# Quantifier variance

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The term quantifier variance refers to a position in meta-ontology that claims there is no uniquely best ontological language with which to describe the world.[1] According to Hirsch, it is an outgrowth of Urmson's dictum:

 ‘ “If two sentences are equivalent to each other, then while the use of one rather than the other may be useful for some philosophical purposes, it is not the case that one will be nearer to reality than the other...We can say a thing this way, and we can say it that way, sometimes...But it is no use asking which is the logically or metaphysically right way to say it.”[2] ’ —James Opie Urmson, Philosophical Analysis, p. 186

The rather technical term "quantifier variance" arose from the philosophical term 'quantifier', more particularly existential quantifier. A 'quantifier' in logic originally was the part of statements involving the logic symbols ∀ (for all) and ∃ (there exists) as in an expression like "for all‘such-and-such’ P is true" (∀ x: P(x)) or "there exists at least one ‘such-and-such’ such that P is true" (∃ x: P(x)) where ‘such-and-such’, or x, is an element of a set and P is a proposition or assertion. However, the idea of a quantifier has since been generalized.[3] There are a variety of statements involving quantifiers that serve the same purpose in various ontologies, and they are accordingly all quantifier expressions.[1] Quantifier variance is then the topic concerning exactly what expressions can be construed as quantifier expressions, and just which arguments in a quantifier expression are acceptable, that is, which substitutions for ‘such-and-such’, are permissible.[4]

Hirsch's quantifier variance has been connected to Carnap's idea of a linguistic framework as a 'neoCarnapian' view, namely, "the view that there are a number of equally good meanings of the logical quantifiers; choosing one of these frameworks is to be understood analogously to choosing a Carnapian framework."[5] Of course, not all philosophers (notably Quine and the 'neo'-Qunineans) subscribe to the notion of multiple linguistic frameworks.[5] See meta-ontology.

## Mereology

An important question is the role of composite objects in quantifier variance. For example, a "school" of fish is something that exists in one formulation, while only "fish" exist in another.

A composite object is referred to as a mereological construction or mereological sum, or a fusion, or an aggregate.[6] In an unrestricted view of mereology, any combination of any objects whatever — however arbitrary it may seem to consider them together — constitutes a composite object of which those objects are parts. There is debate over the utility of this notion, and what restrictions should be placed upon what kind of assemblies are meaningful.[6] For example, perhaps it would be useful to place some requirement for one or another relationship between the parts.[6][7]

There are unsettled issues regarding 'quantifier restriction', for example, the role of context in limiting the domain of a quantifier.[8] According to Lewis, context means a quantifier cannot refer to just 'any' mereological sum.[9] Hirsch disagrees with Lewis, and places quantifier variance outside this debate.[10]

Without going into these details, it suffices here to note that quantifier variance does allow the flexibility to speak of the "existence" of an object that is an assembly of components in one formulation or 'language' (perhaps limited to some special kinds of assembly), while in another the assembly may be said not to exist, but only the components.

 ‘ Putnam, for example, writes that “[T]he logical primitives themselves, and in particular the notions of object and existence, have a multitude of different uses rather than one absolute ‘meaning’.”[Putnam] This thesis — the thesis that there are many meanings for the existential quantifier that are equally natural and equally adequate for describing all the facts — is often referred to as “the doctrine of quantifier variance”[Hirsch], [Sider][11] ’ —Ryan Wasserman, Material constitution; The author's references are Putnam,[12] Hirsch,[1] and Sider.[13]

## Realism and antirealism

This flexibility of quantifier variance seemingly conflicts with the realist idea of things existing independent of language and thought.[14][15][16] According to Hirsch and to Thomasson, there is no conflict:[15][16] "But to say that the meaning of the term “object” or “exists”—or of sentences framed using those terms—depends on our conceptual scheme is not at all to say that objects (the term now being used in accord with the rules of an established language, say English) depend on our conceptual scheme."[16] This view is known as the 'cookie-cutter' metaphor: the world is like dough, and concepts are like cookie cutters, carving cookies from the dough.[17] Hirsch says:[1]

 ‘ Quantifier variantism makes no claim at all about what the world is like, not even a claim about what the world is “really” like “in itself”. It claims that whatever the world is like, there can be no uniquely best ontological language to describe it. One should not detect in this claim even a hint of an anti-realist view that what exists in the world somehow depends upon language ’ —Eli Hirsch, Introduction to "Quantifier variance and realism", p. xiv

That viewpoint is not universally accepted, however.[18][9][19][20] Just how do our perceptions and concepts color our view of reality? If an idea exists in our mind, does it have some kind of real existence? Are real things necessarily mind-independent? Answering such questions is part of the subject–object problem and the role of things like mental representations.[21]

## References

1. Eli Hirsch (2011). “Introduction”, Quantifier Variance and Realism : Essays in Metaontology: Essays in Metaontology. Oxford University Press, p. xii. ISBN 0199732116.
2. J.O. Urmson (1967). Philosophical analysis: its development between the two world wars. Oxford University Press, p. 186.  Quoted by Eli Hirsch.
3. Dag Westerståhl (April 19, 2011). Edward N. Zalta, ed:Generalized Quantifiers. The Stanford Encyclopedia of Philosophy (Summer 2011 Edition).
4. Theodore Sider (2011). Writing the book of the world. Oxford University Press, p. 175. ISBN 0199697906. “Quantifier variance, on my formulation, says that "there are" many candidates for being meant by quantifiers; but the quantifier variantist needn't take this quantification [that is, this variety of quantifier-meanings] seriously...we could choose to use the sentence "There exists something that is composed of [such-and-such composite object]" so it comes out true, or we could choose to use it so it comes out false; and under neither choice would our words [get closer to reality] than under the other.  [Italics added, 'cute' in-group phrases replaced, as indicated by brackets]”
5. Helen Beebee, Nikk Effingham, Philip Goff (2012). Metaphysics: The Key Concepts. Taylor & Francis. ISBN 0203835255.
6. For a discussion of mereology, see for example: Kathrin Koslicki (2008). “Ordinary objects as mereological sums”, The Structure of Objects. Oxford University Press, pp. 23 ff. ISBN 0191609137.
7. Jonathan Schaffer (2009). “Chapter 12: On what grounds what”, David Chalmers, David Manley, Ryan Wasserman, eds: Metametaphysics: New Essays on the Foundations of Ontology. Oxford University Press, pp. 347-383. ISBN 0199546045. “The notion of grounding may be put to further use to capture a crucial mereological distinction (missing from classical mereology) between an integrated whole which exhibits a genuine unity, and a mere aggregate which is a random assemblage of parts”
8. Jason Stanley, Zoltàn Gendler Szabó (2000). "On quantifier domain restriction". Mind & Language 15: pp. 219–261.
9. David Lewis (1986). On the Plurality of Worlds. Wiley, p. 70. ISBN 0631224262. “What, then, is the difference between a sum of possible individuals that is a possible world, and one that is not?...How are the worlds demarcated from one another? Why don't all the possibilia comprise one big world?”  See On the Plurality of Worlds.
10. See Chapter 5: Quantifier Variance and Realism, p. 86 “It is essential to distinguish between quantifier variance and what David Lewis calls quantifier restriction. That the quantifier is often contextually restricted in the way Lewis says may be granted by all. The question of quantifier variance, however, pertains to the unrestricted quantifier, to our concept of "existence simpliciter"”
11. Wasserman, Ryan (April 5, 2013). Edward N. Zalta, ed:Material Constitution. The Stanford Encyclopedia of Philosophy (Summer 2013 Edition).
12. Hilary Putnam (June, 1987). "Truth and Convention: On Davidson's Refutation of Conceptual Relativism". Dialectica 41: pp. 69–77. DOI:10.1111/j.1746-8361.1987.tb00880.x. Research Blogging. See p. 71.
13. Theodore Sider (2009). “Chapter 13: Ontological Realism”, D. Chalmers, D. Manley, and R. Wasserman, eds: Metametaphysics: New Essays on the Foundations of Ontology. Oxford University Press, pp. 384–423.
14. Jonathan Schaffer (2009). “Chapter 12: On what grounds what”, David Chalmers, David Manley, Ryan Wasserman, eds: Metametaphysics: New Essays on the Foundations of Ontology. Oxford University Press. ISBN 0199546045. “The dispute is over mind-dependence: are entities, like rocks, grounded in ideas, or independent of them?...[Are numbers] independent of the mind, or based upon our concepts?”
15. Amie L Thomasson. Carnap and the prospects for easy ontology. Retrieved on 04-28-2013. To be published in Ontology after Carnap Stephan Blatti & Sandra Lapointe (eds.)
16. Matti Eklund (2008). “Chapter 9.2: The picture of reality as an amorphous lump”, Theodore Sider, John Hawthorne, and Dean W Zimmerman, eds: Contemporary Debates in Metaphysics, pbk.. Blackwell Publishing, p. 382 ff. ISBN 978-1-4051-1229-1.
17. Many of the essays in Hircsh's book are debates over such matters.
18. Hilary Putnam (1987). The Many Faces of Realism, 2nd. Open Court, p. 33. ISBN 0812690427.
19. Ladyman, James (May 22, 2009). Edward N. Zalta, ed:Structural Realism. The Stanford Encyclopedia of Philosophy (Summer 2013 Edition).
20. Pitt, David (December 11, 2012). Edward N. Zalta, ed:Mental Representation. The Stanford Encyclopedia of Philosophy (Winter 2012 Edition).