The Logic of Plurals

The course will roughly follow the structure of my book, Logic & Natural Language (I keep an updating version of the book on my site, here). We shall first introduce the idea of plural reference and show that various kinds of expression in natural language are sometimes used as plural referring expressions. I shall then argue that common nouns in quantified noun phrases are frequently used as plural referring expressions; e.g., ‘my students’ in ‘Some of my students are Serbian’ or ‘women’ in ‘John loves many women’. This position is closer to Aristotle’s than to Frege’s, and we shall discuss and criticize Frege’s reasons for rejecting it and for claiming that common nouns are always predicative. My position also leads to a conception of quantification very different from Frege’s and its modern descendants; we shall try to show its advantages, also over restricted quantification and recent work on plural quantification. We shall then develop a deductive system for natural language, comparable in its power to first order logic; for that purpose we shall first discuss various issues relating to sentences containing several quantifiers or bound anaphora; for instance, ‘Every man loves some women’ and ‘If a man is in Rhodes, then he’s not in Athens’. The soundness of the system will be proved, and its completeness and relation to the first order Predicate Calculus discussed. Depending on our progress, we may also discuss identity statements, modality, and the analogue of second order logic on my approach.

Breakdown into Units

 Unit I: Plural Reference

We shall introduce and explain the idea of plural reference, and then criticize some attempt to reduce constructions that apparently use plural reference to constructions that do not.

This unit is structured around Part I Chapter 2 of my book.

Recommended additional readings:

Black, Max (1971), ‘The Elusiveness of Sets’, The Review of Metaphysics, 24, pp. 614-36.

Oliver, Alex and Smiley, Timothy (2001), ‘Strategies for a Logic of Plurals’, The Philosophical Quarterly, 51, pp. 289-306.

In his influential paper, which marked the beginning of philosophers and logicians’ interest in the logic of plurals, Black powerfully introduces, towards the end, the idea of plural reference. Oliver and Smiley criticize several reductive attempts.

Unit II: Common Nouns as Plural Referring Expressions

We argue that common nouns are often used as plural referring expressions. E.g., ‘students’ in ‘some students were late’. We show how this explains away an alleged ambiguity of the copula in natural language, an alleged ambiguity the discovery of which was taken to be an achievement of modern logic.

This unit is structured around Part I Chapter 3 Sections 1-2 of my book.

Recommended additional readings: 

Geach, P.T. (1962), Reference and Generality: An Examination of Some Medieval and Modern Theories, emended edition 1968, Cornell University Press.

Strawson, P.F. (1950), ‘On Referring’, reprinted in his (1971), Logico-Linguistic Papers, Methuen, London, pp. 1-27.

Something like this idea is found in the last section of Strawson’s classic paper; Geach introduces it in the last, neglected sections of his book.

Unit III: Criticising Frege

Although common nouns in the grammatical subject position were taken by Aristotelian logic to be logical subject terms, Frege claimed that they are predicative. Later logic and philosophy of language have usually simply accepted his position without argument. We shall criticise Frege’s reasons and also show how he was misled to his position through the influence of the mathematical conception of function.

This unit is structured around my paper, ‘A Critique of Frege on Common Nouns’, which is an elaboration of Part I Chapter 4 of my book.

Recommended additional readings:

The relevant passages of Frege’s work will be distributed.

Unit IV: The Nature of Quantification

We discuss the nature of quantification in general, and show how it is realized in natural language according to our approach. We mark the important departure between our approach and Frege’s.

This unit is structured around Sections 6.1-6.3 of my book.

Unit V: Plural Quantification Logic

This is an approach introduced by Boolos (1984), and later developed by Yi, McKay, Oliver and Smiley, and others. We try to show that it does not capture the nature of quantification in natural language.

This unit is structured around my paper, ‘Plural Quantification Logic: A Critical Appraisal’ (2009).

Recommended additional readings:

Boolos, G. (1984), ‘To Be is to Be a Value of a Variable (or to Be Some Values of Some Variables)’, reprinted in his (1998), Logic, Logic, and Logic, Harvard University Press, pp. 54-72.

Linnebo, Ø. (2004), ‘Plural Quantification’, in the Stanford Encyclopedia of Philosophy.

McKay, T. J. (2006), Plural Predication, Oxford: Clarendon Press.

Oliver, Alex and Smiley, Timothy (2006), ‘A Modest Logic of Plurals’, Journal of Philosophical Logic, 35, pp. 317‑48.

Yi, Byeong-uk (2005-6), ‘The Logic and Meaning of Plurals’, Journal of Philosophical Logic, 34, pp. 459-506 and 35, pp. 239‑88.

Boolos’ paper introduces plural quantification logic, although with his, unlike some of the later literature, it is a reinterpretation of second order logic. Linnebo’s is an introduction and survey of the literature. I do not of course expect that you read all the three other works, but you should consult at least one of them to see how it introduces plural quantification; as McKay uses generalized quantifiers, which we shall discuss later, the other two works might be more accessible at this stage. 

Unit VI: Binary Generalized Quantifiers and Restricted Quantification

The analysis of quantification in natural language by means of generalized quantifiers spread mainly in formal semantics during the eighties (although some precedents, by means of binary quantifiers, appeared in philosophy already in the seventies). It is a generalization of Frege’s original approach. We present the analysis, show in what respect it is superior to Frege’s original analysis of quantification in natural language, and then show the problems it still faces. We then show why our approach is superior to the generalized quantifiers approach, and how it can explain its successes and failure.

This unit is structured around my paper, ‘Generalized Quantifiers, and Beyond’ (2009). I have touched this subject all too briefly in my book, Section 6.4.

Recommended additional readings:

Barwise, J. and Cooper, R. (1981), ‘Generalized Quantifiers and Natural Language’, Linguistics and Philosophy, 4, pp. 159-219.

Keenan, Edward L. and Stavi, J. (1986), ‘A Semantic Characterization of Natural Language Determiners’, Linguistics and Philosophy, 9, pp. 253-326.

Keenan, E.L. and Westerståhl, Dag (1997), ‘Generalized Quantifiers in Linguistics and Logic’, in Johan van Benthem and Alice ter Meulen (eds), Handbook of Logic and Language, Elsevier and The MIT Press, pp. 837-93.

Peters, Stanley and Westerståhl, Dag (2006), Quantifiers in Language and Logic, Oxford: Clarendon Press.

Westerståhl, Dag (2001), ‘Quantifiers’, in Lou Goble (ed.), The Blackwell Guide to Philosophical Logic, Blackwell, Oxford, pp. 437-60.

The first two papers are the classical introduction of the generalized quantifiers approach to formal semantic. The third and fifth are shorter surveys. The fourth (Peters and Westerståhl) is the most detailed treatment of the subject. 

Unit VII: Multiple Quantification

We discuss the semantics of sentences involving several quantifiers, such as ‘Some students take five courses’. This leads us to the discussion of the semantic necessity of passive–active voice, converse relation terms, or some other permutation-mechanism in natural language, by contrast to the predicate calculus.

This unit is structured around Chapter 7 of my book.

Unit VIII: Bound Anaphora

We now extend our semantic analysis to sentences containing bound anaphora, e.g., ‘If a man buys a donkey he vaccinates it’. The nature of anaphora is discussed, and the problem with anaphora bound with the indefinite article is presented and resolved.

This unit is structured around much of Chapter 8 of my book.

Recommended additional readings:

Evans, Gareth (1977 & 1980), ‘Pronouns, Quantifiers, and Relative Clauses (I)’, reprinted in his (1985), Collected Papers, Oxford University Press, pp. 76-152.

Evans, Gareth (1980), ‘Pronouns’, reprinted in his (1985), Collected Papers, Oxford University Press, pp. 214-48.

Neale, Stephen (1990), Descriptions, The MIT Press.

Evans analyses the nature of bound anaphora, presents the difficulty with the indefinite article, and tries to resolve it with his E-type anaphors. Neale, following Evans with slight variations, presents criticisms of his theory and tries to answer them.

Unit IX: Deductive System

We proceed to develop a deductive system, apply it to various cases, and prove its soundness. We also discuss some of its natural extensions. The completeness of the system is explained, and its relation to first order logic, based on Lanzet’s work, is also presented.

This unit is structured around Part III of my book.

Recommended additional readings:

Lanzet, Ran (unpublished), ‘A Generalization of First-Order Logic Preserving the Square of Opposition’.

Lanzet, Ran and Ben-Yami, Hanoch (2004), ‘Logical Inquiries into a New Formal System with Plural Reference’, in Hendricks, V. et al. (eds.), First-Order Logic Revisited, Logos Verlag, Berlin, pp. 173-223.

In our co-authored paper we develop a formal language on the basis of the analysis and system found in my book. Lanzet proves there its completeness. In his unpublished work Lanzet elaborates the system to a three-valued logic that manages to capture defining clauses of natural language. In both the relation to the first-order Predicate Calculus is analysed.

Annotated Additional Bibliography

These are additional works that elaborate of several issues touched, or ignored, in the units above.

Aristotle (4th Century BC), Categories, On Interpretation, Prior Analytics, Peripatetic Texts, Athens. [Aristotle’s classic development of his semantics and logic, to which my system is closer than it is to Frege’s.]

Ben-Yami, H. (2001), ‘The Semantics of Kind Terms’, Philosophical Studies, 102, pp. 155-84. [This is the first place where I published some of my ideas on plural reference, in which I also show how they improve on current theories of kind terms.]

Frege, Gottlob (1879), Begriffsschrift: Eine der Arithmetischen nachgebildete Formelsprache des reinen Denkens, Verlag von Louis Nebert, Halle A/S. [The essential classical work where Frege develops his logic. We shall often refer to its first sections in our course. Highly recommended.]

Geach, P.T. (1968), ‘History of the Corruptions of Logic’, reprinted in his (1972), Logic Matters, Blackwell, Oxford, pp. 44-61. [Geach both argues for an analysis of common nouns similar to ours and explains what he takes to be the major difficulty facing it.]

Lewis, David (1991), Parts of Classes, Blackwell, Oxford. [Lewis argues clearly for the idea of plural reference.]

Lockwood, Michael (1975), ‘On Predicating Proper Names’, The Philosophical Review, 84, pp. 471‑98. [A defence of the use of proper names as predicates, a topic to which we may get late in our course.]

McCawley, James D. (1981), Everything that Linguists have Always Wanted to Know about Logic (but were ashamed to ask), The University of Chicago Press. [McCawley develops a deductive system close to ours in some ways.]

Russell, Bertrand (1903), The Principles of Mathematics, second edition 1937, George Allen and Unwin, London. [Here Russell still argued for plural reference (the class as many), and thought it removes his famous paradox. Frege, regrettably, managed to convince Russell that the class as many is incoherent.]

Simons, Peter (1982), ‘Number and Manifolds’ and ‘Plural Reference and Set Theory’, in Barry Smith (ed.), Parts and Moments: Studies in Logic and Formal Ontology, Philosophia Verlag, München and Wien, pp. 160-260. [This is another early attempt, different than Boolos’, to add plurals to logic.]

Sommers, Fred (1969), ‘Do We Need Identity?’, The Journal of Philosophy, 66, pp. 499-504. [Like Lockwood, Sommers argues for the possibility to predicate proper names.]

Sommers, Fred (1982), The Logic of Natural Language, Clarendon: Oxford. [A criticism of the predicate calculus as a system for analysing natural language semantics, and a development of a different system.]

Strawson, P.F. (1952), Introduction to Logical Theory, Methuen, London. [Contains Strawson’s famous defence of Aristotelian logic.]

Wiggins, D. (1981), ‘“Most” and “All”: Some Comments on a Familiar Programme, and on the Logical Form of Quantified Sentences’, in Mark Platts (ed.), Reference, Truth and Reality, Routledge and Kegan Paul, London, pp. 318-46. [A philosophical attempt to analyse natural language quantification by means of binary quantifiers.]