Abstract
This paper investigates the relation between Carnap and Quine’s views on analyticity on the one hand, and their views on philosophical analysis or explication on the other. I argue that the stance each takes on what constitutes a successful explication largely dictates the view they take on analyticity. I show that although acknowledged by neither party (in fact Quine frequently expressed his agreement with Carnap on this subject) their views on explication are substantially different. I argue that this difference not only explains their differences on the question of analyticity, but points to a Quinean way to answer a challenge that Quine posed to Carnap. The answer to this challenge leads to a Quinean view of analyticity such that arithmetical truths are analytic, according to Quine’s own remarks, and set theory is at least defensibly analytic.
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Notes
Quine does, however, claim that even this rests ultimately on synonymy. This remark is discussed later in this paper.
Since Quine uses the conditional instead of the biconditional, Quine must be thinking of \(\langle \hbox{x, y}\rangle\) as a function of x and y. In this way the other direction of the biconditional becomes unnecessary.
Quine also notes that ordered pairs need not be defined as sets at all. In arithmetic the ordered pair \(\langle \hbox{x, y}\rangle\) can be defined as \(2^{x} * 3^{y}\).
On Quine’s view explication is also elimination in the sense that in general we also eliminate a type of entity. For example when we define ordered pairs as sets we have eliminated the class of ordered pairs. The present essay will, however, not deal with ontological questions.
There is also a point about state-descriptions being insufficient to define the general notion of analyticity, but this is not an argument against Carnap, since, as Quine acknowledges, state-descriptions were never meant to play this role.
Grice and Strawson (1956) tentatively attribute to Quine the view that there is no clear explicandum. They do not more than tentatively attribute this view to Quine, since they think it is highly implausible that our use of the term ‘analytic’ is not guided by some coherent informal notion. But Quine need not deny that there is a coherent notion to be explained. What he denies, as will be argued, is that we can identify an important feature of this notion that needs to be preserved by any explication. I think Grice and Strawson are right in giving prominence to the task of giving an analysis. However, without a detailed understanding of what Carnap and Quine each meant by giving an analysis (explication), the exact nature of the problem is unclear.
Except in a very restricted sense which will be discussed below.
I mentioned George’s reading of the Carnap and Quine debate. I would like, now, to make a brief point about George on the analytic/synthetic distinction. George shows that Carnap can easily adopt a Carnapian attitude towards the A/S distinction. That is, Carnap can see it as a matter of choice whether we accept a system that includes the analytic/synthetic distinction. Quine, however, on George’s view, is in a very difficult position. George dubs the rejection of the A/S distinction a ‘double-edged sword’. Quine can neither hold that the existence of the A/S distinction is an empirical question—since he claims that there has been no empirical meaning given to the A/S distinction—nor can he claim with Carnap that it is a matter of choice. Quine’s seems then to have cut his own legs out from beneath him, and fallen into incoherence. George then claims that Quine avoids incoherence by relying on a delicate balance between his somewhat obscure notion of ‘linguacentrism’ and his empiricism. George concludes that “[d]ebate about whether the dispute is empty or instead substantive is, for Quine, itself lacking content.” (George 2000, p. 22) Despite, George’s best efforts, the position he saddles Quine with is, if not incoherent, dangerously close to incoherent. The problem I see with George’s analysis is that he takes the participants to be addressing the question ‘Is there an A/S distinction?’ instead of the question ‘Can analyticity be successfully analyzed?’ As we will see, once the participants in the debate are seen as addressing the second question (as their writings on the subject suggest) the problems that George struggles with simply do not arise.
Of course, he also thinks it can be fruitful. For instance if it could be shown that all mathematical truths are analytic.
The talk of identifying what it is about the notion that makes it worth troubling about comes from § 53 of Word and Object (as quoted above).
Of course, if we could choose our community as we wish this could come out as accepted by the entire community. But the notion of Quinean-analyticity remains different from community-wide acceptance. If we select a small enough community a feature that is not Quinean-analytic but a ‘don’t care’ might be accepted by the entire community.
Since these are the examples Quine himself discusses.
I am not claiming that this is the only possible definition of analyticity. Others, for instance Juhl and Loomis (2010) or Russell (2008), have recently put forward their own definition of analyticity. I have no interest in arguing that the definition given here is better than theirs. My goal is merely to show that there is at least one interesting definition of analyticity that satisfies Quine’s demands for a successful explication. Showing that it counts much of mathematics as analytic is the way I intend to show that the definition provided above is of interest. If the reader feels that I have given merely one interesting definition of analyticity among many possible, I will be sufficiently satisfied.
Of course, they do not use the term Quinean-analytic, but are arguing about what features any specific account of number must have. This is exactly what we have been calling Quinean-analytic.
I don’t use the terminology of ‘Quinean-analytic’ in this article, but I do argue that what we identify by analyzing our informal mathematical notions is true.
Of course, Quine wanted to put forward an epistemic view that did not include the notion of analyticity. It is certainly not being claimed that Quine himself would be completely satisfied with the views presented in this paper.
That is we might give an account according to which this sentence is not analytic—not an account according to which it is not true.
References
Benacerraf, P. (1965). What numbers could not be. Philosophical Review, 74, 47–73.
Carnap, R. (1950). Logical foundations of probability. Chicago: University of Chicago Press.
Carnap, R. (1956). Meaning and necessity: A study in semantics and modal logic (2nd ed.). Chicago and London: University of Chicago Press.
Carnap, R. (1996). The elimination of metaphysics through a logical analysis of language. In: S. Sarkar (Ed.), Science and philosophy in the twentieth century, (Vol. 2). New York: Garland Press.
Creath, R. (2007). Quine’s challenge to Carnap. In: Friedman, M., Creath, R. (Eds.), The Cambridge Companion to Carnap, Chap. 14, (pp. 316–335). Cambridge: Cambridge University Press.
Dummett, M. (1978). The significance of Quine’s indeterminacy thesis. In: Truth and other enigmas (pp. 375–419). Cambridge: Harvard.
George, A. (2000). On washing the fur without wetting it. Mind, 109(433), 1–23.
Grice, P., & Strawson, P. F. (1956). In defense of a dogma. Philosophical Review, 65, 141–158.
Hallett, M. (1984). Cantorian set theory and the limitation of size. No. 10 in Oxford Logic Guides. Oxford: Oxford University Press.
Juhl, C., & Loomis, E. (2010). Analyticity. London & New York: Routledge.
Kreisel, G. (1967). Informal rigour and completeness proofs. In: I. Lakatos (Eds.), Problems in the philosophy of mathematics (pp. 138–186). New York: Humanities Press.
Lavers, G. (2008). Carnap, formalism, and informal rigour. Philosophia Mathematica, 16(1), 4–24.
Lavers, G. (2009). Benacerraf’s dilemma and informal mathematics. Review of Symbolic Logic, 2(4), 769–85.
Quine, W. V. O. (1953). From a logical point of view. Cambridge: Harvard University Press.
Quine, W. V. O. (1960). Word and object. Cambridge: MIT press.
Quine, W. V. O. (1963). Carnap and logical truth. In: P. A. Schilpp (Ed.), The philosophy of Rudolf Carnap, vol. XI of Library of Living Philosophers, (pp. 385–406). La Salle: Open Court.
Quine, W. V. O. (1969). Epistemology naturalized. In: Ontological relativity and other essays, (pp. 69–90). New York: Columbia University Press.
Quine, W. V. O. (1973). Roots of Reference. La Salle: Open Court.
Quine, W. V. O. (2008). Confessions of a confirmed extensionalist and other essays. Cambridge: Harvard University Press.
Quine, W. V. O., & Carnap, R. (1990). Dear Carnap, Dear Van: The Quine-Carnap correspondence. California: University of California Press.
Russell, G. (2008). Truth in virtue of meaning: A defence of the analytic/synthetic distinction. Oxford: Oxford University Press.
Stein, H. (1992). Was Carnap entirely wrong after all? Synthese, 93(1–2), 275–295.
Acknowledgments
Thanks to Gerald Callaghan, Michael Cuffaro, William Demopoulos, Sona Ghosh, and Parzhad Torfehnezhad for their comments and suggestions. I would also like to thank the anonymous referees who read this paper for their helpful advice. Finally, I would like to thank the many people in the various audiences to whom I presented this paper for their comments and suggestions.
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Lavers, G. On the Quinean-analyticity of mathematical propositions. Philos Stud 159, 299–319 (2012). https://doi.org/10.1007/s11098-011-9709-2
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DOI: https://doi.org/10.1007/s11098-011-9709-2