Putnam's model theoretic argument

May 6, 1997

Tetsuji Iseda

Cautions: My illustration of Putnam's argument is a simplified version of his argument in the appendix of Reason Truth and History (Putnam 1981, 217-218). He uses different versions of the argument in "Models and Reality" and in Chapter 2 of Reason, Truth and History. In addition, my illustration is a somewhat 'creative' interpretation. Putnam does not necessarily agree with me.

1. Correspondence theory of truth

To start with, please look at Figure 1. This is a crude picture of a "model", in the sense used in predicate logic. In this model there are only three words and five individual entities ([[alpha]] ~ [[epsilon]]) associated with the words. The truth value of a sentence composed with these words (like "Nana is a cat", "No cats are electrons", "Some electrons are not Nana" etc.) is determined by the relationship among the sets associated with the words.

This figure also captures the basic idea of the correspondence theory of truth, when combined with the following premise.

Premise 1: [[alpha]] ~ [[epsilon]] are mind-independent entities in the world.

Figure 1

2. Model-theoretic argument

Putnam's model-theoretic argument is supposed to show that the correspondence theory of truth does not work, in the sense that we can never determine the intended interpretation of the words. To show this, we need to add another premise.

Premise 2: All we can use to determine interpretation are operational and theoretical constraints.

What are operational and theoretical constraints?

Operational constraints: To stipulate that a certain sentence is to be true iff a certain test result is observed (Putnam 1981, 29-30)

Theoretical constraints: Constraints which refer to formal properties of the theory. Putnam's example: 'an admissible interpretation is such that it turns out to be true that different effects always have different causes'; other examples: an admissible interpretation should be simple; an admissible interpretation should be conservative; see Putnam 1981 30-31.

Please note that both constraints are expressible in linguistic form.

Putnam's argument is that under premise 1 and premise 2, there are many unintended interpretations that we cannot exclude by neither operational nor theoretical constraints.

Putnam's own proof of the argument is kind of technical, but the basic idea is fairly simple. Figure 2 illustrates the idea.

Figure 2

Please study Figure 2 carefully. Interpretation I and Interpretation J are different interpretations of the same language. A sentence true under interpretation I is also true under interpretation J, and vice versa (check the sentences "Nana is a cat", "No cats are electrons", "Some electrons are not Nana" under each interpretation). Still, What "Nana" means and what "electron" means are totally different in two interpretations. As you can easily see, this is no accident, and Putnam proved that this kind of unintended interpretations can be constructed for almost any model.

These unintended interpretations cannot be eliminate by operational constraints, because interpretations I and J make exactly the same sentences true.

Theoretical constraints cannot discriminate unintended interpretations either, because as far as these constraints are expressible in the language, the content of the constraints also allows unintended interpretations.

Our intention to use interpretation I does not help either, because intentions also allow several interpretations.

Figure 3

Causal theory of reference does not solve the problem. Let us consider the constraint 'an admissible interpretation is such that each word and its referents are causally related in an appropriate way'. The phrase "causally related" (and if you want, "appropriate", also) allows different interpretations, as is shown in Figure 3. 3. What does this argument mean?

a. Refutation of the correspondence theory of truth

First of all, this argument is supposed to show that the correspondence theory of truth is wrong. The correspondence theory of truth requires that truth value of sentences is determined by unique correspondence between the world and words.

The above argument shows that we cannot single out such unique relationship. For example, when I use the word "Nana" and make many different sorts of true assertions about Nana, still "Nana" can mean anything in the world. This seems to mean that what "Nana" really refers to is not essential to the truth value of the sentences with the word "Nana". Therefore the unique correspondence does not determine the truth value.

We can save the correspondence theory of truth by saying that there is a unique correspondence, but we do not know which. According to this view, we can neither control nor investigate the correspondence.

b. Refutation of realism

Putnam himself is hopelessly unclear about the relationship between the above refutation of the correspondence theory of truth and the realism issue (but, see Putnam 1990, p.31 for his effort to make clear). So the following argument is partly my reconstruction from his remarks and partly my invention.

First, suppose that we do not give up the correspondence theory of truth despite the difficulty. Then, we have the same underdetermination problem for sentences like "cats exist", "electrons exist". To dramatize, let us introduce possible world semantics (Figure4). Figure 4 is arranged so that every sentence has the same truth value under interpretation I and interpretation J. But the sentence "electrons exist" under interpretation J is equivalent to the sentence "unicorns are animals" under interpretation I. So, according to this picture, when a professor declares "electrons exist!", we (including the professor himself) never know if he means that electrons exist, or he really means that unicorns are animals. Such an ambiguous sentence cannot establish the commitment to realism.

Figure 4

Of course, we can partly avoid this problem by abandoning the correspondence theory of truth. But the problem with the correspondence theory of truth is that it is too weak to establish the relationship between language and world necessary to express the commitment to realism. Thus, the real solution of the problem is to provide a stronger semantic theory, not to try to establish the commitment to realism without a semantic theory. When a space shuttle is not good enough to go to Mars, what we should to do (to go to Mars) is to construct a better space ship, not to try to go there without a space ship.

As my reconstruction shows, this criticism is not directed to realism itself, but to would-be realists who need to establish their realist commitment using language. But, of course, if no one can be a realist, then the realism issue disappears. This is what Putnam (seems to) claim.

4. Criticisms and replies

Criticism 1: Isn't this a Cartesian skepticism?

Reply: No. Putnam's argument shows that (1) we do not have control over interpretations, and (2) there are overwhelmingly many unintended interpretations. Under these conditions, it is quite unlikely that we can mean what we want to. Imagine that I am going to toss a coin 100 times, and I intend to get 100 heads in a row. You might say, "but it is quite unlikely that you get 100 heads in a row, unless you can control the coin". Is this a Cartesian skepticism? I do not think so, and the situation Putnam describe is similar to this.

Criticism 2 (Brueckner 1984 and others; see Devitt 1997, 332): When causal theorists say "references are determined by causal relationship", they does not mean our expression of "causal relationship", but causal relationship itself.

Reply: Well, there are two options. Do you think that we can control or investigate the causal relationship? If the answer is yes, that "causal relationship" should be also underdetermined. If the answer is no, then such a causal relationship does not help you to express realism.

Criticism 3 (Devitt 1997, 226-227): The expression "x refers to y and to nothing else" can be explicated with the following formulation:

Term x is causally related in way A to object y and to nothing else.

Now of course the problem is what the expression 'causally related' refers to. The answer is:

'Causally related' is causally related in way B to causal relations and to nothing else.

In short, the causal relations themselves fix the reference of the causal relations, so there is no need of further basis for reference. Putnam can keep asking the same question, and we can use the same reply. The point is that "that an answer can be questioned in this way does not show that it was not a good answer to its questions" (227).

Criticism 4 (Devitt 1997, 227-229): Putnam is begging the question when he assumes that all we can use to determine interpretation are operational and theoretical constraints. What he need to show is that causal relation does not determine reference, and he presuppose the conclusion in this premise.

Reply (Partly from Putnam 1983, xi-xii): If Putnam begs the question, Devitt also does, in criticism 3. He presupposes that causal relation can determine reference, and this is what he is supposed to show. In this mutual question begging situation, the question is which side should have the burden of proof. The asssessment of this problem requires a deeper analysis of causal theory of reference, which is out of the scope of this class.

Criticism 5 (Devitt 1997, 334-336): Why are we interested in semantic questions? Because we want "to explain intentional behaviors or actions" (334). From this point of view, unintended interpretations become irrelevant. Suppose that there is a person who strokes Nana and says "I love this cat". This utterance allows an unintended interpretation that I hate that dog, but this interpretation does not explain her behavior. Therefore unintended interpretations are inferior to intended interpretations.

Reply: It seems to me that Devitt's argument is vulnerable to Davidsonian underdetermination thesis. Devitt's natural explanation presupposes many beliefs and desires inside the speaker: she believes that Nana likes to be stroked, she believes that dogs do not care whether Nana is stroked or not, she wants to say something relevant, she do not want to tell a lie, etc. But we do not have clues to determine these beliefs and desires individually. So if we attribute to her another set of beliefs and desires, maybe the interpretation that I hate that dog can be a natural explanation of her behavior.


Brueckner,A.L.(1984) "Putnam's model-theoretical argument against metaphysical realism" Analysis 44,134-140.

Devitt,M.(1997) Realism and Truth second edition with new afterword. Princeton University Press.

Putnam, H. (1981) Reason, Truth and History. Cambridge University Press.

--. (1983) Realism and Reason. Cambridge University Press.

--. (1990) Realism with a Human Face. Harvard University Press.