### 1. Brief *Vita*

Willard Van Orman Quine was born on June 25, 1908 in Akron, Ohio. He majored in mathematics with honors reading in mathematical philosophy at Oberlin College (BA), and later in philosophy in Harvard (M.A. and Ph.D.). He worked under the supervision of Whitehead on a generalization of *Principia Mathematica*. After his Ph.D. thesis, he was awarded a Sheldon Traveling Fellowship. This trip to Europe in 1933 would have a profound influence on his further intellectual development. In Harvard, the prominent logicians, such as Whitehead, C.I. Lewis, and Sheffer, were unaware of recent advances in Europe. In Vienna, he attended some of the Vienna Circle lectures. In Prague, he had long discussions with Carnap on the drafts of *Der* *logische Syntax der Sprache*, which would result in a lifelong friendship and later in one of the central debates in American philosophy. In Warsaw, Quine was able to become acquainted with the work of, among others, Tarski, Łukasiewicz and Leśniewski. A few years later, in the advent of World War II, he helped to solve immigration problems for many of the philosophers and scientists he met in Europe.

Back in Harvard, Quine was elected to the newly established Society of Fellows; the behaviorist B.F. Skinner was another Junior Fellow. He was appointed faculty instructor in 1936, and stayed in Harvard for the rest of his career, with the exception of three years of Navy service (1943-1946), a few sabbaticals (e.g., Oxford and Princeton) and innumerable travels all over the world. Before his Navy service, he had worked mainly on problems in logic. Soon after his return, he published some of his most influential philosophical papers, namely “On what there is” (1948), in which he formulated the criterion of ontological commitment “to be is to be the value of a variable”; and “Two dogmas of empiricism,” in which he attacked Carnap’s distinction between analytic and synthetic statements. He continued work in logic, and wrote a few critical articles in the new field of modal logic.

Since the beginning of the fifties, Quine became gradually more interested in the philosophy of language, which would result in his major work *Word and Object* (1960). In this book, he argued that meaning and synonymy cannot be part of austere (logically regimented) science. In order to illustrate this “indeterminacy of translation” he discussed the case of radical translation, i.e. a field linguist trying to translate a completely unknown language. He claimed that several translation manuals are possible, without there being a single “right” one. Quine’s philosophy of language was strongly influenced by the behaviorist tradition, and was soon strongly criticized by Chomsky at the dawn of the cognitivist era in linguistics and psychology.

In 1968, Quine published “Epistemology naturalized.” Though the ideas in the paper were not radically new, and in fact more clearly expressed in later works such as *The Roots of Reference* (1974) or his latest work *From Stimulus to Science *(1995), nevertheless the paper revitalized the naturalistic tradition in epistemology. The foundational program was abandoned, and epistemology was to be studied as a chapter of psychology. Thus the idea of an external point of epistemic evaluation was given up, and the traditional distinction between science and philosophy was blurred.

Over the decades, Quine has contributed to many other fields in philosophy. In philosophy of science, the Quine-Duhem thesis, formulated in “Two dogmas of empiricism” is still well-known, as well as the thesis of the underdetermination of theories by facts (1975). In philosophy of mathematics, the contemporary positions of naturalism, structuralism and nominalism owe much to Quine’s work. Over the years, he became aware of the problems of his earlier ontological views in “Ontological relativity” (1968) and most forcefully in “Whither physical objects?” (1976).

In 1978 he retired from Harvard. He continued to work and published a few more books. He received the first Rolf Schock prize (1993), the Kyoto prize (1996), and in 1997 his 18^{th} honorary degree. Quine died in Boston on December 25, 2000.

### 2. Systems of Logistic

In 1931, Quine embarked on an extension of *Principia Mathematica* (*PM*) under the supervision of Whitehead. He considered the second edition of *Principia Mathematica *as a major intellectual achievement, and still the last word in mathematical logic at that time, but saw room for many technical improvements. The following decade, he spent most of his time reorganizing and tidying the framework of *PM*, resulting in the peculiar logical systems *NF* (“New foundations for mathematical logic,” 1937) and *ML (Mathematical Logic*, 1940).

The major task he set himself for his Ph.D. was to generalize the system so that it could express theorems irrespective of the sort of propositional function it contained. In the logical framework of *PM*, classes and dyadic relations were given separate treatment, while relations beyond the dyadic relations were left out. Quine wanted a more general system containing also theorems with polyadic propositional functions containing any number of variables from any type. To this end, he used the notion of sequence, hence the title of the work “*The Logic of Sequences: A Generalization of Principia Mathematica.*” Throughout the work, he systematically demonstrates that the theorems of *PM* follow from his system.

A sequence is a typed set of terms with a certain order; one can form a new sequence from two existing sequences, namely the ordered pair containing both. The basic building blocks of sequences are individuals (Urelements) and (propositional) functions, which take as arguments sequences of a certain type. It is clear that the sequences can be ordered in a type hierarchy similar to the one in *PM*. The major innovation thus consists in taking together the separate arguments (e.g., “a,” “b,” “c”) of a propositional function (“f (a,b,c)”) into a single sequence “(a,b,c)”. Sequences are the “active primitive ideas” of the system. The “passive primitive ideas” or logical constants are catenation, predication, superplexion and assertion. Catenation (pairing) is needed in order to form new sequences; and is basically the formation of ordered pairs. Predication (membership) is necessary to form propositions from a function and a sequence. Assertion is taken directly from *PM*. Superplexion is an inelegant operation from which the more common implication, conjunction, etc. can be derived. Quine’s system enabled a general formulation of the theorems of *PM* without specifying the argument type. For example, he did not have to repeat the law of double negation for monadic predicates, dyadic predicates and so on, but could formulate it as a single theorem applying to an arbitrary n-adic function and sequence (§4.33).

With hindsight, Quine concluded that the generalization was a good strategy, but not a really surprising one. However, he thought the work was still important for other reasons. In addition to the introduction of sequences, he had made other changes to *PM*. He strove for economy in basic concepts and notation, and could avoid many superfluous notations in *PM*. He was very careful on use-mention confusions. In many places in *PM*, names and the objects they stand for are not neatly distinguished. He would discover a decade later that Frege had been careful in his seminal work on logic, but that Whitehead and Russell had become sloppy. But the most important innovation, which would prove to be of major importance for his later work, was the replacement of intensional propositional functions by extensional “functions,” in fact classes. In *PM*, it was possible to have two propositional functions with the same extension, which were yet different; classes were only introduced as imperfect symbols. Later his philosophical objections to intensions would prove to be substantial.

Quine had finished the Ph.D. thesis of 290 pages full of laborious notations in one year. At first, in their two-weekly meetings, Whitehead was pleased to discuss new ideas in logic. Soon, however, as Quine later complained, he would become distracted by metaphysical issues. According to Quine, his contribution was the suggestion of two technical terms, namely “length” for the length of a sequence, and, remarkably, “essence,” the essence of *x* being the class of all classes containing *x*. Whitehead and the Harvard department were impressed with Quine’s work and would subsidize the publication. During his trip to Europe, he made numerous revisions and reorganized the work completely. The basic structure of the logical system remained the same, but the presentation was streamlined. In view of the contemporary developments in Europe, he made a few notable amendments. Abstraction was added as a primitive idea, the redundant assertion was dropped, and predication was subsumed under ordination (previously catenation). Thus he could regard a proposition as a sequence, viz. an ordered pair of a function and a sequence. As extensionalism had become generally accepted in Europe, its philosophical defense had been dropped. The reworked text appeared as *A System of Logistic* in 1934.

In the following years, he developed several systems of logistic. After discussions with Leśniewski, he tried to develop nominalistic systems, i.e. logical systems in which quantification over abstract objects, e.g. propositions, propositional functions, classes, etc. can be avoided. The project failed, but would remain a lingering concern, and come at the surface in discussions with Tarski in 1940, and in a joint paper with Goodman in 1947. In “Set-theoretic foundations for logic” (1936), he integrated logic and the Zermelo-Fraenkel set theory (ZFC), which later would become the standard view on logic and set theory. However, in the following year, he abandoned this system and returned to *PM*, which he transformed in a radical way, yielding his famous “New Foundations” (1937). Even after various modifications, Quine was still unhappy with several inelegant features of *PM*. Especially the reduplication of the null set, the natural numbers, and the quantifiers at every level in the type hierarchy, and the postulation of an infinite set of individuals were to be avoided. To this end, he jettisoned the type hierarchy and ended with a single universe with a universal quantifier as a logical primitive, next to alternative denial and membership. In order to avoid Russell’s paradox, the major worry of set-theorists at the time, he took the stratified formulas of *PM*, and used these in an abstraction axiom schema. Instead a being a criterion of meaningfulness of expressions, stratification became an ontological criterion, determining which expressions do determine a set. The expression “*x*Ï*x*” was not meaningless, but yet did not determine a set in NF, and thus the immediate route to Russell’s paradox was blocked. NF is an intriguing but counterintuitive system. It is based on the common logical constants, which are combined with a purely grammatical abstraction axiom schema that is copied from *PM*. Deriving mathematical results in the systems proved to be irksome. Three years later, he transformed the system again, and added proper classes. He was worried about expressions that can be predicated of several objects, but yet do not determine a set or class. In the new system *Mathematical Logic* (1940), the expression “*x*Ï*x*” was said to determine a proper class, which could not be a member of any other class, thus barring Russell’s paradox in another but still ontological way.

Shortly after finishing *Mathematical Logic*, his major logical work, he was asked to write a contribution to the Schilpp volume on Whitehead. The article is a rather dry overview of *A Treatise of Universal Algebra* and *Principia Mathematica*. Quine’s major objections to *PM* are all clearly expressed. The paper is written at a decisive stage in his career. He had finished his major logical work, which he would summarize in an introductory book (1941) and in a book written in Portuguese (1944), before joining the army. Later Quine would gain fame as a philosopher, but his development in logic was at an end. The mathematical community had wholly embraced ZFC, while Quine was unable to opt for a single list of set-theoretic axioms. This is particularly clear in *Set Theory and its Logic*. He considered this as one of his most important works, but already at the time it appeared to be obsolete. His adherence to the ideas and structure of *Principia Mathematica* had become an impediment in his logical and set-theoretic work.

### 3. Logic and Metaphysics

Quine had an ambivalent appreciation of Whitehead. In Oberlin, he had discovered modern logic on recommendation of Bill Bennett, a senior in English, who had told him that Russell had a “mathematical philosophy.” He made further inquiries, but nobody in Oberlin knew modern logic. The chair of the mathematics department provided the works of Venn, Peano, Couturat, Russell’s *Principles of Mathematics*, and of course *Principia Mathematica*, which his mother later bought him as a Christmas present. For his honor’s thesis, Quine generalized a formula from Couturat, and provided an elaborate derivation of it from the axioms of *Principia Mathematica*. It was only natural that he would move to Harvard to study with Whitehead.

At the time Quine arrived in Harvard, Whitehead’s interest had shifted towards metaphysics, and he was no longer immersed in logic; the logic course was taught by Paul Sheffer. Quine was, as other students, impressed by Whitehead who “radiated greatness and seemed old as the hills.” He took Whitehead’s courses on “Science and the Modern World” and “Cosmologies Ancient and Modern,” but was soon bored, and wrote the exam paper on the more technical method of extensive abstraction. He never appreciated Whitehead’s metaphysical leanings, and even less his religious temper. There is a revealing anecdote that once after one of Whitehead’s lectures, Quine exclaimed, “Alfred North Whitehead. Mary Baker Eddy. Jesus H. Christ.” (Mary Baker Eddy was the founder of the Christian Science Church.) The difference in philosophical orientation is particularly clear in two places, namely in Whitehead’s foreword to Quine’s *A System of Logistic* and at the end of Quine’s contribution to the volume on Whitehead in Schilpp’s Library of Living Philosophers. In the foreword of Quine’s first book, Whitehead praised Quine’s contribution to logic and then concluded: “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 prescribes the shapes of metaphysical thought.” Quine, on the other hand, in his piece on Whitehead’s philosophy, exclusively concentrated on the logical work, and ended drily with “this [*MCMW*] was the beginning of a quest for the broadest, most basic concepts and principles of nature, and in the decades since *Principia* the quest has issued in a metaphysics.” This is the only sentence he ever wrote with regard to Whitehead’s non-logical work.

Though Quine never embraced process philosophy, or even the philosophy of organism or the method of extensive abstraction, Whitehead’s work was important for his own later philosophical work, because the influence of *Principia Mathematica* extended far beyond his early logical work. Many of Quine’s important contributions on ontology, in philosophy of language, philosophy of science, or in epistemology, can be directly related to his technical work on *Principia Mathematica*. I will illustrate this with the two clearest examples, namely the influence of Quine’s early work on his ontological views and on his central tenet of extensionalism.

The first expression of Quine’s ontological criterion can be found in two articles from 1939, namely “Designation and existence” and “A logistical approach to the ontological problem,” which are very similar to the far more famous “On what there is.” These articles appeared shortly after NF. The change from a semantic way to block Russell’s paradox to an ontological restriction on the abstraction of classes, and the introduction of quantification as a logical constant rather than a function at many stages in the type hierarchy, brought along several ontological problems that forced him to reflect on ontology. Problems to prove the existence of an infinite set in NF, problems with the axiom of choice, reflection on the status of universals and predicates that do not determine a set, and problems with the Burali-Forti paradox in the first version of *ML* (to be solved three years later by Hao Wang by means of a restriction on quantification in the stratification axiom) all led him to take existential problems in set theory very seriously. His early ontological views are the extension of these issues to non-logical contexts (i.e. frameworks with non-logical predicates). One can safely say that Quine’s ontological views are a natural outcome of his early work on *PM*.

A result that is probably even important is extensionalism. According to Quine, extensionalism is one of the two major tenets of his philosophy, naturalism being the other. He would later describe the replacement of the intensional propositional functions by extensional classes as one of the most important changes to *PM*. In the rest of his career, he would strongly object to the reintroduction of intensions. He criticized Carnap’s intensional semantics in the forties and fifties. He opposed modal logic, also after Kripke’s work on completeness proofs for certain models for modal logics. He did not believe in meanings as genuine identifiable entities, which had a large impact on his view on language, as is readily seen in *Word and Object*. Even in the nineties, he would write as one of his last articles “In defense of extensionalism.” Quine transformed *PM* into an extensional system, soon learned that extensionalism had become common in Europe, and in the following decades, against the tide, never considered returning to intensional logic. The work under the supervision of Whitehead thus had a huge impact on the rest of Quine’s philosophical career.

### Works Cited and Further Readings

#### Books by Quine

1934. *A System of Logistic* (Cambridge MA, Harvard University Press).

1940. *Mathematical Logic* (New York, Norton).

1941. *Elementary Logic* (Cambridge MA, Harvard University Press).

1944. *O Sentido da Nova Lógica* (São Paulo, Martins).

1950. *Methods of Logic* (New York, Holt).

1953. *From a Logical Point of View* (Cambridge MA, Harvard University Press).

1960. *Word and Object* (Cambridge MA, MIT Press.

1963. *Set Theory and its Logic* (Cambridge MA, Harvard University Press).

1966. *The Ways of Paradox and Other Essays* (New York, Random House).

1966. *Selected Logic Papers* (New York, Random House).

1969. *Ontological Relativity and Other Essays* (New York, Columbia).

1970. *Philosophy of Logic* (Englewood, Prentice Hall).

1974. *The Roots of Reference* (La Salle, Open Court).

1981. *Theories and Things* (Cambridge MA, Harvard University Press).

1985. *The Time of My Life* (Cambridge MA, Bradford).

1990. *Pursuit of Truth* (Cambridge MA, Harvard University Press).

1990. *The Logic of Sequences* (Harvard Dissertations in Philosophy) (New York, Garland).

1995. *From Stimulus to Science* (Cambridge MA, Harvard University Press).

Quine, W.V.O. & Ullian, J.S. 1970. *The Web of Belief* (New York, Random House).

Creath, R. (ed.). 1990. *Dear Carnap, Dear Van. The Quine-Carnap Correspondence and Related Work* (Berkeley, University of California Press).

*In Conversation W.V.O. Quine*: 7 videos. 1994. Philosophy International London.

#### Selected articles and chapters

1934. “Ontological Remarks on the Propositional Calculus,” *Mind*, 43, 472-76.

1934. “Report on Whitehead’s Logical Definitions of Extension, Class, and Number,” *American Mathematics Monthly***,** 41, 129—131.

1936. “A Theory of Classes Presupposing No Canons of Type,” *Proceedings of the National Academy of Sciences of the United States of America*, 22**,** 320-26.

1936. “On the Axiom of Reducibility,” *Mind*, 45, 498-500.

1936. “Toward a Calculus of Concepts,” *Journal of Symbolic Logic*, 1, 2-25.

1936. “Set-Theoretic Foundations for Logic,” *Journal of Symbolic Logic*, 1, 45-57.

1937. “Truth By Convention,” in *Philosophical Essays for A. N. Whitehead*, edited by O. H. Lee (New York, Longmans), 90-124.

1937. “New Foundations for Mathematical Logic,” *American* *Monthly*, 44, 70-80.

1938. “On the Theory of Types,” *Journal of Symbolic Logic*, 3,** **125-39.

1939. “Designation and Existence,” *Journal of Philosophy*, 36, 701-709.

1939. “A Logistical Approach to the Ontological Problem,” *Erkenntnis*, 9, 84-89. [Preprints only due to German invasion; reprinted in W. V. Quine’s *The Ways of Paradox *(1966)].

1941. “Whitehead and the Rise of Modern Logic,” in *Philosophy of A. N. Whitehead*, edited by P. A. Schilpp (La Salle, Open Court), 125-63.

1947. “On Universals,” *Journal of Symbolic Logic*, 12, 74-84.

1947 (with N. Goodman). “Steps toward a Constructive Nominalism,” *Journal of Symbolic Logic*, 12, 105-122.

1948. “On What There Is,” *Review of Metaphysics*, 2, 21-38.

1951. “Two Dogmas of Empiricism,” *Philosophical Review*, 60, 20-43.

1956. “Unification of Universes in Set Theory,” *Journal of Symbolic Logic*,** **267-79.

1969. “Epistemology Naturalized,” in *Ontological Relativity and Others Essays*, 69-90.

1975. “On empirically equivalent systems of the world,” *Erkenntnis*, 9, 313-28.

1976. “Whither physical objects?” in *Essays in Memory of Imre Lakatos*, edited by R. Cohen *et* *al*. (Dordrecht), 497-504.

1985. “Autobiography of W.V. Quine,” in *The Philosophy of W. V. Quine*, edited by L. Hahn & P. Schilpp (La Salle, Open Court), 1-46.

1993. “The Inception of ‘New Foundations’,” *Bulletin de la Société Mathématique* *de* *Belgique*, 45, 325-28.

1994. “Promoting Extensionality,” *Synthese*, 98, 143-51.

#### Readings on Quine

Barrett, R. and R. Gibson (eds.) 1990. *Perspectives on Quine* (Cambridge MA/Oxford, Blackwell).

Davidson, D. and J. Hintikka. 1975. *Words and Objections* (Dordrecht, Reidel).

Decock, L. 2002. *Trading Ontology for Ideology. The Interplay of Logic, Set Theory and Semantics in Quine’s Philosophy* (Dordrecht, Kluwer)*.*

Decock, L. and L. Horsten (eds.). 2000. *Quine. Naturalized Epistemology, Perceptual Knowledge and Ontology* (Amsterdam, Rodopi).

Føllesdal, D. 2001. *Philosophy of Quine*, Vols. I-V (New York, Garland).

Forster, T.E. 1992. *Set Theory with a Universal Set* (Oxford, Clarendon Press).

Gibson, R. 1982. *The Philosophy of W.V. Quine* (Tampa, University Press of Florida).

_____. 1988. *Enlightened Empiricism. An Examination of W.V. Quine’s Theory of Knowledge* (Tampa, University Press of Florida).

Gochet, P. 1978. *Quine en perspective* (Paris, Flammarion).

_____. 1986. *Ascent to Truth* (München, Philosophia Verlag).

Hahn, L. & Schilpp, P. (eds.). 1998. *The Philosophy of W. V. Quine* (La Salle, Open Court).

Hookway, C. 1988. *Quine* (Cambridge, Stanford University Press).

Koppelberg, D. 1987. *Die Aufhebung der analytischen Philosophie. **Quine als Synthese von Carnap und Neurath* (Frankfurt-am-Main, Suhrkamp).

Laugier, S. 1992. *L’anthropologie logique de Quine* (Paris, Vrin).

Leonardi, P. & Santambrogio, M. 1995. *On Quine* (Cambridge, Cambridge University Press).

Orenstein, A. 2002. *W.V. Quine* (Princeton, Princeton University Press).

Orenstein, A. and P. Kotatko. 2002. *Knowledge, Language and Logic* (Dordrecht, Kluwer).

### Author Information

Lieven Decock

Faculty of Philosophy

Free University Amsterdam

De Boelelaan 1105, 1081 HV Amsterdam, The Netherlands

www.ph.vu.nl

LB.Decock@ph.vu.nl

### How to Cite this Article

Decock, Lieven, “Willard Van Orman Quine (1908–2000)”, last modified 2008, *The Whitehead Encyclopedia*, Brian G. Henning and Joseph Petek (eds.), originally edited by Michel Weber and Will Desmond, URL = <http://encyclopedia.whiteheadresearch.org/entries/bios/contemporaries/willard-van-orman-quine/>.