putnam's indeterminacy argument: the skolemization of absolutely everything

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CARSTEN HANSEN PUTNAM'S INDETERMINACY ARGUMENT: THE SKOLEMIZATION OF ABSOLUTELY EVERYTHING* (Received 25 April, 1986) I. THE ARGUMENT PRESENTED The purpose of this paper is to examine to what extent considerations put forward in Putnam's well known paper 'Models and Reality' con- stitute an indeterminacy argument against realism. 1 The argument is presented as an 'extension' of Thoralf Skolem's arguments in connec- tion with the L6wenheim-Skolem paradox. According to Putnam, it establishes that the "centrality of the classical notions of truth and reference" cannot be preserved without postulating non-natural mental powers. (M & R, p. 1) I shall discuss the argument in its general form where it concerns the possibility of giving a satisfactory realistic account of what it is that gives language a determinate interpretation. Part of 'Models and Reality' attempts to establish that the indeter- minacy result specifically affects realism in set theory, but I shall not be discussing this matter. 2 The argument which I take to be the core of 'Models and Reality' merits a detailed analysis. Presenting it Putnam in effect claims that model theoretic results - notably the L6wenheim-Skolem theorems (henceforth L-S theorems) - can be used to adjudicate in the semanti- cal and ontological discussions of realism and anti-realism. As a matter of fact, a first reading of 'Models and Reality' leaves one with the (incorrect) impression that only model theoretic premises figure in the argument. That such an argument, if possible, should appeal to philos- ophers is obvious. Given seemingly incontestable premises (namely some of the most basic rest/Its in a mathematical discipline) sound arguments leading to progress in a controversial philosophical disci- pline might finally be attainable. At least part of what has drawn philosophers towards mathematics, throughout the history of Western philosophy, has been the hope of importing the clarity and stringency Philosophical Studies 51 (1987) 77-99. 1987 by D. Reidel Publishing Company.

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Page 1: Putnam's indeterminacy argument: The Skolemization of absolutely everything

CARSTEN HANSEN

P U T N A M ' S I N D E T E R M I N A C Y A R G U M E N T : THE

S K O L E M I Z A T I O N OF A B S O L U T E L Y E V E R Y T H I N G *

(Received 25 April, 1986)

I. THE ARGUMENT PRESENTED

The purpose of this paper is to examine to what extent considerations put forward in Putnam's well known paper 'Models and Reality' con- stitute an indeterminacy argument against realism. 1 The argument is presented as an 'extension' of Thoralf Skolem's arguments in connec- tion with the L6wenheim-Skolem paradox. According to Putnam, it establishes that the "centrality of the classical notions of truth and reference" cannot be preserved without postulating non-natural mental powers. (M & R, p. 1) I shall discuss the argument in its general form where it concerns the possibility of giving a satisfactory realistic account of what it is that gives language a determinate interpretation. Part of 'Models and Reality' attempts to establish that the indeter- minacy result specifically affects realism in set theory, but I shall not be discussing this matter. 2

The argument which I take to be the core of 'Models and Reality' merits a detailed analysis. Presenting it Putnam in effect claims that model theoretic results - notably the L6wenheim-Skolem theorems (henceforth L-S theorems) - can be used to adjudicate in the semanti- cal and ontological discussions of realism and anti-realism. As a matter of fact, a first reading of 'Models and Reality' leaves one with the (incorrect) impression that only model theoretic premises figure in the argument. That such an argument, if possible, should appeal to philos- ophers is obvious. Given seemingly incontestable premises (namely some of the most basic rest/Its in a mathematical discipline) sound arguments leading to progress in a controversial philosophical disci- pline might finally be attainable. At least part of what has drawn philosophers towards mathematics, throughout the history of Western philosophy, has been the hope of importing the clarity and stringency

Philosophical Studies 51 (1987) 77-99. �9 1987 by D. Reidel Publishing Company.

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78 CARSTEN HANSEN

of mathematical reasoning to the domain of philosophical thought. Given this, the need arises for a clarification of the conditions for the application of mathematical results to the philosophical problems under discussion. Even those who from the outset are convinced that Putnam's model theoretic argument couldn't be cogent should recognize the importance of being able to state why this is so. This paper can therefore be seen as a case-study of the relevance of the L-S theorems for the semantical and ontological problems of realism and anti-realism. 3 As several authors have noted, 'Models and Reality' isn't exactly a paradigm of clarity; for this reason I shall devote some space to exegetical matters. Needless to say, my ambition is to present the strongest and most interesting argument possible on the basis of Putnam's suggestions.

Putnam claims that the argument applies to realism in its most interesting form - moderate, naturalistically minded realism. Empiri- cally questionable appeals to "non-natural mental powers of grasping objects" are rejected as "unhelpful epistemology and almost certainly bad science as well". (M & R, p. 5) If the argument is in fact to apply to an interesting form of realism any one of its premises must either be implied by realism, or be justifiable independently of realism. Also, since the argument is supposed discriminately to refute realism, at least one of its premises must be one to which the realist, distinctively, is committed. Unless the anti-realist can renounce allegiance to at least one of the premises of the argument, it will amount to a general 'paradox' rather than a problem specifically applying to realism.

By realism I here understand minimal realism, which I take to be a position committed to the notion of a mind-independent world and to a non-epistemic notion of truth. (I shall discuss this in greater detail later). I take these two commitments to embody what is central to all realist positions. Of course, to justify adequately the above as an acceptable characterization of realism, as the term has been used in various contexts, far exceeds the scope of this paper. Suffice it to say that it squares with Michael Dummett 's characterization of the central tenets of realism:

The primary tenet of realism, as applied to some given class of statements, is that each statement in the class is determined as true or not-true, independently of our knowledge, by some obiective reality whose existence and constitution is, again, independent of our knowledge2

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PUTNAM'S INDETERMINACY ARGUMENT 79

Given Putnam's acknowledged debt to Dummett in connection with the problems concerning realism, this seems reasonable. Putnam does not direct his argument against minimal realism, but rather against a position he labels 'metaphysical realism'. However, ! do not believe that 'metaphysical realism' is suitable as a basic position in terms of which the problems pertaining to realism and anti-realism can fruit- fully be discussed. Apart from being committed to the minimal realist theses, the metaphysical realist is, as Colin McGinn has noted, credited with a variety of beliefs that are not implied by minimal realism. 5 In so far as the premises of the argument figure among the theses con- stituting metaphysical realism and the argument is valid, the meta- physical realist is refuted by Putnam's argument. I shall argue, how- ever, that some of the premises of that argument are neither implied by minimal realism, nor intrinsically reasonable. My claim, then, is that by refuting 'metaphysical realism' Putnam fails to refute the minimal realist's theses, and that there are extended positions more plausible than metaphysical realism available to anyone attracted to the basic tenets of realism. I shall be arguing for the following main points: (i) pace Putnam's explicit remarks, the Lrwenheim-Skolem theorems do not contribute in any substantial way to his argument against realism; (ii) the argument presupposes a highly restrictive epistemological premise that does not follow from realism and which furthermore can- not be given much in the way of independent support, Therefore, the argument does not constitute a cogent challenge to realism.

The aim of the argument, in its general form, is stated most clearly in the section entitled 'The Skolemization of absolutely everything'. It establishes, we are told, that on realist premises:

...even terms referring to ordinary material objects - terms such as ~ and 'dog' - are interpreted differently in the different 'intended' models. It seems, this time, as if we can- not even refer to ordinary middle-sized physical objects except as formal constructs variously interpreted in various models... It seems to be absolutely impossible to fix a determinate reference (without appealing to non-natural mental powers) to any term at all. (M & R, p. 16)

The argument has the following general form:

(1) On realist premises, being unable to single out the intended interpretation of an ideal theory Timplies indeterminacy of interpretation for natural languages. (Premise)

(2) A position implying indeterminacy of interpretation must be rejected. (Premise)

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8 0 CARSTEN HANSEN

(3) It is not possible, on realist premises, to single out the intended interpretation of an ideal theory T, without making use of non-natural mental powers of grasping objects. (Lemma to be established)

(4) A realist position which does not make use of non-natural mental powers of grasping objects implies indeterminacy of interpretation. (Lemma- follows from (1) and (3)) (5) A realist position which does not make use of non-natural mental powers of grasping objects, must be rejected. (Lemma- follows from (2) and (4)). QED

The indeterminacy argument is thus formulated in terms of whether or not it is possible, on realist premises, to fix, or to single out, the intended interpretation of an ideal theory T. Putnam's main concern in 'Models and Reality ' is to establish Lemma (3). He has on several occasions remarked that the notion of an (epistemically) ideal theory is far from clear. 6 That the argument is nonetheless formulated in terms of such a theory seems to reflect Putnam's desire to make as many concessions to the realist as he finds he reasonably can. I shall not at tempt to discuss this notion at any great length, but will instead concentrate on the properties of T that are of relevance to the argu- ment.

In order to render plausible the claim that the conclusion of the argument applies to language in general, T is taken to represent "a formalization of total science.., or even a formalization of all of our beliefs (whether they count as science or not).. ." (M & R, p. 3) Since the L-S theorems are regarded as applicable to it, T must be first order and satisfiable in an infinite domain. Furthermore, T is assumed to be partially interpreted since it satisfies a set of so-called 'operational constraints'. These constraints are expressed in terms of a valuation OP, which assigns (correct) extensions to each n-place O-term (obser- vation-term) of the language in terms of the n-tuples of elements of S on which it is defined, where S is the liberally construed set of macro- scopically observable things and events. (M & R, pp. 11-13) The ideal theory also satisfies a set of ' theoretical constraints' - those we would impose in the ideal limit of inquiry. (M & R, pp. 12-13) It is somewhat confusing to be told that the theoretical constraints are represented by the theory itself. (M & R, p. 16) Apparently, the theoretical constraints are the axioms of the theory, or axiomatic extensions of it. 7 There is a more extensive discussion of ' theoretical constraints' in Reason, Truth and History, where they are characterized as constraints that refer to

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PUTNAM'S INDETERMINACY ARGUMENT 81

the formal properties of theories. 8 A clear delineation of the notion of a formal property would be obtained by stipulating that the notion is to apply only to those properties that can be attributed to theories when they are considered as purely syntactic structures. Theoretical con- straints would then be those that refer to the presence or absence of such formal properties. Putnam uses the notion of a formal property in a much wider sense, however. Examples of theoretical constraints which he offers are simplicity, conservatism, and the 'principle of determinism' - the principle that "an admissible interpretation is such that it turns out to be true that different effects always have different causes". (RTH, p. 31) It would seem that a theory could satisfy a con- straint such as the latter only relative to an interpretation. 1 shall not, therefore, rely on the account given in Reason, Truth and History, but only on the remarks in 'Models and Reality'.

The conditions that have to be satisfied for Tto be 'ideal' restrict the class of acceptable models somewhat, but taken alone they fail to single out an intended model. Since T is assumed to be a formalization of 'total science' it will contain the axioms of Peano arithmetic, and the class of acceptable models of T will be restricted in that finite models are ruled out. By the L-S theorems, however, T will admit models of all infinite cardinalities. All acceptable models of T also have to be Compatible with the partial valuation OP. The 'non-observational' terms of Tare not fixed by OP, however, nor can OP place restrictions on the cardinality of acceptable models of T. Given that the theoretical constraints are, in effect, the axioms of T, or axiomatic extensions of it, they will not serve to single out an intended model either - if the L-S theorems apply to a theory, they will apply to any axiomatic extension of it. Putnam's claim that realism implies indeterminacy of interpreta- tion presupposes that a realist cannot legitimately appeal to any inter- pretation fixing resources over and beyond the operational and theo- retical constraints formulated by him:

If we have more available with which to fix the intended model than merely theoretical and operational constraints, then the problem disappears. (M & R, p. 14)

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82 CARSTEN HANSEN

2, M O D E L - T H E O R E T I C ASSUMPTIONS AND THE IMPORT OF THE

L-S THEOREMS

The challenge to realism, then, is that of singling out the 'intended' model of T, and Putnam's claim is that there are no legitimate means of doing so available to a realist. The fact that a language, when represented as an ideal formalized theory, will have many models should only cause concern, however, if realism is committed to con- sidering the interpretation of a natural language to consist in, or in some vaguer sense to be representable by, a model in the standard model-theoretic sense. Putnam does not state precisely the extent to which he considers realism to be committed to the use of model- theoretic notions and model-theoretic semantics. I shall not, however, attempt to determine the precise nature of the model-theoretic assumptions that must be attributed to realism for the argument to work. Nor will I discuss in any detail whether any interesting form of realism is, in fact, committed to considering the interpretation of a natural language to be representable by a model in the standard model-theoretic sense. It should be noted, though, that the application of model-theoretic notions to natural languages is far from un- problematic. If the interpretation of a language is taken to consist in, or to be representable by, a model, then one is presumably committed to explaining a particular speaker's understanding of the language in terms of knowledge of such a model. Such an explanation might well make communication a mystery: we might speak with different models in mind and never become aware of the fact. 9

As far as I can see, Putnam views anti-realism as unaffected by the problem of 'skolemization' precisely because it is not committed to accounting for the semantics of natural language in terms of models in the standard model-theoretic sense. At one point he remarks that the correct reaction to his argument is "to develop a theory on which inter- pretations are specified other than by specifying models". (M & R, p. 14) It seems quite natural to ask why a realist should be precluded from adopting a similar position. Espousing the central tenets of realism, i.e. minimal realism, does not logically commit one to specifying inter- pretations in terms of models. Voicing this objection amounts to rejecting premise (1) in the argument, a premise for which Putnam

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PUTNAM'S INDETERMINACY ARGUMENT 83

does not give any argumentative support. Plantinga expresses this in the following manner:

The facts Putnam points to would be presently significant, I think, only if we had some reason to think that the terms of our language get their meanings or extensions, somehow, by virtue of the set theoretical models of first order formalizations of the body of our beliefs? ~

Though 1 agree with Plantinga, I shall, for the sake of argument, accept the manner in which Putnam has formulated the challenge and con- sider his reasons for claiming that a realist cannot legitimately single out a model other than by formulating theoretical and operational constraints in Putnam's sense.

Given Putnam's repeated insistence on the centrality of the L-S theorems, an assessment of their contribution to the argument is needed. Briefly, they concern the existence and cardinality of models of syntactic structures which are satisfiable in infinite domains. Collec- tively they state that if a set of first order sentences has an infinite model it will have models of every infinite cardinality.l I Glossing the import of the L-S theorems with respect to his argument, Putnam states:

What Skolem really pointed out is this: no interesting theory (in the sense of first order theory) can in and of itself determine its objects up to isomorphism. (M & R, p. 23)

The inability to determine its models up to isomorphism follows, of course, from the fact that any theory to which the L-S theorems apply will have models of different cardinalities. The function of the L-S theorems in the argument seems, then, to be to establish that signifi- cantly different (non-isomorphic) models of even an ideal theory T actually exist. The rest of the argument would then be the attempt to establish that a realist cannot legitimately appeal to anything beyond T, axiomatic extensions of T, and a partial interpretation OP, when attempting to single out an intended model.

A strong case can, however, be made for the claim that the L-S theorems do not, contrary to Putnam's claims, Contribute to the argu- ment in any direct sense. This is most clearly seen by focusing once again on the nature of the challenge to realism. Putnam's emphasis on the fact that T fails to determine its models up to isomorphism, suggests that he regards the challenge as that of singling out a class of isomorphic models.

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84 C A R S T E N H A N S E N

In mathematical contexts this would, to all intents and purposes, suffice as a specification of the intended interpretation of a theory. Mathematical theories are, in general, designed to characterize struc- tures rather than particular realizations of structures. When an iso- morphism class of models can be singled out this means that a unique characterization of the structure shared by the members of the iso- morphism class has been obtained. The elements and subsets of the domain are, in this connection, considered solely as abstract, mathe- matical objectsJ e Having determined a model up to isomorphism would not, however, suffice to answer the challenge as it is formulated in the general claim cited initiallyJ 3 To use one of Putnam's examples, a model in which the extension of the term 'cat' was a set of dogs or cherries would be unacceptable even if the truth-values of the sen- tences of the language happened to be the same as under the intended interpretation. The general point is that the extensions of the terms of the language are not fixed within a class of isomorphic models. Per- muting the domain of a model results in a change of the extensions of the terms of the theory, but since a permutation is simply a one-to-one mapping of every element of the domain on to another, isomorphism is preserved. In Putnam's argument this means that all terms whose extensions are not fixed by the valuation OP will be indeterminate within a class of isomorphic models. What is needed to meet the challenge formulated in the general claim, then, is the ability to distin- guish between isomorphic models, or the ability to fix the extensions of individual terms. Since this challenge could be formulated even if it were granted that a realist had the means of determining a model of T up to isomorphism, it seems reasonable to conclude that the L-S theorems do not contribute substantially to the argument. This inter- pretation is supported by the fact that Putnam, in Reason, Truth and History (pp. 217-18), presents another argument, dispensing with the L-S theorems, which also purports to establish that it is not possible, on realist premises, to fix the reference of the individual terms of the language. This argument differs from the general indeterminacy argu- ment of 'Models and Reality' in that it is formulated within the frame- work of a possible-worlds semantics. Putnam claims that:

. . .even if we have constra ints o f whatever na ture which determine the t ru th-value of

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PUTNAM'S INDETERMINACY ARGUMENT 85

every sentence in every possible world, still the reference of the individual terms remains indeterminate. (RTH, p. 35)

The argument shows that for an arbitrary possible world wj, given any non-trivial interpretation I of a language L, there will exist a second interpretation J, which assigns extensions to the predicates in a manner differing f rom/ , but which makes the same set of sentences of L true as does L What is important in the present context is that the argument focuses on the problem of distinguishing between iso- morphic models of the same set of sentences of a language L - the model correlated with the interpretation J is obtained by a permuta- tion of the domain of/ . This means that even ifa realist had the means of determining a model up to isomorphism, the indeterminacy charge could still be upheld. Given that a realist is restricted to determining the interpretation of a language on the basis of truth-value distribu- tions alone, then realism does imply indeterminacy in the manner stated. I myself, however, agree with Simon Blackburn when he suggests that:

...the facts which do determine the interpretations of subsententiat components do not lie where Putnam looked for them. They lie in the psychology of the speaker and the history of their terms. 14

This time the crucial point is that the realist is restricted in being allowed to appeal only to truth-value distributions when attempting to single out a model, a restriction for which Putnam gives no argu- mentative support. I shall not attempt to develop Blackburn's objec- tion to the argument in Reason, Truth and History into a detailed reply. My primary reason for referring to that argument is to support the claim that the L-S theorems do not, pace Putnam, contribute to the argument in 'Models and Reality' in any substantial way.

It could be argued, however, that the L-S theorems make it possible to express with greater precision the degree to which realism implies indeterminacy of interpretation by specifying the extent to which T can determine the properties of its models - given, of course, the correctness of the indeterminacy claim, and that one accepts the model theoretic setting of the argument. This would accord well with their status as limitative theorems. Given Putnam's characterization of T, however, the determinative power of T is significantly weaker than

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86 CARSTEN HANSEN

indicated by Putnam's explicit remarks. Being unable to determine its models up to isomorphism, T cannot be categorical. As the following

argument shows, however, T will not even be alpha-categorical for

any cz >~ ~0:

(1) G6del's first incompleteness theorem. If a theory T has sufficient expressive power to permit G6del-coding, then Tis either incomplete or inconsistent.

(2) If T is a consistent theory with no finite models and which is alpha-categorical for some infinite ~z, then Tis completeJ 5

(3) No axiomatic theory, with an interesting expressive power in the abovementioned sense, can be alpha-categorical for any ~r >~N0. QED

Not only will T have non- isomorphic models of different cardinali- ties, it will also have non- isomorphic models of the same cardinali ty

~z for every ~ >/~0. Tha t this argument does, in fact, apply to the ideal theory T is apparent for the following reasons: a sufficient condi-

t ion for T having expressive power as required in (1) is that T con- tains the axioms of Robinson's ari thmetic, which forms a subset of Peano ar i thmet ic ) 6 Since T is supposed to be a formalization of

' total science', this condit ion is obviously satisfied. T could then be

complete only if it is formalized using second order logic, or if it is not axiomatic (i.e., such that there is a decidable subset of T whose con- sequences are just the theorems of T). The former possibility is not

an available option, since the L-S theorems, which do not hold for second order logic are taken to apply to T. As for the second option, Pu tnam remarks, in 'Models and Reality' , about the implausibili ty of ever obtaining a complete set of axioms for set theory:

...for one thing, a complete set of axioms would have to be non-recursive, and it is hard to envisage coming to have a non-recursive set of axioms in the literature or in our heads even in the unlikely possibility that the human race went on forever doing set theory. (M & R, p. 5)

These considerations apply to any non-axiomatizable theory. Allow- ing T to be non-axiomatizable would involve attributing to those who are assumed to grasp it means of doing so that could hardly be 'naturalistically acceptable' . This would not accord with Putnam's intent ion of arguing against 'moderate ' realism. Thus the latter possibility of securing the completeness of T is not an available opt ion either. Fur thermore , under the abovement ioned assumptions (consistency, axiomatizabil i ty and sufficient expressive power),

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PUTNAM'S INDETERMINACY ARGUMENT 87

the ideal theory T will not even be able to single out a class of ele- mentarily equivalent structures. (Elementary equivalence being, of course, a much weaker notion than isomorphism). Two structures MI and M2 are elementarily equivalent (MI-= M2) if, for any L-sen- tence 6, we have M1 ~ 6r ~ 6. Under the given assumptions,

M1, M2 ~ T implies Ml -= M 2

only if the deductive closure CI(T) of T is maximally consistent. ~7 That structures that are not elementarily equivalent will satisfy T is shown in the following argument:

(1) Robinson's arithmetic is essentially incompleteJ 8

(2) An ideal theory will contain the axioms of Robinson's arithmetic and will there- fore also be incomplete.

(3) I f p is a sentence of L such that neither T~-p nor T~-~p, then both TUp and T tO =p will be consistent.

(4) Structures satisfying T tOp and T tO ~ p respectively will, of course, also satisfy T, but will not be elementarily equivalent. QED

Given the premises of the argument, realism implies a degree of indeterminacy significantly greater than that indicated by Putnam. The conclusions to be drawn from this discussion are first, that the L-S theorems do not contribute to the argument since the chal- lenge to realism can be formulated even if the realist is granted to com- mand resources for determining a model up to isomorphism; and secondly, that the L-S theorems are not an accurate expression of the degree of indeterminacy that would be implied by realism, given the cogency of Putnam's argument.

3. THE SCOPE OF THE INTERPRETATION FIXING RESOURCES

OF REALISM

The core of Putnam's argument against realism, then, is his claim that a realist cannot legitimately invoke model-fixing resources that improve upon T's own ability to determine the properties of its models. It is not immediately obvious how Putnam supports this claim. The only 'moderate' realist move considered at any length is the appeal to a causal theory of reference to explain how exten- sions are fixed, and even this move is rejected in a quick and off-hand

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88 CARSTEN HANSEN

manner. 19 Putnam considers such an appeal to be question-begging. According to him, a causal theory of reference cannot be legitimately considered as anything more than an uninterpreted extension of the ideal theory:

IT]he problem is that adding to our hypothetical formalized language of science a body of theory entitled 'Causal Theory of Reference' is just adding more theory. But Skolem's argument, and our extensions of it, are not affected by enlarging the theory. [And,] ...unless the word 'causes' (or whatever the causal predicate or predicates may be) is already glued to one definite relation with metaphysical glue, this does not fix a deter- minate extension for 'refers' at all. (M & R, p. 18)

Putnam's objection is that in giving an account of how extensions are fixed the realist is using language which is illegitimately assumed to have a determinate interpretation. It is important to know exactly what question is being begged. There are two distinct ways in which the challenge to single out the 'intended model' may be construed. First, the challenge might be interpreted in such a manner that it would be met by providing an account of how, i.e. by what means, extensions are fixed. Secondly, the challenge might be construed as the demand that some sort of proof be provided that our terms do have determinate extensions (interpretations).

As Putnam himself notes, causal theories of reference address the former challenge. (M &R, pp. 17-18) In the context of giving an account of the means by which our terms acquire determinate exten- sions there can be nothing wrong in assuming that the language in which the account is given has a determinate interpretation. No attempt to explain how extensions are fixed could fail by begging the question whether language has a determinate interpretation, and this is what Putnam's objection amounts to. Only in the context of attempting to establish that our terms do have determinate extensions would it be illegitimate to assume that that is the case.

There can hardly be any doubt that no account of how extensions are fixed could ever do the job of establishing that language has a determinate interpretation. With respect to the latter challenge any such account, and specifically also an anti-realist account in terms of verification procedures, would just amount to 'adding more theory' in Putnam's sense. An anti-realist account of what it is that gives language a determinate interpretation would also be part of his or her

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PUTNAM'S IN D E T E R MIN A C Y ARGUMENT 89

'total' theory and therefore equally unable to restrict the number of models of T. The mistake involved in trying to put any such account to work in this manner can be put as follows: it amounts to trying to get the account itself (the set of sentences explaining by what means language acquires an interpretation) to fix reference, whilst what such a theory aims to do is to provide an account of what it is that deter- mines the extensions of the terms. Putnam might be interpreted as making just this point. If so, the following questions present them- selves quite naturally. Under what circumstances does the problem of establishing that language actually has a determinate interpretation arise? More specifically, why does Putnam think that it is incumbent upon realism to prove that our terms have determinate extensions? One might very well ask how any position could prove that language has a determinate interpretation. A legitimate parallel to this question is to ask how anyone could prove that it is possible to refer deter- minately. What could anyone do except give examples of successful acts of reference (which presuppose the possibility of referring deter- minately)?

What must be at issue, rather, is whether a given philosophical position - in casu realism - involves a commitment to some thesis that precludes the possibility of language having a determinate inter- pretation. It is only by pointing to such a thesis that it can be argued that no account could be provided, given the premises of that posi- tion, of how extensions are fixed. It is my claim that Putnam's argu- ment must be construed in this manner; and I will proceed to isolate the crucial thesis to which realism is purportedly committed.

In gupport of his claim Putnam endorses two constraints on what a realist can appeal to when trying to meet the challenge set by him: (1) What determines the interpretation o f a term can only depend on cir- cumstances known by the linguistic community and therefore in prin- ciple accessible to each individual member. As Dummett has pointed out, there are substantial reasons for maintaining this constraint owing to the relations between meaning (interpretation), understand- ing ("knowing the meaning of') and knowledge. For this reason the constraint may be viewed as motivated independently of realism. 2~ At one point, Putnam explicitly states that an appeal to a causal theory of reference violates this constraint:

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90 CARSTEN HANSEN

In the context of a twentieth-century world view, by contrast, to say in one's most intimi- dating tone of voice 'I believe that causal connections determine what our words corre- spond to' is only to say that one believes in a one-knows-not-what which solves our problem one-knows-not-how. (R & R, p. xii)

In addi t ion to this, P u t n a m cla ims that there are substant ia l con-

straints on w h a t can be k n o w n and, on realist premises , appea led to.

This restr ic t ion is given express ion in the c la im that : (2) we have knowledge of, or access to, (mental) representations alone. T h e clear-

est s t a t emen t o f this cons t ra in t occurs in the i n t roduc t ion to Realism and Reason, where it is expl ici t ly related to the i n d e t e r m i n a c y argu-

m e n t o f 'Mode l s and Real i ty ' . I shall quote at length since this is

cent ra l to m y in te rpre ta t ion o f P u t n a m ' s a rgumen t :

Early philosophical psychologists - for example, Hume - pointed out that we do not literally have the object in our minds. The mind never compares an image or word with an object, but only with other images, words, beliefs, judgements, etc. The idea of a comparison of words or mental representations with objects is a senseless one. So how can a determinate correspondence between words or mental representations and exter- nal objects ever be singled out? How is the correspondence supposed to be fixed?... If we limit psychology, for the moment, to 'solipsistic' description, description of what happens in the individual considered in isolation from his environment, then no psychological facts in this narrow sense, no facts about introspectible mental phenom- ena (or even unconscious mental phenomena) and no facts about brain processing can fix any correspondence between a word or ~representation' and anything external to the mind or brain.

The only paper in this work that makes serious use of technical logic ('Models and Reality') is not an attempt to solve this problem (how the 'correspondence' is fixed), but rather a verification that the problem really exists. (R & R, pp. viii-ix)

'Mode l s and Real i ty ' , then, is supposed to verify tha t reference is

radica l ly i nde t e rmina t e on realist premises - given the thesis tha t the

m i n d has access on ly to its own representa t ions . U n d e r s t a n d i n g the

no t ion o f men ta l r epresen ta t ion invo lved is therefore crucial . O n the

basis o f the above q u o t a t i o n it seems reasonable to in terpre t 'men t a l

r ep resen ta t ion ' as m e a n i n g ' p sycho log ica l state in the na r row sense ' - as cha rac te r i zed by P u t n a m in ' T h e m e a n i n g o f " m e a n i n g " '. A

psycho log ica l state in this sense is one tha t does no t p re suppose the

exis tence o f " a n y indiv idual o the r than the subject to w h o m that state is asc r ibed" . 21 Fu r the r c o m p a r i s o n with P u t n a m ' s earl ier views in

' T h e m e a n i n g o f " m e a n i n g " ' is cal led for. T h e sentence in the

second to last quo t a t i on beg inn ing " I f we l imit p sycho logy . . . " echos the po in t P u t n a m was c o n c e r n e d to establish wi th his T w i n Ear th

a rguments . T h e point , e m b o d i e d in the s logan tha t 'mean ings a ren ' t

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PUTNAM'SINOETERMINACY ARGUMENT 91

in the head ' , is that the psychological states of individual speakers

(taken individual ly or collectively) fail to de termine the extensions of

the te rms of the language. Tha t Pu tnam, in 'Models and Real i ty ' and

Reason , Truth a n d His tory does not discuss his previous work in

' T h e meaning o f " m e a n i n g " ' is quite puzzling. For in the latter, after

having argued that 'meanings a ren ' t in the head ' , he goes on to give a

realistically acceptable account of the de te rmina t ion of extensions in

a non-solipsistic, social setting:

We have now seen that the extension of a term is not fixed by a concept that the indi- vidual speaker has in his head, and this is true both because extension is, in general, determined socially - there is division of linguistic labor as much as of'real' labor - and because extension is, in part, determined indexically. The extension of our terms depends upon the actual nature of the particular things that serve as paradigms, and this actual nature is not, in general, fully known to the speaker. (MoM, p. 245)

It needs to be explained why P u t n a m no longer considers this

explanat ion to be avai lable to a realist. Seeing why this is so will

i l luminate the not ion of menta l representa t ion involved in 'Models

and Real i ty ' . In the in t roduct ion to R e a l i s m & R e a s o n he considers

the following object ion to the a rgument of 'Models and Real i ty ' :

Your argument only shows that reference is not fixed by anything 'inside the head'. But that is no problem - why can't reference be fixed by something non-psychological?

To which P u t n a m responds:

The answer, quite simply, is that the idea that the 'non-psychological' fixes reference - i.e. that nature itself determines what our words stand for - is totally unintelligible. [And he continues, as I have already quoted], ... to say in one's most intimidating tone of voice 'I believe that causal connections determine what our words correspond to' is only to say that one believes in a one-knows-not-what which solves our problem one- knows-not-how. (R & R, p. xii)

It is remarkable that when discussing whether the 'non-psychologica l '

can de termine reference, the only possibil i ty P u t n a m considers is

whether 'na ture itself ' can provide the desired result. He comple te ly

ignores the possibili ty of appeal ing to the role of the linguistic com- mun i ty and the p h e n o m e n o n of indexicali ty in de termining the inter- pre ta t ion of the language - at least when discussing what he now con- siders to be legit imate realist options. It is also remarkable that causal connect ions are rendered comple te ly inaccessible in P u t n a m ' s account - an appeal to them would be an appeal to a " o n e - k n o w s -

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not-what which solves our problem one-knows-not-how". (It is thus because of failing to meet the second constraint that an appeal to 'causal connections' violates the first).

A radical interpretation of the claim that we have access only to our own representations is called for, if the realist's predicament is to be regarded as that sketched above. The challenge set to realism must be construed as that of establishing correspondence-relations between two sets of things given access to one set only. This manner of formu- lating the challenge is clearly expressed in the following passage from Reason, Truth and History, where the realist's d i lemma is explained:

To pick out just one correspondence between words or mental signs and mind indepen- dent things we would have already to have referential access to mind independent things. You can't single out a correspondence between two things by just squeezing one of them hard (or doing anything to just one of them; ... (RTH, p. 73)

The representations to which we have access, if this is to be an account of the realist's situation, are in a significant respect like Lockean 'ideas' - they are mental entities of a sort that form a 'veil ' between us and the external world. Only on such a construal would it be plausible to claim that an exhaustive division between the 'psycho- logical' and the 'non-psychological ' could be drawn with the 'psycho- logical' understood as something to which we have direct access, and the 'non-psychological ' as something to which we, in effect, have no access. 22

When trying to form an interpretation of the argument that takes account of what Putnam actually writes, the realist must be seen as constrained in having knowledge of, or access to, mental representa- tions only. The things with which determinate correspondences are to be established, if language is to have a determinate interpretation, are accessible only indirectly, via representations. What constrains realism in the argument is thus an epistemological thesis as to what can be known, and therefore appealed to, on realist premises. This epistemological premise leaves the argument with hardly any force against realism - it does not seem at all plausible in its own right, nor does it follow from realism. To interpret Putnam's argument as con- taining what may reasonably be called a Lockean epistemological premise requires some justification. It can be provided, however. First, it is the only interpretation which takes account of the con-

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spicuous occurrence in Putnam's writings of terms and phrases such as singling out 'a determinate correspondence between words or mental representations and external objects' and grasping 'what is outside the mind'. Taking a closer look at Putnam's initial sketch, in 'Models and Reality', of the "three main positions on reference and truth" it seems plausible to say that they are defined mainly by how they choose to react to this thesis. (M & R, p. 1) The platonist reacts to the problem by positing 'mysterious' and 'non-natural' mental powers of grasping the non-representational. The moderate realist, not wanting to operate with such mysterious powers of reference and yet accepting that the mind has access only to its representations, is forced to accept defeat. The verificationist reacts by claiming that the semantics of the language ought to be explained in another manner. Abandoning 'referential semantics' is suggested as the correct reaction to the threat of indeterminacy.

Secondly, the 'access only to representations' thesis explains Putnam's quick and off-hand rejection of realist attempts to invoke further means of specifying interpretations. Given the thesis that the immediate objects of knowledge are mental representations, then it will, as Putnam claims, be question-begging to assume the possibility of having knowledge of the external, non-representational objects in terms of which the language is to be interpreted. What seems to have become something of a standard answer to Putnam, namely, to claim that "causal relations between the world and the use of language" and "non-linguistic facts about the world" contribute substantially to giving language a determinate interpretation cannot be correct unless the 'access only to representations' thesis is repudiated. 23 (The repudiation of which, of course, will not suffice to make the causal account correct).

Finally, the 'access only to representations' thesis explains Putnam's very brief rejection of second order formalizations as a way of meeting the challenge. In an instructive article, Pearce & Rantala claim that Putnam, by disregarding the possibility of higher order formalizations, fails to consider the very thing to which a realist would appeal:

It is an ad hoc move, however, for Putnam to assume without further argument that all theoretical constraints must be formalized in first order logic.., this assumption...

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should be carefully questioned; for it turns Putnam's argument into a thesis practically devoid of any content. 24

Putnam does remark on the choice between formalizing in first and second order logic, albeit only once, at which point he rejects the latter as being useless:

Some have proposed that second-order formalizations are the solution, at least for mathematics; but the 'intended' interpretation of the second order formalism is not fixed by the use of the formalism (the formalism itself admits so-called 'Henkin- models', i.e., models in which the second order variables fail to range over the full power set of the universe of individuals), and it becomes necessary to attribute to the mind special powers of'grasping second order notions'. (M & R, p. 23)

On the face of it, this rejection of the utility of second order formali- zations is inconclusive. As Putnam notes, non-standard 'Henkin- models' only appear under the assumption that the variables fail to range over the fullpower-set of the universe of discourse. What needs to be established is the range of our epistemic access to the power-set of the universe of discourse; and it appears possible for a realist to claim that we do have access to the full power-set, whence the domain of the second order variables is not restricted. The 'access only to representations' thesis is, however, in itself a radical rejection of this possibility and, in effect, reduces the realist to trying to deter- mine the intended interpretation with the aid of the formalism alone

- as Putnam claims in the quotation just provided. On the assump- tion of the 'access only to representations' thesis, it would therefore not be possible to distinguish between standard models and non- standard 'Henkin-models', and so Putnam's argument does not really depend on a restriction to first order logic. Furthermore, it should be noted that even under the assumption that the variables do range over the full power-set of the domain, a second-order formalization would only suffice to secure determinacy up to isomorphism. Putnam's challenge of singling out a member of a class of isomorphic models, and thus fixing the extensions of the terms, would not thereby have been met.

I have argued that Putnam justifies his rejection of all realist attempts to single out an intended interpretation by claiming that realist commitments preclude any possibility of fixing a model. Putnam does not, however, state that the crucial 'access only to repre-

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sentations' thesis is one to which realism is distinctively committed. The question remains, then, why the argument should not also be applied to non-realist positions if they endorse that thesis. As I quoted earlier, Putnam states that the proper reaction to his argument is to "develop a theory on which interpretations are specified other than by specifying models". (M & R, p. 14) Putnam's view seems to be that the 'access only to representations' thesis does not imply indeterminacy if one is not committed to specifying a model in terms of 'non-representational' objects. However, I shall not examine this claim any further. For the purposes of this paper it suffices that the crucial thesis on which the argument depends has been shown to be one to which realism is not committed, and which furthermore is highly questionable.

4. REALISM AND EPISTEMOLOGY

The general indeterminacy argument of 'Models and Reality' depends crucially, therefore, on there being prior restrictions on what a realist can legitimately appeal to when trying to meet Putnam's challenge. Owing to the first constraint, appeal can be made only to what is known by the linguistic community to contribute to determining the interpretation of the language - to what is, in principle, accessible to the speakers of the language. Claiming further that there are specific restrictions on what can be appealed to, on realist premises, amounts to propounding an epistemological thesis concerning the scope of our capacities for knowledge. The epistemological thesis appealed to in 'Models and Reality' (and RTH) is neither intrinsically plausible, nor does it follow from realism. Thus, the argument does not have much force against realism. In the following, I shall try to indicate that realism - understood as a fundamental semantical and ontological position - does not, in itself, have epistemological implications of the sort that lays that doctrine open to an indeterminacy argument against it, of the type envisaged by Putnam.

Disregarding the quite metaphorical locutions of 'inner representa- tions' and 'external reality', and the Lockean construal of the notion of 'representation' which apparently lies behind them, there does seem to be a sense in which a realist is committed to a claim

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involving the relation between the world and our representations of it. Being committed to the n6tion of a mind-independent world, a realist must insist on reality being (in general) quite independent of our representations of it. The question, then, is whether the sense in which realism is committed to there being a 'gap' between reality and our representations of it makes it question-begging to assume the possibility of access to the (largely non-representational) objects in terms of which the interpretation of the language is to be specified.

This question can be discussed in terms of the realist's views on the relation between truth and our means of recognizing truth. A realist is (I suggest) committed to truth being non-epistemic in the sense that necessary and sufficient conditions for truth cannot be formulated in epistemic terms, e.g., in terms of our means of ascer- taining truth. This commits realism to the further claim that it is not our being in possession of the means of ascertaining the truth of a statement that makes the statement true. But from this it does not follow that, on realist premises, we never have sufficient means of ascertaining truth and, therefore, of acquiring knowledge. For this to follow realism would have to imply fallibilism or scepticism across the board. I take fallibilism, with respect to the cognitive situation of some group of beings, to be the position that there are no standards of epistemic justification, satisfiable by that group, the satisfaction of which is sufficient to imply the truth of statements of a given class. A position implying fallibilism with respect to non-representational reality could not legitimately assume the possibility of access to, or knowledge of, any non-representational aspect of reality, and would therefore be vulnerable to the type of criticism that Putnam levels against realism. Presumably, no one would disagree that fallibilism (or scepticism, I am not distinguishing between the two) presupposes, and therefore implies, realism. However, I am claiming that the converse implication does not hold. Realism does not, by itself, imply fallibilism, though it is consistent with it. Any attempt to argue against realism in the manner set out by Putnam therefore pre- Supposes that a highly restrictive account of our epistemic capacities can be established independently of realism. Putnam hasn't advanced any independent arguments for scepticism or fallibilism, and surely the question whether our means of ascertaining truth are ever suffi-

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cient is one that ought to be answered on a case by case basis, with respect to specific classes of statements or aspects of reality. Putnam cannot, therefore, simply claim that realism is, in general, incom- patible with the possibility of access to non-representational reality. And such a claim is what is needed for it to be reasonable to reject a realistic attempt to explain how extensions are fixed on the ground that it presupposes the possibility of reference and the determinacy of interpretation. Taken alone the basic ontological and semantical claims of realism do not, therefore, have epistemological implications of the sort presupposed by an indeterminacy argument of the type propounded by Putnam.

The upshot of this discussion would seem to be that the considera- tions set forth in 'Models and Reality', do not constitute a new and cogent argument against realism. Putnam does not provide a sound argument to the effect that realism implies indeterminacy of interpre- tation. From this it does not follow that there is an acceptable account, immediately available to someone espousing the basic tenets of realism, of what it is that gives language a determinate interpreta- tion. The question how interpretations are fixed remains. Any suggested account needs to be scrutinized and rejected if found in- adequate. What Putnam has failed to do, if the argument of this paper is correct, is to provide an argument establishing that no such account could be provided on realist premises.

NOTES

* For helpful comments and criticism I would like to thank Michael Dummett, Alexander George, Stig Andur Pedersen, Stig Alstrup Rasmussen and Anne Varty. 1 'Models and Reality' was originally published in Journal of Symbolic Logic XLV (1980), pp. 464-82. I refer to it, abbreviated as 'M & R', as it appears in H. Putnam, Realism and Reason: Philosophical Papers Vol. 3 (Cambridge: Cambridge University Press, 1983), pp. 1-25. Page references following 'R & R' are to other papers in that volume. 2 See Alvin Plantinga, 'How to be an anti-realist', Proceedings of the American Philo- sophical Association 56 (1982), pp. 57-58, and esp. D. Pearce & V. Rantala, 'Realism and Reference: Some comments on Putnam', Synthese 52 (1982), pp. 42-44, for remarks on Putnam's indeterminacy argument applied specifically to realism in set theory. 3 For a different case study, examining Thoralf Skolem's attempts to use the L-S theorems to decide between competing philosophies of mathematics, see Alexander George, 'Skolem and the L6wenstein-Skolem Theorem: A Case Study of the Philo-

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sophical Significance of Mathematical Results', History and Philosophy of Logic 6 1985), pp. 75-89. Michael Dummett, The Interpretation of Frege's Philosophy (London: Duckworth,

1981), p. 434. 5 Colin McGinn, 'Ideal Justifications', rev. of Realism and Reason, by H. Putnam, Times Literary Supplement, 30 Nov. 1983, p. 1307. 6 See, e . g . , M & R , p . 1 2 a n d R & R , p . 161. 7 See, e.g., M & R, pages 3, 5, and 25. 8 H. Putnam, Reason, Truth and History (Cambridge: Cambridge University Press, 1981), pp. 29-32. Henceforth page references to this book will be abbreviated as 'RTH'. 9 The possibility, on these grounds, of genuine, undetectable differences in interpreta- tion among speakers presupposes that the indeterminacy argument is suspended with respect to the understanding of individual speakers. I owe this point to Stig A. Rasmussen. 10 Plantinga, Ol). cit., p. 60. I I For precise formulations of the L-S theorems see, e.g., John Bell and Mosh6 Macbover, A Course in Mathematical Logic (Amsterdam: North Holland, 1977), pp. 168-173. 12 See, e.g., Paul Benacerraf, 'What numbers could not be', Ph[losophicdl Review 74 11965), PP. 47-73.

See this paper p. 78. 14 Spreading the Word (Oxford: Clarendon Press, 1984), p. 301. 15 Bell & Machover, Mathematical Logic, p. 187. 16 Robinson's arithmetic is presented in detail in Tarski et al., Undecidable Theories (Amsterdam: North-Holland, 1953), and cursorily in R. Rogers, Mathematical Logic and Formalized Theories (Amsterdam: North-Holland, 1971). 17 Cl(T)={p/T~-p}, a set �9 is maximally consistent if it cannot be consistently extended by adding to it any sentence p, p e L. 18 That is, Robinson's arithmetic is incomplete, and every consistent axiomatic exten- sion of it is also incomplete. 19 Putnam also briefly considers the possibility of employing second order formaliza- tions. I shall return to this possibility later on in the paper. 2o For precisifications and qualifications, however, see Michael Dummett, Truth and Other Enigmas (London: Duckworth, 1978), pp. 427-28; and The Interpretation of Frege's Philosophy, pp. 74-84. 21 H. Putnam, 'The meaning of "meaning" ', in Mind, Language and Reality (Cambridge: Cambridge University Press, 1975), p. 220. Subsequent references to this ~2aper will be abbreviated as 'MoM'.

The claim that we have access only to representations might be interpreted as the claim that we do not have access to 'unconceptualized' reality - that when thinking of anything it is always thought of as given or identified in a particular way. This matches Frege's notion of the sense of a proper name as a "particular way of identifying an object as the referent of the name" (M. Dummett, Frege: Philosophy of Language (London: Duckworth, second edition, 1981), p. 95.) This claim does not, however, provide support for the thought that we have knowledge of mental representations only and not of reality itself. 23 See, e.g., A. Brueckner, 'Putnam's model-theoretic argument against metaphysical realism', Analysis 44 (1984), pp. 134-40. 24 Pearce & Rantala, 'Realism and Reference', p. 43.

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