Inception of Quine's ontology

Abstract

This paper traces the development of Quine's ontological ideas throughout his early logical work in the period before 1948. It shows that his ontological criterion critically depends on this work in logic. The use of quantifiers as logical primitives and the introduction of general variables in 1936, the search for adequate comprehension axioms, and problems with proper classes, all forced Quine to consider ontological questions. I also show that Quine's rejection of intensional entities goes back to his generalisation of Principia Mathematica in 1932.

Notes

In this paper, I analyse Quine's ontological criterion from a historical point of view. Quine scholars interested in my own interpretation and assessment of Quine's views may consult Decock 2002 , which is a critical assessment of Quine's ontological views from a philosophical perspective.

Already in 1951 there was a symposium on the paper with Peter Geach and Alfred Ayer; see Quine et al. 1951 . The paper was soon discussed by, among others, Church, Hochberg and Strawson. Since its first printing, the paper has been reprinted over thirty times.

For example, in 1951 Quine published ‘Two dogma's of empiricism’ (reprinted in Quine 1953 , pp. 20–46) and ‘On Carnap's views on ontology’ (reprinted in Quine 1966a , pp. 203–211), in 1953 ‘On mental entities’ (Quine 1966a , pp. 221–227), and in 1954 ‘Carnap and logical truth’ (Quine 1966a , pp. 107–132).

Quine ( 1953 , p. 1). This is Quine's standard use of the word ‘ontology’. Ontology is concerned with the existence or non-existence of certain entities. In another use of the word, it is a synonym for ‘universe’, i.e. the collection of all the entities that exist, or, in other words, of all the values of a variable.

Quine has emphasized that it was a real extension, since Russell did not approve of it; see Quine et al. ( 1951 , p. 153).

In ‘On What there is’, Quine presents Russell's theory of descriptions in a more informal way. There is not one occurrence of a variable in the entire paper.

For a critical note on this procedure, see Decock 1999 .

The quantifier may be an existential or a universal quantifier. See Quine ( 1948 , p. 13; 1960, p. 242).

Only very seldom does Quine countenance the possibility of explicating ontology in a different way, and when he does it results in the demise of ontology; see e.g. Quine ( 1990 , p. 36).

The views of Quine 1948 are repeated in Quine 1960 . He presents the criterion of ontological commitment (1960, pp. 232–243), and explains the need for a discussion of ontology in a linguistic framework in the final section, ‘Semantic ascent’ (1960, pp. 270–276).

Nominalism is the doctrine that there are no abstract objects at all (Quine 1953 , p. 15). This implies that mathematical entities such as sets and numbers do not exist, but also that attributes, propositions, etc. do not exist. For an analysis of Quine's attitude towards nominalism, see Decock ( 2002 , pp. 27–46).

This formulation did not occur in Quine 1960 , though the ideas expressed in the work can thus be summarized. The motto is quoted in Quine ( 1969a , p. 23; 1980, p. 107; 1995a, p. 75).

For additional complications with respect to these reductions, see Decock ( 2002 , pp. 92–106).

This began with the third chapter of Word and Object, and has played a role in Quine's writings ever since. See Quine ( 1973 , pp. 81–142; 1980, pp. 1–23; 1984; 1990a , pp. 23–31; 1990c ; 1994 ; 1995a , pp. 15–42; 1995b ). For a general exposition, see Decock ( 2002 , pp. 175–191).

An exception is his set theoretic system NF. There is a continued interest in this axiomatic system, and the problem of its consistency remains unsolved. See Forster 1992 , Boffa and Pétry 1993, Boffa and Casalegno 1985 , Holmes 1991 , 1993 , Vayl 1990 . For a general overview see Forster 1997 .

In addition, Quine's attitude towards modal logic is still famous. The objections that have survived are mainly of a philosophical nature. Some of the technical problems in Quine 1943 and Quine 1947a , b were already refuted to by Smullyan in (1948), although Quine remained unconvinced for a long time. See Burgess 1997 and Neale 2000 for a careful philosophical and historical analysis.

Quine also mentions having read Venn's Symbolic Logic, Peano's Formulaire de Mathématiques, Couturat's Algebra of Logic, Keyser's Mathematical Philosophy, Russell's Principles of Mathematics, and Whitehead's Introduction to Mathematics; see Hahn and Schilpp ( 1986 , p. 8). He did not know Frege, and had not even found a copy of the Begriffschrift when editing Mathematical Logic in 1940; see Hahn and Schilpp ( 1986 , p. 21).

Quine later expanded on this concept and obtained some technical results, namely a definition of substitution, and a basis for arithmetic; see Quine 1936f and Quine 1946 . Gödel's incompleteness theorem is proven in protosyntax in Mathematical Logic. This crucially involves the syntactical notion of concatenation. See Quine 1937b , Quine ( 1940 , ch. 7).

The superplex of two classes, a and b, is the relation such that the relation bears c to d if in the case that c is a subclass of a, then it follows that d is a subclass of b. For the technical formulation, see Quine ( 1990b , p. 34).

In this, Quine's system differed from PM. All Quine's functions were predicative, while in PM there were non-predicative propositional functions. In PM it was possible to have propositional functions more than one type higher than its highest free variable, if such a propositional function was constructed by means of a polyadic propositional function in which the intermediate variables were bound by quantifiers. Quine reinterpreted these functions as predicative, and could thus avoid PM's contested axiom of reducibility. See Quine ( 1936g ; 1963 , pp. 253–254; 1990b , p. 6).

The fact that the status of the quantifier was crucial for Quine's ontological development can be seen in two comments on discussions with Leśniewski; see Quine 1985 : ‘With Leśniewski I would argue far into the night, trying to convince him that his system of logic did not avoid, as he supposed, the assuming of abstract objects. Ontology was much on my mind.’, and Hahn and Schilpp 1986 : ‘With Leśniewski I would argue far into the night, trying to convince him that his quantification over all syntactic categories carried ontological commitment.’

For an overview of the historical evolution of the quantifier, especially in Europe, see Goldfarb 1979 .

In the foreword to Quine 1934a , Whitehead concludes: ‘Dr. Quine does not touch upon the relationship of Logic to Metaphysics. He keeps strictly within the boundaries of his subject. But—if in conclusion I may venture beyond these limits—the reformation of Logic has an essential reference to Metaphysics. For Logic describes the shapes of metaphysical thought.’

In later work, the influence of Quine's stay in Europe on his logical work is far more visible. A precise analysis of this Polish influence goes beyond the scope of this paper.

In 1947 he embarked anew on a nominalistic project in the paper with Goodman ‘Steps Toward a Constructive Nominalism’. In Quine 1947b he minutely discussed the ontological commitments to abstract objects that are indispensable for certain logical or mathematical purposes. Quine had great sympathy for nominalism until the late fifties, though he never confessed to being a nominalist; see Quine ( 1960 , pp. 243, n. 5; 1969a, p. 75; Quine 1980 , pp. 173–174; Hahn and Schilpp 1986 , p. 26). Nominalism meant the rejection of abstract entities. Because of extensionalism the only acceptable abstract entities were sets. Later Quine explicitly rejected nominalism; see Quine ( 1976 , p. 500): ‘The admission of numbers and other abstract objects is an eventuality that has to be faced, melancholy though it be.’ In the paper Quine even countenanced an ontology of pure sets only.

See Schönfinkel 1928 . Quine has often taken up the issue and published several articles on combinatorial logic and the elimination of variables over the years; see Quine ( 1960 ; 1971 ; 1995a , pp. 33–35; 62–66; 101–105).

In later work, namely Quine ( 1971 , p. 304; 1995a, p. 35) Quine discusses the repercussions of the elimination of variables for ontology. He says that in standard logic the role of the variable is linked to the quantifier. If there is a translation from standard logic to another logical system in which it is clear how the combined role of quantifier and variable is carried over, then the usual criterion of existence might thus be translated. This is the case for combinatorial logic, and the criterion becomes (Quine 1995a , p. 35): ‘[T]o be is to be denoted by a one-place predicate.’

In Quine ( 1937a , p. 82; 1940, p. 123), membership was regarded as a logical notion, but in later work, Quine clearly separated logic and set theory. Membership became a primitive extra-logical predicate. See Quine ( 1953 , p. 116; 1954, p. 46; 1963, p. 257).

It was ZFU rather than ZF; see Quine ( 1936d , p. 85).

Quine ( 1936d , p. 85): ‘Though Γ lacks two chief features of the standard logic, viz., types and the null class, we shall see that within Γ we can construct definitionally a derivative theory Δ in which both the null class and the familiar stratification into types are ostensibly restored and all theorems of the standard logic are forthcoming.’

Quine ( 1937a , p. 81): ‘The variables are to be regarded as taking as values any objects whatever; and among these objects we are to reckon classes of any objects, hence also classes of any classes.’

See Quine ( 1987 , p. 287). For the technical elaboration see Quine ( 1956 ; 1963 , pp. 266–286).

This is also to be found in Mathematical Logic; see Quine ( 1940 , 146–152).

Quine has used both terms; see e.g. Quine ( 1934a , p. 19); or 1938 where the formulae are called meaningless, or Quine ( 1966b , 127), where they are called ungrammatical.

Of course, this does not mean that ontological questions are entirely superfluous in type theory, though they are less pressing. In ‘Russell's ontological development’, Quine even blames Russell for neglecting ontological questions: ‘A moral of all this is that inattention to referential semantics works two ways, obscuring some ontological questions and creating an illusion of others’; see Quine ( 1966c , p. 78). Moreover, it is striking that in this discussion of Russell's ontological views, Quine discusses propositional functions, classes, propositions, and also the more empirical sense data at length.

However, in a reply to Dreben's comment on the content and historical context of Quine's doctoral thesis, Quine explicitly minimised the ontological relevance of the type theory in the thesis and the book A System of Logistic; see Barrett and Gibson ( 1990 , p. 96); ‘Struck by my ‘dread word “ontology” ’ in a A System of Logistic and by my protracted belaboring on the subject in later years, Burt perhaps overestimates the philosophical intent on pages 12 and 28 of that early book. I was stipulating my range of variables and my usage of ‘sequence’.’.

In this paper I concentrate on Quine's work before 1948, but the role of comprehension axioms in Quine's ontology is also extremely prominent in Set Theory and its Logic. The book is one of Quine's last contributions to mathematics, and it is a comparison of various axiomatic set theories. The book has not received its due attention, mainly because ZFC had become standard set theory in the early 1960s. Now it is more important from a philosophical point of view. Quine's view on ontology becomes especially clear. At every point Quine is extremely meticulous about the existential assumptions that are made. The role of comprehension or abstraction is discussed on page 35, and he speaks either of ‘commitment to there being at least one class’, (p. 38), or a ‘non-committal version of class abstraction’ (pp. 35, 39). The title of the section in which abstraction is discussed is ‘The virtual amid the real’ (p. 34).

The naïve abstraction principle is presented in a modern logical notation. For Quine's original notation, see Quine ( 1937a , p. 89 R3), or Quine ( 1963 , p. 35 (i)).

This is the usual form. Quine used a slightly modified version of it, viz. ‘∃y ∀x: x∈y ⇔ (x ⊂ z ∧ Fx)’. See Quine ( 1936d , p. 87).

In Quine ( 1938 , p. 133), Quine comments on dropping type theory in favour of formal restrictions: ‘The type ontology was at best only a graphic representation or metaphysical rationalization of the formal restrictions; and though some such rationalization may well be desired, it seems clear in particular that the type ontology afforded less help than hindrance.’

For a dissenting voice, see Forster ( 1997 , pp. 840–841).

See Quine 1937c . A later result is that NF is incompatible with the axiom of choice, which is related to the existence of non-Cantorian classes. See Specker 1953 ; Rosser 1954 ; Quine (1963, pp. 292–299).

Quine has commented on this; see Quine ( 1953 , pp. 98–99; 1966b, pp. 114–120). Specker found a proof in 1953; see Specker 1953 .

Quine ( 1963 , p. 3). In Quine 1940 this term does not appear, and neither does the term ‘proper class’. In Quine 1955 , Quine still calls them non-elements. See Quine ( 1955 , p. 157).

See Quine ( 1963 , p. 302): ‘In order to keep the ontological content of the old comprehension schema of NF uncorrupted we have to relativize it to [V]; what was existence for NF becomes sethood.’

See Quine 1941a . Quine 1942 is a comparison of the curtailed system (ML minus *200) and Zermelo's theory with proper classes added.

I borrow the expression from Quine ( 1990b , p. ii).

See Quine ( 1963 , p. 20): ‘The virtual theory of classes … talks much as if there were classes, but explains talk without assuming them’, or Quine ( 1986 , p. 71): ‘But note still that the procedure only simulates; it does not provide for genuine classes.’

In a recent interview with Fara, Quine answered that naturalism and extensionalism were the two main tenets of his philosophy (Quine et al. 1994 ).

Whitehead and Russell ( 1962 , p. 22). This example already shows that Russell and Whitehead were careless in their use of terminology. They made a distinction between ambiguous sentences and propositional functions, but did not consistently apply it. Ambiguous sentences are sentences with an ambiguous subject x, and they take the form of an open sentence, e.g. ‘x is Greek’. Propositional functions are not linguistic expressions, but the abstract objects designated by them. They are named by an open sentence with a caret, e.g. ‘ is Greek’.

See Whitehead and Russell ( 1962 , p. 40): ‘It is true that, conversely, a function can be apprehended without its being necessary to apprehend its values severally and individually. If this were not the case, no function could be apprehended at all, since the number of values (true and false) of a function is necessarily infinite and there are necessarily possible arguments with which we are unacquainted. What is necessary is not that the values should be given individually and extensionally, but that the totality of values should be given intensionally, so that, concerning any assigned object, it is at least theoretically determinate whether or not the said object is a value of the function.’

Quine ( 1990b , pp. 138–140). Two sequences are identical if the one belongs to the unit function of the other. The unit function of a sequence is the product function of the essence of this sequence. The essence of a term is the function satisfied by all the properties of the term. The product function φ of a function of functions ψ is the function that is satisfied by all the sequences X, so that X satisfies all the functions that satisfy ψ. This is a complicated way of saying that two sequences are identical if they satisfy the same functions.

Since functions gave way to classes, the propositions of the form ‘φx’ were replaced by ‘x∈α’. The new term for predication was membership; see Quine ( 1990b , pp. 6, 26–27).

The index refers to pages 11, 16, 29, 33 and 73f that deal with truth-functionality; and to pages 120f that deal with extensionality in connection with classes: ‘If there is any difference between classes and properties, it is merely this: classes are the same when their members are the same, whereas it is not universally conceded that properties are the same when possessed by the same objects. … For mathematics certainly, and perhaps for discourse generally, there is no need of countenancing properties in any other sense.’

They appear briefly in ‘Whitehead and the rise of modern logic’, a historical overview of Whitehead's work. See Quine ( 1941b , pp. 11–12, 22).