Abstract
Putnam famously attempted to use model theory to draw metaphysical conclusions. His Skolemisation argument sought to show metaphysical realists that their favourite theories have countable models. His permutation argument sought to show that they have permuted models. His constructivisation argument sought to show that any empirical evidence is compatible with the Axiom of Constructibility. Here, I examine the metamathematics of all three model-theoretic arguments, and I argue against Bays (2001, 2007) that Putnam is largely immune to metamathematical challenges.
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Notes
Note that Bays does not think that Bays’ dilemma is the only problem that Putnam’s arguments face; nor, perhaps, that it is the main problem.
There are various ways to add classes “harmlessly” to set theories; for an excellent philosophical and technical summary, see Potter (2004, Appendix B). Our classes are what Potter calls “virtual classes”. I follow Potter’s typographical distinction between classes and sets: when talking about classes, I use square-brackets rather than curly-brackets, and “\(\varepsilon\)” rather than “\(\in\)”.
In fact, Putnam has another use for the Completeness Theorem: to argue that the metaphysical realist cannot make sense of the claim that an ideal theory might be false (1978, p. 126; 1980, pp. 472–474; 1989, p. 215). In the interests of brevity, I do not discuss this use of the Completeness Theorem directly.
\(\hbox{WKL}_0\) is a subsystem of second-order arithmetic which contains Weak Kőnig’s Lemma as an axiom, i.e.: “every infinite subtree of the full binary tree has an infinite path”. The other axioms of \(\hbox{WKL}_0\) are those of \(\hbox{RCA}_0\). \(\hbox{RCA}_0\) contains the basic axioms of arithmetic, i.e. the existence of 0, and axioms governing + and × . \(\hbox{RCA}_0\) also has \(\Upsigma^0_1\)-induction (i.e. induction for any \(\Upsigma^0_1\)-formula), and \(\Updelta^0_1\)-comprehension (i.e. for any \(\Updelta^0_1\)-formula ϕ, “\({\{n \in {\mathbb{N}} \mid \phi(n)\}}\) exists” is an axiom). See Simpson (1999, pp. 23–24, 92–93).
For technical details, see Franzen (2004, pp. 172–176)
For technical details, see Franzen (2004, pp. 187–197).
Bays (2007, pp. 126–127). I should emphasise that Bays does not commit himself to this thought; he merely suggests it as a possible response on behalf of the metaphysical realist. Furthermore, Bays is not here considering iterated consistency sequences, but sequences of theories formed by adding increasingly large axioms of infinity of at each stage (in response to the constructivisation argument). So the response that I am here considering is an adaptation of a suggestion made by Bays.
Bays suggested something like this to me in conversation on 7.xi.2008. I am not certain that he had exactly this in mind but, even if he did not, I wish to thank him for making me consider the idea.
References
Bays, T. (2001). On Putnam and his models. The Journal of Philosophy, 98, 331–350.
Bays, T. (2007). More on Putnam’s models: A reply to Bellotti. Erkenntnis, 67, 119–135.
Bays, T. (2009). Skolem’s paradox. In E. N. Zalta (Ed.), Stanford encyclopedia of philosophy. [http://plato.stanford.edu/entries/paradox-skolem/]—visited on 14.vi.2010.
Bellotti, L. (2005). Putnam and constructibility. Erkenntnis, 62, 395–409.
Benacerraf, P. (1985). Skolem and the Skeptic. Proceedings of the Aristotelian Society, 59, 85–115.
Boolos, G. S., & Jeffrey, R. C. (1989). Computability and Logic, 3rd ed. Cambridge: Cambridge University Press.
Franzen, T. (2004). Inexhaustibility: A non-exhaustive treatment. Wellesley, MA: Association for Symbolic Logic, Lecture Notes in Logic 16.
Hodges, W. (1993). Model theory. Cambridge: Cambridge University Press.
Kunen, K. (1980). Set theory: An introduction to independence proofs. Volume 102, Studies in logic and the foundations of mathematics. Oxford: North-Holland.
Lewis, D. (1984). Putnam’s paradox. Australasian Journal of Philosophy, 62, 221–236.
McIntosh, C. (1979). Skolem’s criticisms of set theory. Noûs, 13, 313–334.
Mostowski, A. (1969). Constructible sets with applications. Amsterdam: North-Holland, Studies in Logic and the Foundations of Mathematics.
Potter, M. (2004). Set theory and its philosophy. Oxford: Oxford University Press.
Putnam, H. (1978). Meaning and the moral sciences. London: Routledge and Kegan Paul.
Putnam, H. (1980). Models and reality. Journal of Symbolic Logic, 45, 464–482.
Putnam, H. (1981). Reason, truth and history. Cambridge: Cambridge University Press.
Putnam, H. (1983). Realism and reason: Philosophical papers (Vol. 3). Cambridge: Cambridge University Press.
Putnam, H. (1989). Model theory and the ‘factuality’ of semantics. In A. George (Ed.), Reflections on chomsky (pp. 213–232). Oxford: Basil Blackwell.
Putnam, H. (1992). Replies. Philosophical Topics, 20, 347–408.
Simpson, S. G. (1999). Subsystems of second-order arithmetic. Berlin: Springer-Verlag
Skolem, T. (1922). Some remarks on axiomatised set theory. In: J. van Heijenoort (Ed.), From Frege to Gödel: A source book in mathematical logic, 1879–1931 (1967) (pp. 290–301). Cambridge, MA: Harvard University Press.
Skolem, T. (1958). Une relativisation des notions mathématiques fondamentales. In: E. J. Fenstad (Ed.), Selected works in logic (1970) (pp. 633–638). Oslo: Universitetsforlaget.
Velleman, D. J. (1998). ‘Review of Levin’s ‘Putnam on reference and constructible sets’ (1997)’. Mathematical Reviews, 98c:03015, 1364
Wright, C. (1985). Skolem and the Skeptic. Proceedings of the Aristotelian Society, 59, 117–137.
Acknowledgments
I wish to thank Timothy Bays, Michael Potter and Peter Smith for patient suggestions, advice, and comments. I particularly want to thank Gerald Sacks, who taught me model theory, with whom I had many engaging discussions, and without whom I would probably have no proof of the Submodel Skolem Theorem.
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Button, T. The Metamathematics of Putnam’s Model-Theoretic Arguments. Erkenn 74, 321–349 (2011). https://doi.org/10.1007/s10670-011-9270-6
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DOI: https://doi.org/10.1007/s10670-011-9270-6