et a and b denote two atoms in A. Then there are two mo...

Table of contents

• Cover
• Title Page
• Copyright
• Contents
• Preface
• Part I: Philosophy of Logic
• Part II: Philosophy of Mathematics
• Part III: Wittgenstein
• Index of Names
• Index of Subjects