The Model-Theoretic Argument

Draft of March 2022. This is a draft of the front matter and ch. 1, for a new book manuscript on metametaphysics.

There are two ways of reading Newman’s objection to Russell’s structuralism. One assumes that according to Russell, our knowledge of a theory about the external world is captured by an existential generalization on all non-logical symbols of the theory. Under this reading, our knowledge amounts to a cardinality claim. Another reading assumes that our knowledge singles out a structure in Russell’s (and Newman’s) sense: a model theoretic structure that is determined up to isomorphism. Under this reading, our knowledge is far (...) from trivial, for it amounts to knowledge of the structure of the relations between objects, but not their identity. Newman’s objection is then but an expression of structural realism. Since therefore the content of theories is described by classes of structures closed under isomorphism, the most natural description of a theory in structural realism is syntactic. (shrink)

Few if any distinctions are more easily recognisable and assented to than that between objects, that is, things which are some ways, and that which they are, that is, ways for objects to be. In this paper I present an argument designed to show that this distinction is indeterminate in the sense that the truth-conditions of predicational sentences leave open what should count as an object and a way of being. The bulk of the argument is inspired by the celebrated (...) permutation argument advanced by Quine, Wallace, Putnam and others. (shrink)

On a widespread naturalist view, the meanings of mathematical terms are determined, and can only be determined, by the way we use mathematical language—in particular, by the basic mathematical principles we’re disposed to accept. But it’s mysterious how this can be so, since, as is well known, minimally strong first-order theories are non-categorical and so are compatible with countless non-isomorphic interpretations. As for second-order theories: though they typically enjoy categoricity results—for instance, Dedekind’s categoricity theorem for second-order PA and Zermelo’s quasi-categoricity (...) theorem for second-order ZFC—these results require full second-order logic. So appealing to these results seems only to push the problem back, since the principles of second-order logic are themselves non-categorical: those principles are compatible with restricted interpretations of the second-order quantifiers on which Dedekind’s and Zermelo’s results are no longer available. In this paper, we provide a naturalist-friendly, non-revisionary solution to an analogous but seemingly more basic problem—Carnap’s Categoricity Problem for propositional and first-order logic—and show that our solution generalizes, giving us full second-order logic and thereby securing the categoricity or quasi-categoricity of second-order mathematical theories. Briefly, the first-order quantifiers have their intended interpretation, we claim, because we’re disposed to follow the quantifier rules in an open-ended way. As we show, given this open-endedness, the interpretation of the quantifiers must be permutation-invariant and so, by a theorem recently proved by Bonnay and Westerståhl, must be the standard interpretation. Analogously for the second-order case: we prove, by generalizing Bonnay and Westerståhl’s theorem, that the permutation invariance of the interpretation of the second-order quantifiers, guaranteed once again by the open-endedness of our inferential dispositions, suffices to yield full second-order logic. (shrink)

Putnam’s model-theoretic arguments for the indeterminacy of reference have been taken to pose a special problem for mathematical languages. In this paper, I argue that if one accepts that there are theory-external constraints on the reference of at least some expressions of ordinary language, then Putnam’s model-theoretic arguments for mathematical languages don’t go through. In particular, I argue for a kind of descriptivism about mathematical expressions according to which their reference is “anchored” in the reference of expressions of ordinary language. (...) These anchors add enough to the content of mathematical expressions to forestall the radical kind of indeterminacy that model-theoretic arguments are purported to show, while still leaving room for a plausible, moderate kind of indeterminacy. (shrink)

In this article, I have two aims. Firstly, I argue that Hilary Putnam's model theoretic indeterminacy argument against external realism and Saul Kripke's so-called Kripkensteinian argument against semantic realism have the same dialectical structure and the same conclusion---both force the opponent to face the same dilemma. Namely: either adopt meaning minimalism or postulate unobservable semantic facts. Secondly, I analyze more closely the first horn of the dilemma---meaning minimalism. This is the position according to which there are no truth conditions for (...) meaning-ascriptions. It has been suggested that this position is incoherent. However, I argue that there is a coherent option available for the meaning minimalist. As Crispin Wright has proposed, a coherent meaning minimalist has to adopt a structured truth-predicate with at least two levels: one is a minimal or a deflationary truth-predicate for a semantic discourse and the other, more substantial or objective truth-predicate for discourses like natural sciences. Subsequently, this leads to a position close to Huw Price's global expressivism. Thus, the ultimate dilemma that Putnam's and the Kripkensteinian argument establish is the following choice: either meaning minimalism with a structured two-level truth-predicate or robust realism regarding meaning. (shrink)

This is an extended intellectual obituary for Hilary Putnam.

In this paper I want to adjudicate the dispute between those philosophers who do and those who do not think that the philosophy of language can illuminate metaphysical questions. To this end, I take the debate between Devitt and Putnam as a case study and diagnose what I take to be illuminating about their disagreement over metaphysical realism. I argue that both Putnam and Devitt are incorrect in their assessment of the significance of the model theoretic argument for realism. That, (...) whilst Devitt is entitled to claim that truth does not have anything to do with realism, Putnam’s challenge can still gain traction and seriously call into question our ability to engage in realist metaphysics. I argue that even if a completely semantically neutral conception of realism can be successfully articulated, doing so has the potential to bankrupt the methodology of metaphysical realism. Having taken this debate as a case study, I then offer some brief remarks on how to understand the philosopher who claims that realist metaphysicians should care about discussions of metasemantics and truth. Whilst I want to be cautious about generalising on the basis on this case alone, I think there are important lessons to be learned about the way in which considerations to do with language can shed light on the concerns of metametaphysics. (shrink)

Wittgenstein’s rule-following argument suggests that linguistic understanding does not consist in knowing interpretations, whereas Kripkenstein’s version suggests that meaning cannot be metaphysically fixed by interpretations. In the present paper, rule-following considerations are used to suggest that certain ontological questions cannot be answered by interpretations. Specifically, if the aim is to specify the ontology of a language, an interpretation cannot answer what object an expression of L denotes, if the interpretations are themselves L-expressions. Briefly, that’s because the ontology of such interpretations (...) would naturally be in question as much as the expressions they interpret. So in order to settle the question of ontology, the interpretations themselves would need to be interpreted, and thus a regress. I conclude that knowing the answer to what ontology underlies L cannot be a matter of knowing interpretations. The paper ends with a quietist conclusion; the slogan is that empirical science is ontology enough, or rather, it is about all the ontology one should expect. (shrink)

This is a critical notice of Tim Button's book "The Limits of Realism".

A review of The Limits of Realism, by Tim Button. Oxford University Press, 2013.

Tim Button explores the relationship between minds, words, and world. He argues that the two main strands of scepticism are deeply related and can be overcome, but that there is a limit to how much we can show. We must position ourselves somewhere between internal realism and external realism, and we cannot hope to say exactly where.

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 (...) trivial, insofar as from a closed body of observations (or “Ramsey sentence”) one can always generate other equally-satisfactory networks of relations, provided they respect the original set’s cardinality. Since any model thus generated will be empirically adequate, “[t]he defence is therefore driven back from the fairly safe fictitious-real classification to the much less tenable ‘trivial’ and ‘important’” (Newman 1928). Given the definitional rigour afforded by the initial appeal to isomorphism (via one-to-one correspondences in extension), the received assessment, shared by Russell himself, is that retreating to a pragmatic adjudication would betoken a fatal blow. However, I suggest that reliance on “importance” can be avoided if we incorporate an impersonal criterion of diachronic precedence. When collecting observations, an ordinality emerges alongside the cardinality which gives that underived structure an irrevocable epistemological privilege. Hence, I argue that, all other things being equal, any construct parasitic on an antecedent theory ought to be regarded as inferior/dispensable, since it was generated by an algorithm lacking the world-involving pedigree of its host structure. (shrink)

Presento y defiendo dos reinterpretaciones de la prueba ideada por Putnam para demostrar que no somos cerebros en una cubeta. Ambas resaltan explícitamente el papel desempeñado por el autoconocimiento de nuestros propios pensamientos y por el externismo sobre el contenido. La primera asume que el externismo implica que un cerebro en una cubeta no puede pensar el contenido proposicional relevante (constituido por conceptos acerca de cerebros y cubetas). La otra versión invoca una tesis externista más débil, conforme a la cual (...) un sujeto corpóreo y un cerebro en una cubeta no pueden compartir el contenido proposicional relevante. (shrink)

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.

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 (...) followers of the causal theory of reference by insisting that it is "just more theory", which could be erased by argument like any other language structure. In response meta-physical realists, like David Lewis and James Van Cleve, claim that "it is not causal language that fixes reference; it is causation itself". David Leech Anderson agrees with metaphysical realists that in principle causal theory could fix the reference, but in his opinion we do not have reliable and substantive theory of that kind, and it doesn't even seem probable that we will have such a theory in future. In conclusion, I agree with Putnam that the metaphysical realist says that "we-know-not-what fixes the reference relation we-know-not-how", and he doesn't have any chance of avoiding the demolition by the model-theorethic argument, because he still tries to separate sharply the realms of language and reality. (shrink)

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.

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.

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.

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.

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 (...) adequate “solution” to Skolem’s Paradox. So, whatever the philosophical upshot of Skolem’s Paradox may be, the mathematical side of Skolem’s Paradox seems to be relatively straightforward. (shrink)

Model theoretic considerations purportedly show that a certain version of structural realism, one which articulates the nvtion of structure via Ramsey sentences, is in fact trivially true. In this paper we argue that the structural realist is by no means forced to Ramseyfy in the manner assumed in the formal proof. However, the structural realist's reprise is short-lived. For, as we show, there are related versions of the model theoretic argument which cannot be so easily blocked by the structural realist. (...) We examine various ways in which the structural realist may respond, and conclude that the best way of blocking the model theoretic argument involves formulating his Ramseyfied theories using intensional operators. Introduction The model theoretic arguments On Ramseyfying away predicates The model theoretic argument bites back Restricting the second order quantifiers 5.1 Naturalness 5.2 Intrinsic 5.3 Qualitative 5.4 Contingent and causal Intensional operators and relations between properties Conclusion. (shrink)

Barry Taylor's book mounts a major new argument against one of the fundamental tenets of much contemporary philosophy, the idea that we can make sense of reality as existing objectively, independently of our capacities to come to know it. He concludes that there is no defensible notion of truth which preserves the theses of traditional realism, nor any extant position sufficiently true to the ideals of that doctrine to inherit its title. In presenting his case Taylor engages with many key (...) works of contemporary metaphysics, semantics, and philosophical logic, so his book will be of interest to a broad spectrum of scholars and students. (shrink)

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.

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 (...) refer to rabbits" is made true by the meaning of the word "refer." The reference relation is extended to other languages by translation. The explanation for this peculiarly egocentric conception of semantics-questions of others' meanings are settled by asking what I mean by words of my language-is to be found in our practice of predicting and explaining other people's behavior by empathetic identification. I understand other people's behavior by asking what I would do in their place. (shrink)

Two things about Hilary Putnam have not changed throughout his career: some (including Putnam himself) have regarded him as a “realist” and some have seen him as a philosopherwho changed his positions (certainly with respect to realism) almost continually. Apparently, what realism meant to him in the 1960s, in the late seventies and eighties, and in the nineties, respectively, are quite different things. Putnam indicates this by changing prefixes: scientific, metaphysical, internal, pragmatic, commonsense, but always realism. Encouraged by Putnam’s own (...) attempts to distinguish his views from one time to another, his work is often regarded as split between an early period of “metaphysical realism” (his characterization) and a later and still continuing period of “internal realism”. Late Putnam is understood to be a view that insists on the primacy of our practices, while the early period is taken to be a view from outside these, a “God’s Eye view”. As Putnam himself stresses (1992b), this way of dividing his work obscures continuities, the most important of which is a continuing attempt to understand what is involved in judging practices of inquiry, like science, as being objectively correct. Thus Putnam’s early and his current work appear to have more in common than the division between “early” and “late” suggests. In fact, Putnam’s earlier writings owe much of their critical force to his adopting the pragmatic perspective of an open-minded participant in practices of empirical inquiry, a stance not explicitly articulated in these writings but rather taken simply as a matter of course.1 Thus insofaras Putnam’s early writings defend a form of representational realism, they can be regarded as attempts to articulate a realist position at work inside our ordinary practices of making empirical judgments. For this reason, we begin our review of Putnam’s realisms by extracting from the early writings a core of principles that carries over into his current work but underwent significantly different interpretations over time.. (shrink)

: Some twenty years since its publication Putnam's model‐theoretic argument is still much discussed. The present paper aims to defend a reconstruction of the argument but begins by attempting to clarify the form of the argument. Usually, and with good textual grounds, the argument is treated as a reductio argument against metaphysical realism. I argue instead that it should be treated as developing a paradox. I go on to claim that the most promising response to this paradox is to be (...) able to provide a theory of sense, in the style recommended by Dummett. So, according to this reasoning, the argument is not an argument against metaphysical realism but an argument against positions which reject the notion of sense. (shrink)

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.

Abstract: Hilary Putnam has reformulated his model-theoretic argument as an argument against indirect realism in the philosophy of perception. This new argument is reviewed and defended. Putnam’s new focus on philosophical theories of perception (instead of metaphysical realism) makes better sense of his previous responses to the objection from the causal theory of reference. It is argued that the model-theoretic argument can also be construed as an argument that holders of a causal theory of reference should adopt direct realism in (...) the philosophy of perception. (shrink)

Abstract Curiously missing in the vast literature on Hilary Putnam's so-called model-theoretic argument against semantic realism is any response from would-be proponents of what Putnam would call magical theories of reference. Such silence is surprising in light of the fact that such theories have occupied a significant position in the history of philosophy and the fact that there are still several prominent thinkers who would, no doubt, favor such a theory. This paper develops and examines various responses to Putnam's argument (...) on behalf of the proponent of a magical theory of reference. While Putnam's explicit replies to such responses to his argument seem to involve little more than name calling, I develop arguments that show that there are significant problems facing any would-be proponent of such a view. While magical theories of reference are far from the strawmen Putnam seems to take them to be, there are, I argue, genuine reasons for a semantic realist to prefer a non-magical theory of reference. (shrink)

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: 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; on the other hand, the model-theoretic argument construed as a reductio argument has not - prima facia - (...) ontological consequences, but only restrains our methodology to deal with the intentional realm. (shrink)

Igor Douven has offered an original reconstruction and defence of Putnam's model-theoretic argument against metaphysical realism. Douven's construal has notable exegetical virtues, since it makes sense of some assumptions in Putnam's argument which his opponents have considered question-begging or puzzling. In this article I provide an indirect defence of metaphysical realism, by showing why this new version of the anti-realist argument should also be rejected. The main problems in the Douven-Putnam argument come from ascribing to the realist a distorted view (...) of correspondence truth. The view entails that when no feature selects just one of all the possible interpretations of language the existence of an interpretation suffices to make true a theory. The sensible realist is not committed to this extreme conception of correspondence truth. /// Igor Douven ha presentado y defendido una reconstructión del argumento teórico-modelista de Putnam contra el realismo metafísico. Esta versión tiene ciertas virtudes exegéticas, pues hace inteligible algunos presu- puestos putnamianos que sus críticos han considerado injustificados o sorprendentes. En este artículo proporciono una defensa indirecta del realismo metafisico, mostrando por qué debe también rechazarse el argumento antirrealista bajo esta nueva forma. Los problemas principales del argumento Douven-Putnam proceden de atribuir al realista una concepción distorsionada de la idea de verdad como correspondencia. Conforme a esa concepción, la ausencia de factores que seleccionen una única interpretación del lenguaje de entre las muchas posibles conlleva que la existencia de alguna interpretación ya es suficiente para hacer verdadera a una teoría que sea consistente. El realista sensato no está obligado a aceptar esa tesis. (shrink)

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.

This is a two tiered investigation. On the one hand, the author presents a systematic account of the philosophy of Hilary Putnam. Being the first comprehensive account to be published in the German-speaking world, the author traces the development of Putnam's realism and philosophy of language and their connections from the early 1950's to 2000. Contrary to the popular view of the discontinuity of Putnam's philosophy, he demonstrates that Putnam maintains certain semantic, pragmatic and epistemological foundations for the rational confirmability, (...) fallibility and self-corrigibility of theoretical knowledge in the empirical sciences as a constant around which many other philosophical assumptions are allowed to vary under critical examination. The resulting view of Putnam's philosophical "position" is decidedly "anti-metaphysical" (contrary to the common association of his philosophy of language with e.g. Kripke's understanding of realism), but not fully "deflationary" (contrary to the reclamation of his criticism of realist metaphysics as Quinean or Rortyan skepticism). On the other hand, the author generalizes, in the second and third parts of the book, Putnam's now famous theory of meaning and reference to develop a framework for the pragmatics of semantic continuity and discontinuity, as well as conceptual change within general terms in empirical settings, like natural kind terms, theoretical terms in natural sciences and commonsense classifications. The third part presents, develops and defends the basic normative premisses as a pragmatic alternative to both, metaphysical (Kripke, Devitt) and sceptical approaches (Quine, Kuhn). The main result here is that many (if not all) preconditions for the semantic bahvior of 'rigid' general terms are not, as usually assumed, ontological in nature, but can be usefully framed as part of the pragmatics of general terms in epistemic practices counting with rational revisability. (shrink)

Hilary Putnam has famously undergone some radical changes of mind with regard to the issue of scientific realism and its wider epistemological bearings. In this paper I defend the arguments put forward by early Putnam in his essays on the causal theory of reference as applied to natural-kind terms, despite his own later view that those arguments amounted to a form of 'metaphysical' realism which could not be sustained against various lines of sceptical attack. I discuss some of the reasons (...) for Putnam's retreat, first to the theory of 'internal realism proposed in his middle-period writings, and then to a commonsense pragmatist stance which claims to resituate this whole discussion on ground that has not been trodden into ruts by the contending philosophical schools. In particular I examine his protracted engagement with various forms of anti-realist doctrine, with Wittgenstein's thinking about language-games or meaning-as-use, and with a range of skeptical - relativist positions adopted in the wake of Quine’s influential attack on the two last 'dogmas' of logical empiricism. My paper seeks to show that Putnam has been over-impressed by some of the arguments — from these and other sources — which he takes to constitute a knock-down case against the kind of externalist and causal-realist approach developed in his early essays. It concludes by re-stating that position in summary form and relating it to other, more recent defences of causal realism in epistemology and philosophy of science. (shrink)

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.