Putnam's Model Theoretic Argument
David Banach

(This is an excerpt from a chapter of a longer work)

     Putnam's model theoretic argument can be seen as an extension of Searle's argument to linguistic and representational systems. It argues that even an entire linguistic system taken as a whole cannot determinately refer by itself. It shows once more that the attempt to make representations come alive will be a failure, even if the the representation is as complex and comprehensive as an entire linguistic symbol system.

     Putnam's argument is just another instance of Hegel's insight that representation involves interaction with an object in which different ways of presenting the object are connected and attributed to the object as their causal locus. It should be no surprise, then, that a formal system, a collection of meaningless symbols and rules for combining and manipulating them, should be unable to uniquely determine its own reference. What seems surprising, however, is that model theory, the most powerful tool at the disposal of the attempt to make linguistic representations come alive, should bring about the demise of the attempt. But, again, even this should not surprise one after seeing the structure of Hegel's argument. Hegel's metacritical move was to attempt to represent according to the physical visual model; it will be found that it is impossible to do so according to the presuppositions contained in the model itself. In the same way, the attempt to make linguistic representations come alive in virtue of their formal structure is seen to be impossible because of properties of that very formal structure. We now need to look at Putnam's argument and see why this is so.

     Putnam appropriates the Lowenheim-Skolem Theorem from model theory and extends it to representational systems that include empirical representations. Model theory provides interpretations for formal systems. That is, it provides assignments of individuals, sets, functions, and relations to the various symbols in a formal system.  If such an interpretation makes all the well formed formulas in a system of symbols true, then that interpretation is called a model. Thus truth functional semantics, or the attempt to spell out the meaning of linguistic items by assigning them an extension that preserves the truth of the sentences containing each item is sometimes called model theoretic semantics. It attempts to spell out the meaning of a linguistic system by spelling out a model for it, that is, by defining an interpretation of it that makes all its statements turn out true.

     Putnam's argument is aimed most forcefully, then, at model theoretic semantics, although it has much wider application. It shows that fixing the truth value of a statement in all possible worlds does not fix the reference of the linguistic items that make up the statement. But such an argument would have much wider implications than just the downfall of model theoretic semantics. It would show, in a forceful way, that reference is not determined by truth. That is, it would show that reference is not determined by correctness of representation, by the intrinsic similarity of the representation to its object. It shows that representation is not just similarity, because similarity cannot even determine reference. Even truth cannot bridge the gap between representations and the world and determine a unique relationship between the representation and its intended object. Such an argument would show that representations cannot come alive through their own properties.

     The intuitive idea behind Putnam's argument is quite simple. Even if we know that a statement is true we do not know what it is true of. The standard example here is Quine's gavagai example.1 An anthropologist encountering a culture with an unknown language sees a rabbit go by, upon which a native utters "gavagai". The natives repeat this when ever they see a rabbit, and they assent whenever the anthropologist says "gavagai" in the presence of a rabbit. The anthropologist is pretty sure that "gavagai" is true of the situations in which rabbits are present. Yet they are not exactly sure what it is true of. Should it be translated by "There is a rabbit.", "There is an undetached rabbit part.", "There is a rabbit event.", or "There is the rabbit god."? The point here is that an uninterpreted piece of language cannot determine reference; that is, language seen as a set of meaningless symbols cannot determine its own reference even if it is in some truth or similarity relationship to the world.

     The Lowenheim-Skolem Theorem is the expression of this fact for formal systems. It holds that any formal system that has a model, i.e. any satisfiable system, has a countable (finite or equinumerous with the set of natural numbers 2) model. This was a quite surprising result, since it showed that even systems in which you could prove Cantor's Theorem, which stated the existence of transfinite numbers, had countable models. It showed that there would always be unintended models of any formal system. The formal constraints imposed by the system do not uniquely determine its interpretation. Different interpretations will make the same system true; that is, it will have numerous models. In fact, there is a stronger version of the Lowenheim-Skolem Theorem which requires the Axiom of Choice for its proof that states that every system that has an infinite model has another model which is a subset of the first, so it is easy to see how the number of unintended models could multiply quite quickly.

     Putnam shows that this not only true for the formal systems in number and set theory, but even for a system which incorporated all of our empirical knowledge. This shows that our linguistic representation of the world, even if true, does not determine a definite reference or correspondence relationship to the world. Various different models or ontologies could satisfy the theoretical and operational constraints imposed by our system of knowledge. The theoretical constraints are those imposed by the formal structure of the system. Any model must make all the theorems, or logical truths, of the system true. The operational constraints are the constraints imposed by the inclusion of our empirical knowledge of the world in the system . This is expressed in the system by a set of sentences stating the quantity of all physical magnitudes (mass, heat, electrical charge, gravitational force, etc.) at all space-time points to some arbitrary accuracy. (Putnam 1977, p. 3) Thus, Putnam shows that even a representational system that includes all possible operational constraints, all possible empirical knowledge about the world, would not establish reference to a world beyond our representations. He does this by applying the Lowenheim-Skolem Theorem to a formalization of an ideal empirical theory. Even such a theory would admit of different alternative interpretations that satisfied all the theoretical and operational constraints.

     Putnam shows this by devising a method for constructing unintended interpretations that satisfy all the constraints from the intended model. He does this in Reason, Truth and History (Putnam 1981), giving a technical exposition of the general procedure in an appendix (pp. 217-218) and an example of the method in Chapter Two (pp. 33-35). 3 Rather than go through Putnam's example, which requires a bit of work to understand, I will give a simpler example that illustrates the same feature of representational systems that allows Putnam's argument to work. I will then explain why Putnam's example had to be more complex and how it differs from the one given.

     Consider a very simple formal system. Let it contain only two constants, a and b, two predicate symbols, P1 and P2, and one relation symbol, R. It has no quantifiers, no variables, and no sentential connectives. The operational constraints in the system are exhausted by the only three sentences in the system. We can imagine that the world which it describes existed only for one instant, and the only empirical knowledge possible would be of two objects, their predicates, and their relation at that instant. Let the operational constraints and the sentences of the system , then, be exhausted by: P1a, P2b, and aRb.

     One model for this system would be one that assigned the symbols meanings in a world that consisted of only a circle and a square and in which, at the only instant at which the world existed, the circle was on top of the square. This model would be formally defined in this way:

Circle on Square World:

a- the circular object; call it x.

b- the square object; call it y

P1- circularity, formally defined as {x}.

P2- squareness, formally defined as {y}.

R- on top of, defined as the set of ordered pairs {<x,y>}

This interpretation makes our simple system true. It satisfies all theoretical and operational constraints imposed by the system; hence it is a model of that system.

     But there is another model of the system as well. (in fact, there are indefinitely many.) Consider a world which consists of two dogs, a german shepard and a beagle. At the one instant at which the world is existing, the german shepard is eating the beagle. A model which mapped our system into this world would be defined in this way:

Dog Eat Dog World:

a- a german shepard; call it x.

b- a beagle; call it y.

P1- german shepardness, defined as {x}.

P2- beagleness, defined as {y}.

R- is eating, defined as {<x,y>}.

Each of these models maps the system onto a set of objects, properties, and relationships that satisfy the system; they make the three sentences of the system true. What this shows is that when a set of objects or symbols is taken as a representational system the relationship between the symbols and the objects they are meant to represent is wholly arbitrary. A representational system can be used to represent one set of objects just as easily as another as long as both sets have the same formal structure as the set of objects that is taken to be the symbol system. They need only have an isomorphism to the symbol system that allows them to be mapped onto the system in a one to one correspondence. Any interpretation that mapped the above system onto a world with two objects each with one property and with one relation between them would be a model of the system no matter what the objects, properties, or relation were.

     Even when sentences expressing operational constraints are included in the system it still cannot uniquely determine a model, because the sentences that express the operational constraints are themselves meaningless strings of symbols that can be interpreted by any isomorphic set of objects, properties, and relations. 4

     Putnam gives a general argument exploiting the property of symbols systems shown above and expressed in the Lowenheim-Skolem Theorem to show that any symbol system, no matter how much information it contains, will admit different interpretations. (Putnam 1981, pp. 217-218) The example that he gives (Putnam 1981, pp. 33-34) to exemplify the general procedure used in this proof works exactly the same way as the example above does. 5 His example is considerably more complex because of an extra constraint that he adds to his argument.

     In the example above the two interpretations map the system onto different worlds, different sets of objects. Putnam, however, is interested in showing that the Lowenheim-Skolem results hold even if we limit the interpretations to a single domain. This would show that even if there is a single world with a determinate set of objects, no representational system could determinately refer to any subset of the world. So Putnam's example is of two different interpretations of a sentence that map the symbols onto the same domain of objects. Putnam succeeds in getting interpretations that differ, yet which make the same set of operational constraints true by giving disjunctive definitions of the symbols that allow them to be mapped onto one subset of objects in one situation and onto another subset of objects in other situations.

     In this way a symbol can satisfy the same operational constraint (by being mapped onto the same objects) as an intended interpretation in situations where the operational constraint is operative, while at the same time being a different interpretation (in virtue of mapping the symbol onto other objects in other situations). Even with this added complexity, Putnam`s example is still just an example of the fact that a symbol system does not determine its own interpretation; it supplies only the most meager of formal constraints upon its interpretation, and these constraints allow multiple incompatible interpretations.

     The results of the Lowenheim-Skolem Theorem, then, were inevitable once we began to get precise about how exactly the formal structure of symbolic systems constrains their interpretation. Formal systems are sets of meaningless symbols and can be interpreted as applying to any domain onto which they can be isomorphically mapped. Putnam's use of these results shows that the attempt to make language the representation which can come alive and to see all cognitive representation as linguistic will be a failure in the same way that the attempt to make ideas represent in virtue of their phenomenological character or causal origin was a failure. Putnam's argument is a specific instance, applying to linguistic systems, of Hegel's argument that representations cannot be seen as self-existing objects that determine their relation to their object themselves. They must be seen as one moment or aspect of a process of interaction with an object. Hegel showed that the physical-visual model of representation, the attempt to see representations as objects existing independently of what they represent and yet as determinately referring by themselves, is incoherent. Putnam shows the same thing for the special case of linguistic representation. 6

     Hegel, however, follows the argument in one direction (at least for a little while), and Putnam follows it in another direction. Hegel sees the argument as leading towards a model of representation as a moment in a dialectic interaction with the world, one in which representations can still be seen as representing something outside themselves. He still has a Representational Model of Epistemology.

     Putnam takes another route out of the problem. He holds that the argument shows that models are assignments within our representational systems, and that, therefore, the domains or worlds into which they map our symbol systems are also constructions within the system. He says:

Models are not lost noumenal waifs looking for someone to name them; they are constructions within our theory itself, and they have names from birth. (Putnam 1977, p.25)


   For an internalist like myself, the situation is quite different. In an internalist view also, signs do not intrinsically correspond to objects, independently of how the signs are employed and by whom. But a sign that is actually employed in a particular way by a particular community of users can correspond to particular objects within the conceptual scheme of those users. 'Objects' do not exist independently of conceptual schemes. We cut up the world into objects when we introduce one or another scheme of description. Since the object and the signs are alike internal to the scheme of description, it is possible to say what matches what. (Putnam 1981, p.52)


If, as I maintain, 'objects' themselves are as much made as discovered, as much products of our conceptual invention as of the 'objective' factor in experience, the factor independent of our will, then of course objects intrinsically belong under certain labels; because those labels are the tools we used to construct a version of the world with such objects in the first place. (Putnam 1981, p. 54)

Thus, Putnam solves the problem posed by the model theoretic argument by abandoning the external model of objectivity and the Representational Model of Epistemology. Representations intrinsically refer to objects because they were used in the construction of those objects, the objects being themselves internal to the representational system. Thus objectivity cannot be a matter of our representations being caused by the object, and knowledge cannot be a relation between representations and extra-representational objects.

     It seems to me that Putnam's conclusion is a result of failing to see the full power of Hegel's argument. It does not simply show that the physical-visual model of representation will not work if it is made to apply to extra-representational objects, it shows that it does not work, period. Putnam retains the physical-visual model of representation at the price of the external model of objectivity. On Putnam's view representations intrinsically correspond to objects because they were used in the construction of those objects. (We shall see in Chapter Nine that the same arguments he raises against the external physical-visual model of representation can be brought against his internal version.) In order to retain representations that intrinsically refer, Putnam accepts the counter-intuitive conclusions involved in internalism. It is strange that Putnam's conclusions are a result of the retention of the very model of representation that he argues against.

     Before we can argue that this is in fact what Putnam does and that there was an alternative model of representation open to him, we need to see more clearly what conclusion Putnam draws from his version of Hegel's argument and the arguments he gives for drawing that conclusion.




 1 See Quine 1959, 1960.

 2 The set of natural numbers is the set of positive integers from one to infinity.

3 Putnam gives many versions of his model theoretic argument, but only in Reason, Truth and History is the general nature of his use of the Lowenheim-Skolem Theorem and a particular example of his procedure given. See also Putnam 1977; Putnam 1983, pp. ix-xi; and Putnam 1976, pp. 125-126, pp. 130-131, and pp. 133-135.

 4 The isomorphism needed does not even require that the number of objects in the domain of the model be the same as the number of symbols in the system. For example, an interpretation that mapped the system described above onto a world consisting of one object, a red ball, would be a model of the system if it mapped R onto the identity relation.

 5 Putnam (1981, pp. 33-35) gives an example of this procedure.

6 See Putnam 1977, pp. 16-17 for his application of the model theoretic argument to the thesis that all thought is done in a mental language.




Putnam, Hilary. 1970. "Is Semantics Possible?." In Mind, Language and Reality, Cambridge: Cambridge University Press, 1975, pp. 139-152.

  1973. "Explanation and Reference." In Mind, Language and Reality, Cambridge: Cambridge University Press, 1975, pp. 196-214.

  1975. "The Meaning of 'Meaning'," In Mind, Language and Reality, Cambridge: Cambridge University Press, 1975, pp. 215-271.

  1976. "Realism and Reason." In Meaning and the Moral Sciences; London: Routledge and Kegan Paul, 1978.

  1977. "Models and Reality." In Realism and Reason. Cambrhdge: Cambridge University Press, 1983, pp. 1-25.

  1978. Meaning and the Moral Sciences. London: Routledge and Kegan Paul, 1978.

  1981. Reason Truth and History. Cambridge: Cambridge University Press, 1981.

  1981a "Why there isn't a Ready-Made World." In Realism and Reason, Cambridge: University University Press, 1983. pp. 205-228.

  1983. Realism and Reason. Cambridge: Cambridge University Press, 1983.

Quine, W. V. 1959. "Meaning and Translation." In On Translation, Cambridge: Harvard University Press, 1959.

  1960. Word and Object. Cambridge: M.I.T. Press, 1960.


© 2006 David Banach 

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