# The Model-Theoretic Argument

Edited by Tim Button (Cambridge University)
 Summary Putnam attempted to refute metaphysical realism by the use of a model-theoretic argument (or perhaps arguments, in the plural). For the most part, the required model theory was very simple. The easiest model theoretic argument involved the idea of a permutation over the objects of the world, so that (for example) my word "cat" does not apply to all and only cats, but to cats*, where cats* might just be cherries. (Another model theoretic argument involved appeal to the Completeness Theorem of first-order logic, or the Löwenheim-Skolem Theorem.) The aim of these arguments was to threaten the metaphysical realist with radical indeterminacy of reference (which Putnam did not himself embrace, but took to be a reductio of metaphysical realism). In response to the claim that causation (for example) fixes reference, Putnam always responded by maintaining that this was just more theory. Much of the subsequent literature has turned on the acceptability (or otherwise) of Putnam's just more theory manoeuvre.
 Key works Although there were some anticipations, the model-theoretic argument(s) are most famous from Putnam 1977, 1980 and 1981 (ch.2). Several suggestions have been made, in an effort to rule out the deviant interpretations generated by the use of elementary model theory: Putnam himself considered appealing to causation; Lewis 1984 advanced the idea that some properties are more "referentially magnetic" than others; in a more mathematical context, Shapiro 1991 (ch.8) appealed to second-order (rather than first-order) logic; and McGee 2005 highlighted the fact that certain expressions should be (Kripkean) rigid designators. Putnam's response to all of these considerations was his just-more-theory manoeuvre (which he always presented alongside his model-theoretic arguments). Many authors (including Devitt 1983, Hale & Wright 1997 and Bays 2001) found this manoeuvre entirely question-begging; but Putnam was undeterred. His fullest explanation of why is provided in his 2000; and Putnam's position here -- and the model-theoretic arguments, more generally -- are defended by Button 2013.
 Introductions Putnam 1977; Putnam 1981 (ch.2); Lewis 1984; Hale & Wright 1997; Button 2013 (chs.1-7).
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1. Mario Alai (1989). A Critique of Putnam's Antirealism. Dissertation, University of Maryland, College Park
Many philosophers have shown great interest in the recent anti-realist turn in Hilary Putnam's thought, whereby he rejects "meta-physical realism" in favor of "internal realism". However, many have also found it difficult to gain an exact understanding, and hence a correct assessment of Putnam's ideas. This work strives for some progress on both of these accounts. ;Part one explicates what Putnam understands by "metaphysical realism" and considers to what extent Putnam himself formerly adhered to it. It reconstructs Putnam's arguments for (...)
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2. Putnam's Model- Theoretic Argument has been generally held as invalid. In this work, attention is addressed to two broad facts understated by critics and commentators: (i) there are, at least, two different model-theoretic arguments. One is directed against realism and the other is directed to naturalistic semantics. The general rejection affects the former, but it is open to discussion if it affects the latter; (ii) on the other hand, the model-theoretic argument construed as a reductio argument has not - prima (...)
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3. David L. Anderson (1992). What is Realistic About Putnam's Internal Realism? Philosophical Topics 20 (1):49-83.
Failure to recognize the "realistic" motivations for Putnam's commitment to internal realism has led to a widely shared misunderstanding of Putnam's arguments against metaphysical realism. Realist critics of these arguments frequently offer rebuttals that fail to confront his arguments. Simply put, Putnam's arguments --the brains in a vat argument as well as the model-theoretic argument -- are "reductios" that are intended to show that "metaphysical realism itself is not sufficiently realistic". If that claim can be substantiated then Putnam can go (...)

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4. David Leech Anderson (1993). What is the Model-Theoretic Argument? Journal of Philosophy 60 (6):311-322.
In a recent article, James Van Cleve joins a growing throng who have argued that Hilary Putnam's model-theoretic argument (and his "just more theory" response) begs the question against those who hold externalist theories of reference. Van Cleve has misinterpreted Putnam's argument. Putnam does not demand that the statements which make up the causal theory of reference must, themselves, do the reference-fixing. That would be question-begging. Rather, Putnam's argument is a "reductio", which can only be blocked with a theory of (...)

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5. Timothy Bays (2008). Two Arguments Against Realism. Philosophical Quarterly 58 (231):193–213.
I present two generalizations of Putnam's model-theoretic argument against realism. The first replaces Putnam's model theory with some new, and substantially simpler, model theory, while the second replaces Putnam's model theory with some more accessible results from astronomy. By design, both of these new arguments fail. But the similarities between these new arguments and Putnam's original arguments illuminate the latter's overall structure, and the flaws in these new arguments highlight the corresponding flaws in Putnam's arguments.

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6. Timothy Bays (2007). More on Putnam's Models: A Reply to Belloti. [REVIEW] Erkenntnis 67 (1):119--35.
In an earlier paper, I claimed that one version of Putnam's model-theoretic argument against realism turned on a subtle, but philosophically significant, mathematical mistake. Recently, Luca Bellotti has criticized my argument for this claim. This paper responds to Bellotti's criticisms.

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7. Timothy Bays (2006). The Mathematics of Skolem's Paradox. In Dale Jacquette (ed.), Philosophy of Logic. North Holland 615--648.
Over the years, Skolem’s Paradox has generated a fairly steady stream of philosophical discussion; nonetheless, the overwhelming consensus among philosophers and logicians is that the paradox doesn’t constitute a mathematical problem (i.e., it doesn’t constitute a real contradiction). Further, there’s general agreement as to why the paradox doesn’t constitute a mathematical problem. By looking at the way firstorder structures interpret quantifiers—and, in particular, by looking at how this interpretation changes as we move from structure to structure—we can give a technically (...)

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8. Timothy Bays (2001). On Putnam and His Models. Journal of Philosophy 98 (7):331-350.
It is not my claim that the ‘L¨ owenheim-Skolem paradox’ is an antinomy in formal logic; but I shall argue that it is an antinomy, or something close to it, in philosophy of language. Moreover, I shall argue that the resolution of the antinomy—the only resolution that I myself can see as making sense—has profound implications for the great metaphysical dispute about realism which has always been the central dispute in the philosophy of language.

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9. Luca Bellotti (2005). Putnam and Constructibility. Erkenntnis 62 (3):395--409.
I discuss and try to evaluate the argument about constructible sets made by Putnam in ‘ ”Models and Reality”, and some of the counterarguments directed against it in the literature. I shall conclude that Putnam’s argument, while correct in substance, nevertheless has no direct bearing on the philosophical question of unintended models of set theory.

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10. Paul Benacerraf (1985). Skolem and the Skeptic. Proceedings of the Aristotelian Society 59:85-115.

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11. Anthony L. Brueckner (1984). Putnam's Model-Theoretic Argument Against Metaphysical Realism. Analysis 44 (3):134--40.

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12. Tim Button (2013). The Limits of Realism. Oxford University Press.
Tim Button explores the relationship between words and world; between semantics and scepticism. A certain kind of philosopher—the external realist—worries that appearances might be radically deceptive; we might all, for example, be brains in vats, stimulated by an infernal machine. But anyone who entertains the possibility of radical deception must also entertain a further worry: that all of our thoughts are totally contentless. That worry is just incoherent. We cannot, then, be external realists, who worry about the possibility of radical (...)

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13. Tim Button (2011). The Metamathematics of Putnam's Model-Theoretic Arguments. Erkenntnis 74 (3):321-349.
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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14. Timothy Chambers (2001). Putnam's Paradox: A Less Quick Reply to Haukioja and Kroon. Mind 110 (439):709-714.

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15. Timothy Chambers (2000). A Quick Reply to Putnam's Paradox. Mind 109 (434):195-197.

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16. Marc Champagne (2012). Russell and the Newman Problem Revisited. Analysis and Metaphysics 11:65 - 74.
In his 1927 Analysis of Matter and elsewhere, Russell argued that we can successfully infer the structure of the external world from that of our explanatory schemes. While nothing guarantees that the intrinsic qualities of experiences are shared by their objects, he held that the relations tying together those relata perforce mirror relations that actually obtain (these being expressible in the formal idiom of the Principia Mathematica). This claim was subsequently criticized by the Cambridge mathematician Max Newman as true but (...)

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18. The paper aims at describing the discussion about the model-theorethic argument of Hilary Putnam in the past thirty years. First of all it presents the view of Timothy Bays, who through scrupulous examination of the formal side of the argument demonstrates that in fact it has very little in common with the model-theory. It is rather a simple and purely philosophical argument, which isn't more reliable and conclusive than any other argument in philosophy. Putnam tries to block the answer of (...)
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19. David Alan Davies (1988). The Infirmities of Externalism: Putnam and Dummett on the Realist Programme in Metaphysics and the Philosophy of Language. Dissertation, The University of Western Ontario (Canada)
The traditional "realist" programme in metaphysics and the philosophy of language--what Hilary Putnam has termed "metaphysical realism" or "externalism", on the one hand, and "semantic realism" as an account of the workings of language that draws essentially on the externalist notions of truth and reference, on the other--has been attacked, on independent but closely related grounds, by Putnam and Michael Dummett. I attempt to clarify their arguments against the realist programme, and to assess the force of the various strategies that (...)
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20. Michael Devitt (1984). Review of Putnam's Reason, Truth and History. [REVIEW] Philosophical Review 93 (2):274--7.
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22. The model-theoretic argument known as Putnam´s paradox threatens our notion of truth with triviality: Almost any world can satisfy almost any theory. Formal argument and intuition are at odds. David Lewis devised a solution according to which the very stucture of the world fixes how it is to be divided into elite classes which determine the reference of any true theory. Three claims are defended: Firstly, Lewis´ proposal must be completed by an account of successful referential intentions. Secondly, contrary to (...)

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23. Igor Douven (1999). A Note on Global Descriptivism and Putnam's Model-Theoretic Argument. Australasian Journal of Philosophy 77 (3):342 – 348.

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24. Igor Douven (1999). Putnam's Model-Theoretic Argument Reconstructed. Journal of Philosophy 96 (9):479-490.
Putnam's model theoretic argument against metaphysical realism can be reconstructed as valid, with premises acceptable to the realist. There is no illegitimate assumption that the causal theory of reference is false.

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25. Jürgen Dümont (1999). Putnam's Model-Theoretic Argument(S). A Detailed Reconstruction. Journal for General Philosophy of Science / Zeitschrift für Allgemeine Wissenschaftstheorie 30 (2):341-364.
Two of Hilary Putnam's model-theoretic arguments against metaphysical realism are examined in detail. One of them is developed as an extension of a model-theoretic argument against mathematical realism based on considerations concerning the so-called Skolem-Paradox in set theory. This argument against mathematical realism is also treated explicitly. The article concentrates on the fine structure of the arguments because most commentators have concentrated on the major premisses of Putnam's argument and especially on his treatment of metaphysical realism. It is shown that (...)

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26. Iris Einheuser (2010). The Model-Theoretic Argument Against Quantifying Over Everything. Dialectica 64 (2):237-246.
A variant of Hilary Putnam's model-theoretic argument against metaphysical realism appears to show that our quantifiers do not determinately range over absolutely everything. This paper argues that some recent attempts to respond to the quantificational skeptic are unsuccessful and offers an alternative response: the key to answering the skeptic is not to refute her argument but to realize that the argument's setup prevents it from being convincing to those it is directed at.

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27. Hartry Field (1982). Realism and Relativism. Journal of Philosophy 79 (10):553-567.

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28. Mathias Frisch (1999). Van Fraassen's Dissolution of Putnam's Model-Theoretic Argument. Philosophy of Science 66 (1):158-164.
Bas van Fraassen has recently argued for a "dissolution" of Hilary Putnam's well-known model-theoretic argument. In this paper I argue that, as it stands, van Fraassen's reply to Putnam is unsuccessful. Nonetheless, it suggests the form a successful response might take.

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29. Manuel García-Carpintero (1996). The Model-Theoretic Argument: Another Turn of the Screw. [REVIEW] Erkenntnis 44 (3):305 - 316.

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30. Mark Q. Gardiner (1995). Just More Theory? Australasian Journal of Philosophy 73 (3):421 – 428.

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31. Mark Q. Gardiner (1995). Operational Constraints and the Model-Theoretic Argument. Erkenntnis 43 (3):395 - 400.
Putnam's Model-Theoretic argument purports to show that, contrary to what the metaphysical realist is committed to, an epistemically ideal theory which satisfies all operational and theoretical constraints can be guaranteed to be true. He draws the additional antirealist conclusion that there can be no single privileged relation of reference. I argue that the very possibility of a so-called ideal theory satisfying all operational constraints presupposes a determinate relation of reference, and hence Putnam must assume precisely what he denies.

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32. Alexander George (1985). Skolem and the Löwenheim-Skolem Theorem: A Case Study of the Philosophical Significance of Mathematical Results. History and Philosophy of Logic 6 (1):75-89.
The dream of a community of philosophers engaged in inquiry with shared standards of evidence and justification has long been with us. It has led some thinkers puzzled by our mathematical experience to look to mathematics for adjudication between competing views. I am skeptical of this approach and consider Skolem's philosophical uses of the Löwenheim-Skolem Theorem to exemplify it. I argue that these uses invariably beg the questions at issue. I say ?uses?, because I claim further that Skolem shifted his (...)

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33. J. GrÜnfeld (1980). Putnam's Indeterminacy of Reference. International Logic Review 22:136.
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34. Johannes Hafner (2005). From Metamathematics to Philosophy: A Critical Assessment of Putnam's Model-Theoretic Arguments. Dissertation, University of California at Berkeley
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35. Bob Hale & Crispin Wright (1997). Putnam's Model-Theoretic Argument Against Metaphysical Realism. In Bob Hale & Crispin Wright (eds.), A Companion to the Philosophy of Language. Blackwell 427--57.
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36. Michael Hallett (1994). Putnam and the Skolem Paradox. In Peter Clark & Bob Hale (eds.), Reading Putnam. Blackwell 66--97.
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37. Carsten Hansen (1987). Putnam's Indeterminacy Argument: The Skolemization of Absolutely Everything. Philosophical Studies 51 (1):77--99.

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38. Jussi Haukioja (2001). Not so Quick: A Reply to Chambers. Mind 110 (439):699-702.

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39. Mark Heller (1988). Putnam, Reference, and Realism. Midwest Studies in Philosophy 12 (1):113-127.

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40. Nicholas K. Jones (2014). Realism Behind the Veil. Analysis 74 (4):721-730.
This is a critical notice of Tim Button's book "The Limits of Realism".

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41. John Koethe (1979). Putnam's Argument Against Realism. Philosophical Review 88 (1):92-99.
Hilary putnam has tried to refute the realist doctrine that the operational verification of a theory never logically precludes its objective falsity by means of an argument involving model-theoretic considerations. in this paper it is first shown that this argument does not work if the theory in question is open to revision. next it is argued that a realist need not and should not admit the notion of the sort of unrevisable, ideal theory required for the purposes of putnam's argument.

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42. Adam Kovach (2009). The Return of Taylor's Putnam. Australasian Journal of Philosophy 87 (1):119 – 125.
It is argued that the version of Hilary Putnam's model-theoretic argument developed by Barry Taylor in Models, Truth and Realism poses no threat to the realist claim that an ideal theory may be false.

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43. Frederick Kroon (2001). Chambers on Putnam's Paradox. Mind 110 (439):703-708.

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44. Gregory Landini (1987). Putnam's Model-Theoretic Argument, Natural Realism, and the Standard Conception of Theories. Philosophical Papers 16 (3):209-233.

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45. Byeong D. Lee (2003). Douven on Putnam's Model-Theoretic Argument. Erkenntnis 58 (1):7--12.
The model-theoretic argument, which Putnam employs to argue againstmetaphysical realism, has faced serious objections of many realist opponents.Igor Douven in his recent paper offers a new interpretation of the model-theoreticargument, which avoids the previous objections. The purpose of this paper is toshow that Douven's reconstruction of Putnam's argument is not successful, andhence that the realist objections still stand.

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46. Ernest Lepore & Barry Loewer (1988). A Putnam's Progress. Midwest Studies in Philosophy 12 (1):459-473.

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47. Michael Levin (1997). Putnam on Reference and Constructible Sets. British Journal for the Philosophy of Science 48 (1):55-67.
Putnam argues that, by ‘reinterpretation’, the Axiom of Constructibility can be saved from empirical refutation. This paper contends that this argument fails, a failure which leaves Putnam's sweeping appeal to the Lowenheim–Skolem Theorem inadequately motivated.

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48. David Lewis (1984). Putnam's Paradox. Australasian Journal of Philosophy 62 (3):221 – 236.

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49. Malcolm Foye Macfail (1998). Reference and Realism in Philosophy of Language and Philosophy of Mathematics. Dissertation, University of California, Berkeley
Most realists accept some version of the correspondence theory of truth, and most hold that reference is not subject to much indeterminacy. My central project is to defend these semantic doctrines against some influential recent attacks. ;The body of my dissertation consists of exposition and criticism of sample members of two groups of arguments: Hilary Putnam's model theoretic arguments against mathematical realism, and Donald Davidson's arguments for the thesis that reference in natural language is inscrutable. I also discuss in detail (...)
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50. Vann McGee (2005). Inscrutability and its Discontents. Noûs 39 (3):397–425.
That reference is inscrutable is demonstrated, it is argued, not only by W. V. Quine's arguments but by Peter Unger's "Problem of the Many." Applied to our own language, this is a paradoxical result, since nothing could be more obvious to speakers of English than that, when they use the word "rabbit," they are talking about rabbits. The solution to this paradox is to take a disquotational view of reference for one's own language, so that "When I use 'rabbit,' I (...)

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