The first post in this series examined Hilary Putnam’s famous argument that a “brain in a vat” (BIV) could not know that it was a BIV—or even think or wonder whether it was a BIV—because its words and thoughts would lack the causal-perceptual links to vats and brains in its environment needed for them to refer to those objects. However, as I said in that first post, for Putnam the BIV argument was just a warm-up exercise. He uses the traditional BIV scenario to illustrate what he regards as the key error of “metaphysical realism” (the view that our percepts and thoughts refer to mind-independent things): that it necessarily relies on a God’s Eye perspective from which we can determine what mind-independent things our percepts and thoughts refer to. Of course, there is no God’s Eye perspective available to human beings, and that is why the project of metaphysical realism must end in failure. Thus, Putnam’s real view is that even if the BIV had the same causal-perceptual embedding in its environment that we enjoy, it would make no difference! Its percepts and thoughts would still not refer to mind-independent things. Reference to mind-independent things is impossible in general. The traditional worry about whether you could be a BIV is a useful entrée to these issues because it presupposes metaphysical realism. Only a metaphysical realist would or could worry about being a BIV, because only if the objects of thought were mind-independent would it be possible to be so radically in error about the nature of one’s environment.

Why does Putnam think that only a God’s Eye perspective can determine the reference of our thoughts and percepts? The reason is given in the so-called “model-theoretic argument” that Putnam presents in each of the three works I mentioned in the first post (“Realism and Reason” [R&R], “Models and Reality” [M&R], and Reason, Truth, and History [RT&H]. In the present post, I explain the argument and the “internal realist” view that Putnam advocates on the basis of it. In the next post, we will examine the merits of the model-theoretic argument. (The whole paper on which these posts are based is available here. To skip to the third post in the series, click here.)

The Model-Theoretic Argument

In the previous post, I argued that Putnam’s argument that a BIV’s thoughts about brains and vats could not refer to physical brains and vats goes wrong because he overextends his “causal theory of reference.” The causal theory of reference treats thoughts and percepts as mental signs that refer to whatever normally causes them in their long-term environment. I argued that this theory might work well for reference to substances like water, but not for everything. In particular, our percepts for properties such as shape, size, and distance are dedicated to these properties. The question for such percepts is thus not so much what they refer to as whether they refer accurately. This, in short, is why a BIV’s percepts of this kind continue to refer to properties such as shape, size, and distance, and thus why a BIV can wonder whether it is a BIV.

We will see that the error of treating all mental representations as signs in need of interpretation is at the heart of Putnam’s case for “internal realism.” And so, our exploration in the previous section of Putnam’s use of the causal theory of reference to support his analysis of the BIV scenario hasn’t been a mere distraction. Nevertheless, as is well known, Putnam believes he has a much more potent weapon to hand in the service of his battle to establish internal realism. This weapon is usually called the “model-theoretic argument,” because it relies on theorems of model theory in formal logic. The argument is the centerpiece of his case for internalism in all three of the works in which he announced his new embrace of that view (R&R, RT&H, M&R), and it was at the center of the vigorous discussion his announcement sparked. Each presentation differs somewhat from the others. Putnam always mentions the Löwenheim–Skolem theorems as his inspiration for his new view, but only the argument in “Models and Reality” makes substantial use of them. The other two rely on permutation theorems. Each version of the argument employs the same strategy: first, show that first-order model theory allows any first-order theory to be given radically different interpretations in terms of different sets of objects in the world that nevertheless preserve the truth of all theorems of the theory; second, argue that there is no way consistent with scientific naturalism to limit the profusion of possible interpretations. The result is that no theory can be given a determinate interpretation: there is no determinate answer to the question of what the terms of any theory refer to in the world independent of us.

The technical details of Putnam’s formal proofs are not important, but it will be useful to describe how the proofs work and what they show. I will describe the permutation version and then explain why it fails to refute metaphysical realism.

Model Theory

Here I provide a very brief refresher on model theory in baby terms.

A first-order formal language consists of symbols and rules for their use that are specified entirely syntactically; i.e., without regard to their meaning. The symbols are divided into logical vocabulary, such as ⊃ and ∃, and nonlogical vocabulary, especially individual constants (usually lower-case letters) and predicate symbols (upper case). There are rules for forming expressions and sentences. Thus, Fa predicates F of the individual a, and Rab says that a bears relation R to b, etc. There are also rules for manipulating sentences to perform inferences and a set of purely logical axioms. But again, all this is specified purely syntactically, as though it were an arbitrary and meaningless game. This is the sense in which it is a formal language.

Meaning—semantics—is assigned to the language primarily by assigning objects for the nonlogical symbols to represent. This is called interpretation, and its study is called model theory. We specify a domain of objects (or entities of any kind) to be assigned as values of the nonlogical symbols. The domain could be the natural numbers, the real numbers, sets, spacetime points, elementary particles, animals, people, anything. Individuals in the domain are assigned to the individual constants, and sets of individuals are assigned to the predicate symbols. Thus, Felix could be assigned to a and Garfield to b, and both could be assigned to C along with all the other cats. Then, Cb means that Garfield is a cat. Relational predicates are assigned pairs of individuals (or triples, quadruples, etc., depending on the number of terms in the relation). Thus, if Garfield is bigger than Felix, the ordered pair ⟨Garfield, Felix⟩ would be assigned to B along with all the other ordered pairs of which the first member is bigger than the second, and Bba means that Garfield is bigger than Felix.

Note well that properties and relations are represented in first-order logic merely as sets of individuals. Thus, to be a cat is merely to be a member of the set of cats and to be bigger-than-something is merely to be the first member of an ordered pair (of which the “something” is the second member) in the set of ordered pairs assigned to the predicate “bigger than.” Thus, properties and relations are defined extensionally. In first-order logic, there is nothing more to being a cat or being orange or having mass or electric charge than being a member of the set of cats or orange things or things that have mass or electric charge. Moreover, there is nothing in logic or model theory that governs the assignment of individuals to sets (except consistency). From the point of view of logic and model theory, such assignments are arbitrary. Any constraints on appropriate assignments must therefore come from extralogical considerations, such as the purposes of the theorist.

The key notion of formal semantics is satisfaction. It is the satisfaction relation that sets the criteria of truth for an interpretation. Satisfaction conditions are specified for the logical vocabulary, predicate symbols, and all other symbols of the formal language. There is a universal, standardized satisfaction relation for first-order logic that makes satisfaction of all formulas depend ultimately—in different ways depending on different types of symbols—on the assignments made from the domain to the nonlogical vocabulary. For example, an arbitrary formula of the form Fa is satisfied if and only if the individual assigned to a is a member of the set assigned to F. (The formula that says, “Garfield is a cat” is satisfied if and only if Garfield is a member of the set of cats.) Given a specification of satisfaction conditions for every type of formula of a language, a statement in that language is true under a given interpretation if and only if it is satisfied under the assignments made by that interpretation (as well as any arbitrary values from the domain that might be taken by any variables).

Interpretations may also be given to “theories” stated in a formal language. A theory is a formal language to which to which some nonlogical axioms have been added. For example, these could be axioms of a mathematical or physical theory, such as Newtonian mechanics or quantum mechanics. The formalization of scientific theories, especially of physics, is a longstanding project in mathematical philosophy of science. The axioms of a theory together with any other statements that can be proved from the axioms are called theorems of the theory. An interpretation under which all the theorems of a theory are true is called a model of the theory.

In sum, an interpretation I of a formal language L or theory T is often depicted as a structure ⟨U_i, P_j⟩, where U is the domain and the U_i are the individuals in the domain, and P are the predicates and each P_j is the set of individuals (or pairs or triples, etc.) assigned to that predicate. If all the theorems of L or T are true under I, then I is a model of L or T.

Putnam’s Proof

After all the preceding background, the proof may be disappointingly simple.

Putnam assumes that we have a true theory of nature and that the external world is the domain. Let us suppose that the theory refers to cats and dogs and clouds and trees and so forth. The desired interpretation is that “Garfield” refers to a certain cat, “cat” refers to the set of cats, and so forth. Let us assume that we make the desired interpretation, and it is a model of the theory (as these were defined in the previous section). What prevents us from systematically permuting all the assignments so that every individual is assigned to a new individual constant and new predicates? So long as the swapping is consistent, so that the predicate reassignments are aligned with the reassignments of individual constants, nothing prevents it! Thus, “Garfield” can be reassigned to Odie, who is also reassigned to the set of cats, and “Odie” can be reassigned to Garfield, who is also reassigned to the set of dogs. Now when we say, “Garfield is a cat,” our statement is as true as before, even though we are talking about totally different objects. For, it remains true that the object referred to by “Garfield” is a member of the set that is referred to by “cat,” even though that object isn’t Garfield and the set in question is the set of dogs, not cats.

In symbols, suppose that the desired interpretation I of our true theory is ⟨U_i, P_j⟩. Let us define a new interpretation I′ with structure ⟨U_i′, P_j′⟩ formed by a 1–1 mapping of every individual U_i  in U to a new position U_k  in U′, and carrying the same mapping through to form the sets assigned to the new P_j′ (For example, if Garfield is swapped for Odie, then Garfield will be a member of every set and ordered pair, triple, etc. that Odie was a member of in the original interpretation, and vice versa.) Clearly, the relational structure between individual constants and predicate symbols is exactly preserved in the mapping from I to I′. Only the particular individuals referred to have been permuted. So, every sentence of our theory that was true under I is also true under I′. Therefore, since I was a model of our theory, I′ is also a model. And what we did to form I′ from I can be repeated to form I″ and I‴ and so on.

In RT&H, Putnam’s own example is a mapping of cats to cherries and mats to trees in such a way that it remains true to say that “A cat is on a mat” in exactly the same situations in which it was true to say it before the remapping, even though in most cases the reference is utterly different (cherries being on trees). Further reinterpretation could alter the reference of the “is on” relation, too, of course. And we needn’t stop with dogs and cats and cherries. Clearly, further permutations can change the interpretation of our entire language violently.

What allows this, of course, is the purely extensional nature of the predicate assignments. As noted above, in formal semantics, what it is to be a cat is to be a member of the set of cats, and nothing in formal semantics constrains these assignments except consistency and other merely formal, weak criteria such as having enough objects in the domain to make all the needed assignments.

The conclusion is that the truth conditions for the sentences of a theory are insufficient to determine the reference of the terms in those sentences. Putnam believes this shows that, as long as the referents of our terms are held to be mind-independent external things, we will have no criterion of their identity. In other words, there will be no fact of the matter as to what our theories are about!

Wait a minute. What about terms for things that aren’t mind-independent? The permutation trick can’t be done for them, can it? At least sentences like, “there is a square patch of red in the upper left quadrant of my visual field,” or “I experience a bright flash,” or perhaps, “the thermometer appears to read 30˚ C” contain terms with determinate referents, don’t they? Putnam evidently thinks we have awareness of our sensory states, such as sensations and percepts. He seems to think of their content as consisting of basic sensory qualities such as colors and shapes and smells. He refers to percepts as internal “pictures,” “images,” and “mental signs.” But, although he does not regard these as mind-independent, he also doesn’t think we can directly refer to them without the mediation of “contaminating” concepts (RT&H, 54, 64–65). I will briefly discuss his reasons below. In the meantime, let us consider how his proof would be affected if he were to allow that we have direct access to the properties and relations presented in mental images and sensations. The answer is that reference to them would not be a problem, but this would make no difference to the larger case of reference to external things. The permutations that generate radically alternative interpretations could still be performed only for the mind-independent properties and relations, leaving the referents of our “mental signs” fixed.

Still, if we could determinately refer to properties and relations presented in mental states, like colors and shapes and relations like being inside and being larger than, might we not ascribe them to external things? If so, perhaps we can build a determinate theory of mind-independent reality after all. It might seem that a property like being larger than could be ascribed to external things, but I suspect that Putnam might again deny that this is possible (the worry this time being the Berkeleyite thesis that we don’t know what it would mean for properties of sensation to apply to physical things, cf. RT&H, 58–64). However, even if Putnam would allow it, once again this strategy would not get us very far. For, the assignment of determinate properties to certain predicate symbols would not constrain the interpretation of “theoretical” predicates that are only verbally specified. There is a familiar—indeed, notorious—distinction in philosophy of science between the “observational” predicates of a theory and the “theoretical” predicates, where the former are supposed to be relatively unproblematic in their interpretation. A proof essentially similar to Putnam’s for the nearly total indeterminacy of interpretation of theoretical predicates of a theory, even allowing for observational predicates to apply to external objects, has been given by Ainsworth (2009, 158–160). The problem, again, is that nothing in logic constrains the extensional interpretation of theoretical predicates.

Putnam’s “Internal Realism”

I said above that Putnam believes his model-theoretic proof shows that, if the referents of the terms in our theories—and by extension, the terms of natural language and of all our thoughts—are mind-independent, then our terms and thoughts simply have no determinate referents at all. This would mean, for example, that we are unable to think about anything in particular in the external world. Let us pause to examine how Putnam situates this conclusion and what consequences he draws from it for the relation between our representations and the world outside.

Putnam claims that “there are three main positions on reference and truth” (M&R, 1). First is an “extreme Platonism” in which our minds possess a non-natural direct channel to external reality enabling us to have direct awareness of external objects akin to the direct awareness we have of internal objects such as sensations and mental images. Second is a “moderate realism” that “seeks to preserve the centrality of the classical notions of truth and reference without postulating non-natural mental powers” (M&R, 1). By “classical notions of truth and reference,” he means a correspondence theory of truth in which truth and reference are metaphysical, not epistemological, relations. In saying that truth is metaphysical, I mean for example that your thought that the cat is on the mat—or for that matter, that there is intelligent life in the Andromeda galaxy—is true or false independent of your awareness of that fact. On the “classical” view, your thought is true just if what it represents to be the case is the case. This correspondence exists (or not) regardless of whether you have any awareness of it. This is the essence of what Putnam calls “metaphysical realism.” The “extreme Platonist” holds this too, but for extreme Platonism the doctrine is unproblematic in view of our direct awareness of external objects, which ensures that truth and knowledge can go together. On the other hand, for the moderate realist, the metaphysical nature of truth means at the very least that we can never be certain of having any substantial truth about external reality. Lacking non-natural direct awareness of external objects, it will nearly always be possible to raise doubts about our supposed knowledge of them.

Of course, Putnam means to raise much more serious difficulties than the uncertainty of knowledge for moderate realism. Moderate realism is the target of his attack. The aim of the model-theoretic proof is to show that thoughts about external reality can have no determinate referents if moderate realism is true. The problem is not that our thoughts would have no external referents at all. The problem is rather the opposite. Recall the permutation theorem of the proof as described above: the problem is not that there is no interpretation, but that there is a plethora—indeed, normally an infinity—of interpretations. This means that, on a correspondence theory of truth, any theory is true, as long as its observable consequences are true and it is consistent and there are enough objects in the domain! So, if Putnam is right, the trouble with moderate realism is that truth is too easily achieved, and, by the same token, nothing determinate is said about external objects, except that there must be at least a certain number of them.

Such a view amounts to a kind of phenomenalism or verificationism, and this is the view that Putnam recommends as the third of his three main positions on reference and truth. On this “internalist” or “internal realist” view, there are no mind-independent objects (that we can talk about). Instead, objects exist only within conceptual schemes. So, we have a language or theory that describes cats and dogs and trees and cherries, and perhaps also electrons and quarks and spacetime. These exist in the world outside our bodies—indeed, our bodies exist in the world outside our minds—because our conceptual scheme (of our language, theory, etc.) says so. We speak of them as “real,” “physical,” as “there even when nobody looks,” and so forth, again because our conceptual scheme says so. We do science as before. However, when we say, “the cat is on the mat,” we are not saying that the mind-independent world is any one way rather than another. The “reality” of “physical objects” we talk about is in a real sense merely virtual. Putnam is at pains to stress that this doesn’t mean anything goes. We may form concepts at will, but not our sensations and other experiences. These also are available to us only through our conceptual descriptions of them, which influence and “contaminate” them (RT&H, 54). Nevertheless, they sharply constrain our concepts and beliefs, as we know from the many times our beliefs are contradicted by experience.

For internal realism, the standard of rational belief is determined not by what is liable to produce truth in the correspondence sense, but by coherence of the total system of beliefs. Putnam is vague as to what coherence amounts to, but he indicates that it is not just logical consistency: “Our conceptions of coherence and acceptability are, on the view I shall develop, deeply interwoven with our psychology. They depend upon our biology and our culture; they are by no means ‘value free’” (RT&H, 55). With time and experience, the coherence and thus the rational acceptability of our beliefs may grow. Some epistemic conditions are better than others as regards coherent belief formation. We can conceive of an ideal limit of inquiry along the lines laid down by Peirce. “Truth” for internal realism is what it is rational to believe in, in this ideal limit of inquiry. Here is a marked contrast between metaphysical and internal realism. For the metaphysical realist, it is conceivable that we could reach the ideal limit of inquiry and obtain the epistemologically most well-founded theory possible and still fail to find the truth. For the internal realist, on the other hand, the epistemologically most well-founded theory possible is true by definition.

We can now see as well why ultimately Putnam thinks it is impossible to be a brain in a vat. Assuming the electrical stimulation of the BIV holds good and always presents a train of experience exactly like what the brain would have if it were normally embodied, the conception of being a BIV is never going to be the most rationally acceptable theory in the ideal limit of inquiry. If nothing else, it is more complex than the story that one is a normal embodied human, but no more consistent with experience. Probably it also is less acceptable from the standpoint of our “psychology,” “biology,” and “culture.” Therefore, it is impossible on internal realism to be a BIV—even for a BIV!

A final point to notice about Putnam’s internal realism is his claim mentioned in the previous section that we have no access even to our own mental images, sensations, etc. without concepts. It is worth examining this claim, if only briefly, to expose an element of the pathology at work in generating Putnam’s internalist view. The basic argument for the claim (RT&H, 64–65) begins by assuming that reference to our sensations would be secured by a similitude mechanism: sign E is introduced to refer to sensation X and to all and only those things that are similar to X. Putnam says this won’t work because “everything is similar to everything else in infinitely many respects” (64). For example, any two sensations might be similar in color or shape, but they will also be similar in both occurring after I just blinked, at 30˚ from vertical in my visual field, having a slight tendency to make me think of my dog Fido, and so on. Many properties, sufficiently gerrymandered, can always be generated to provide a basis for similarity among any number of entities. To avoid this problem, it is necessary to specify the respect in which the desired set of items is to be similar. But, Putnam says, if one can do this, then one already can refer to the items in question. That is, one has the relevant concept. Therefore, it is only through a conceptual scheme that we have access even to our own mental images, sensations, and so forth.

The argument just given exemplifies a pattern that Putnam himself points out. Thus, consider a different mechanism for securing reference: causation. We say that sign E refers to what causes it (or normally causes it or causes it in certain conditions, etc.). Perhaps E is “horse.” There will be many different causal chains that cause you to say “horse,” such as the presence of a horse, an illusory percept as of a horse, a picture of a horse, and so on. To restrict the signification of “horse” to the proper objects, one must specify the appropriate type of causal chain. But if one can do that, then one must have a conceptual criterion of the intended referent. Thus, it is our conceptual scheme that tells us what horses are, not causal relations.

This argument pattern occurs repeatedly throughout Putnam’s attack on realism. Let us name it the impotence of natural mechanisms. It goes as follows. To refer to anything directly, without the aid of a conceptual scheme or theory, requires a natural mechanism such as causation or similarity matching. But since there are always many causes or similarities, the method won’t work unless it is refined by introducing a standard of appropriateness that isolates just the right cause or similarity. And any such standard is a concept. QED. Thus, no cognition of any kind can operate without concepts.

Now, I spoke before (perhaps uncharitably) of a pathology. What I meant is the unquestioned assumption that subconceptual mental processes require mechanistic explanations in order to be admissible, but conceptual ones are freely admitted with no explanations needed. For example, the notion that the human mind can attend to qualities of sensations—colors, tastes, pains—as such, remember them, anticipate them, and so forth without concepts is ridiculed as “a mysterious power of ‘grasping Forms’” that would be mooted only by a “Neo-Platonist of antique vintage” (RT&H, 69). The very idea is laughable, it is implied, but can be made respectable by suggesting that perhaps the mind responds causally to properties of sensations, and then we are launched on another repetition of (the causal version of) the impotence of natural mechanisms. On the other hand, once concepts are invoked, all such problems magically disappear. No mechanical explanation is needed of our ability to wield concepts, inferences, rational justifications, coherence relations, theories, and so forth. Evidently it is kosher to just assume that we have these mental abilities. Yet in fact our conceptual abilities are more mysterious, not less, than our sensory abilities.

This prejudice in favor of concepts seems to be a major enabling condition of Putnam’s new philosophy. Without it, he could never suppose that he can escape from the puzzles he raises about reference by cuddling in the warm arms of “conceptualization.” Of course, Putnam is hardly alone in this foible, which was in fact a major force in 20^th century philosophy propounded by many of its most famous names, such as Sellars, Strawson, Quine, Davidson, and McDowell. Tyler Burge has done yeoman’s labor exposing and refuting it in the past couple of decades. (See “Perceptual Objectivity,” 2009, and for the full treatment, Part II of Origins of Objectivity, 2010. For a detailed and psychologically informed exposition of the powers of the nonconceptual sensory mind, see Perception, 2022.)

References

• Ainsworth, Peter M. 2009. “Newman’s Objection.” British Journal for the Philosophy of Science, 60: 135–171.
• Burge, Tyler. 2009. “Perceptual Objectivity.” Philosophical Review, 118: 285–324.
• ———. 2010. Origins of Objectivity. Oxford U.P.
• ———. 2022. Perception: First Form of Mind. Oxford U.P.
• Putnam, Hilary. “Realism and Reason.” 1976. In Meaning and the Moral Sciences, Routledge & Kegan Paul, 1978: 123–138.
• ———. 1977. “Models and Reality.” In Realism and Reason: Philosophical Papers, Vol. 3, Cambridge U.P., 1983: 1–25.
• ———. 1981. Reason, Truth, and History, Cambridge U.P.