Formal theories, as in logic and mathematics, are sets of sentences closed under logical consequence. Philosophical theories, like scientific theories, are often far less formal. There are many axiomatic theories of the truth predicate for certain formal languages; on analogy with these, some philosophers (most notably Paul Horwich) have proposed axiomatic theories of the property of truth. Though in many ways similar to logical theories, axiomatic theories of truth must be different in several nontrivial ways. I explore what an axiomatic theory of truth would look like. Because Horwich’s is the most prominent, I examine his theory and argue that it fails as a theory of truth. Such a theory is adequate if, given a suitable base theory, every fact about truth is a consequence of the axioms of the theory. I show, using an argument analogous to Gödel’s incompleteness proofs, that no axiomatic theory of truth could ever be adequate. I also argue that a certain class of generalizations cannot be consequences of the theory.
KeywordsTruth Axiomatic theories Deflationism Paul Horwich
To a mathematician or a logician, a theory is a formal thing, a set of axioms and theorems, and perhaps a list of rules and definitions. To a philosopher, a theory is an explanation, a way of understanding some phenomenon of philosophical interest. There is nothing to say a philosopher’s theories cannot be made up of axioms and theorems; only that, for the most part, they are not.
Truth may be an exception. Even as Tarski proved that a truth predicate for a language cannot be defined in that language, he emphasized the connection between truth and logic. More recently, deflationists, who often cite Tarski as an influence, have said that truth is a “logical” property. It is perhaps for reasons such as these that it has seemed plausible that truth might be susceptible to an axiomatic theory. Because of Tarski’s influence, and perhaps in part because of the deflationists’, several axiomatic theories of the truth predicate for formal languages have been proposed. (A good overview of these theories is Halbach and Horsten 2002; a defense of the axiomatic method is given in Heck 1997.)
A dramatically different kind of axiomatic theory was presented by Paul Horwich (1990, 1998b). Most of the debate over his theory, as with other deflationist theories of truth, is about the content of those theories. I wish instead to discuss their form or method. The idea of an axiomatic theory of a philosophically interesting property is novel and interesting, and even though Horwich applies his method only to truth, there is nothing inherent in the method that makes it appropriate only as a theory of truth. That is, there might be a Horwichian theory of knowledge or of causation—a theory that is formally the same as Horwich’s theory of truth, but with different content.
A Horwichian axiomatic theory is different in important ways from a logician’s axiomatic theory. These theories involve a formal language and a “truth” predicate in that language, along with the claim that this notion is an explication of our pretheoretical notion of truth, or a well-behaved replacement for it. A Horwichian theory differs in that it is not formal and rejects this split between the “truth” predicate of the theory and truth itself.
In this paper, I explain how a Horwichian axiomatic theory works, and I expose its flaws as a theory of truth. These flaws are tragic indeed—because the theory is appealing and it is unfortunate that it should be flawed, and because, as it was for Oedipus, it is in some sense its strength that in the end undoes it. Sections 2 and 3 outline the metaphysics and mechanics of a Horwichian axiomatic theory and distinguish it from a formal theory. I do this at some length because Horwich’s critics often are too quick to understand his theory in terms of other kinds of accounts of truth that are more familiar.
Section 6 argues that these difficulties stem from what might have been seen as an attraction of Horwichian theories—that they are not, like the languages of logicians, couched in formal language; they are instead composed directly of propositions.
2 What is the Axiomatic Method?
What I have been calling the Horwichian axiomatic method differs both from the formal axiomatic method on the one hand and from various other philosophical methods on the other. Horwich himself does not go out of his way to distinguish this method from its rivals. He is clearest about it in the Postscript to his 1998b (especially 135–39) and in footnote 7 to his 2002. Failure to make these distinctions has led to many criticisms that do not hit the mark (Davidson’s, for example—see below, and McGee’s). In this section, I will outline the intuition behind Horwich’s method and contrast it with its rivals.
We want a theory of truth because we want an explanation of what truth is. It is not enough merely to list the extension of truth; an explanation must explain the properties of the phenomenon that we want explained. To see how this method of explanation differs from more familiar methods, let us look at a common toy example—water. Compare these two definitions of “water”: “the actual colorless liquid most commonly drunk by humans,” and “H2O.” They both get the extension right, and they even get the intension. But the second seems to explain water in a way the first does not. We want an explanation of water to explain its color, its viscosity, its melting and boiling points, its specific density, and so on. The chemical definition explains these; the contrived one does not.
A philosophical explanation works by explaining the properties of that phenomenon that are deemed in need of explanation. A method of explanation includes both the explanans and a particular process to explain these properties in terms of the explanans. An explanation of water, for example, explains its properties (color, viscosity, and so on) in terms of H2O, via certain chemical equations.
The “H2O” definition works well as an explanation because it is a property that, in some sense, explains all the properties of water. As such it provides an ideal example: the first step in a good explanation involves picking out a property or small group of properties that suffice to explain all the properties of the phenomenon. This is the first insight leading to an axiomatic theory. The second is that propositions might serve proxy for properties. Instead of explaining in some way every property of the phenomenon, we might explain every fact—every true proposition—about the phenomenon. This allows a precise notion of explanation: logical consequence. We have an explanation of some phenomenon if we have a set of propositions such that every true proposition about that phenomenon is a logical consequence of them. And so in the case of truth: an axiomatic theory of truth will provide some set of propositions from which, together with a suitable base theory that contributes the relevant worldly facts, all the true propositions about truth follow.
Given this, we have a theory of truth. It is a theory in the sense of a class of propositions closed under logical consequence, and it is of truth in that the propositions that are classed together are intended to be—all and only—the facts about truth. For example, a simple fact about truth is this: it is true that water melts at a lower temperature than iron. We do not require that a theory of truth provides us this insight all by itself, for a theory of truth need not also be a theory of water and iron, and hence the need for a base theory. In this case, the theory of truth provides only a thin patina of information: once we know that water melts before iron, the theory of truth tells us this is true.
This kind of theory is very different from formal theories of truth. These theories include the axioms, which explicitly state in a formal language the properties their target object is to have, and the intended interpretation, which applies the formal theory to the real propositions and such. It is the interpretation that does the heavy philosophical work. This distinction of the formal theory from the interpretation allows for precision and rigor, but may lead to ontological problems.1 The Horwichian axiomatic method takes a different approach: it attempts to maintain the precision and rigor without this divorce. The axioms are written neither in a formal language with its artificial precision nor in a natural language with its ambiguities and eccentricities. The theory will consist of all the propositions in the appropriate domain—all the propositions that contain the propositional constituent [true];2 some of these propositions, from which all the other propositions follow logically, will be the axioms.
Horwich’s axiomatic theory of the property of truth must be distinguished from his superficially similar account of the meaning of “true.” In rough outlines, the latter theory is this: Words in general get their meaning from a set of principles that explains their use. In the case of truth, the principle that explains the use of “true” is that anyone who understands the word is inclined to accept on no evidence every instance of the schema “the proposition that p is true iff p.” The axiomatic theory and this other theory are about different things (the property of truth vs. the meaning of “true”); they have different explananda (propositions containing [true] vs. uses of the word “true”); and they have different methods (the axiomatic method vs. the principle that the explanatorily basic feature of a word’s use explains its meaning).
Since the axiomatic method relies on propositions exclusively and not in any way on language, it has a prima facie appeal as a method for doing metaphysics. Conceptual analysis, it turns out, is hard to do, and our language may be only an imperfect guide to the shape of reality. It can be argued that in dispensing with language in favor of propositions, a Horwichian theory moves us one step closer to the phenomena that metaphysics is concerned with.
[if [p] is the content of one of S’s factive mental states, then S knows [p]].
3 How to Axiomatize Truth
This axiomatic approach has a certain appeal as a method of metaphysics. If it is possible to give a full theory of some metaphysically interesting property, philosophy would have a useful tool. Let us look, then, at Horwich’s axiomatic theory of truth, to see if it is possible.
As a first pass, we can take the axioms of Horwich’s theory, which he calls MT, to consist of the propositions expressed by sentences of the form “the proposition that p is true iff p.” This is best seen as a loose heuristic, however, since propositions do not match up one-to-one with sentences. One argument for this conclusion is as follows: Since there are uncountably many propositions but at most countably many names in natural language, not every proposition can be named.
A similar thing can be said of McGee’s (1992) criticism that the concessions Horwich needs to make to the Liar paradox make it so that we cannot construct the set of axioms. Even if this conclusion can be shown to apply to a propositional axiomatic theory, rather than (as McGee does) a formal theory, it does not have the bite it would have applied to Horwich’s account of the meaning of the truth predicate. There it is critical that the account make of “is true” something we can understand. But an axiomatic theorist can very easily say—and this line has a certain plausibility—that truth itself is somewhat beyond our grasp, that we can never know all the properties of truth and so never understand more about truth than coincides with our concept. Thus, all that is required is to demonstrate that MT exists; it is not required to demonstrate how to construct the set.
MT is one example of the axiomatic method, and since it is Horwich’s own, it is salient and convenient. But it is not the only conceivable such theory of truth. In any case, my criticisms in Section 4, and to a lesser extent in Section 5, are criticisms not of the content of MT itself, but of the method. Two of the criticisms depend on a certain assumption about propositions: that for every object and every property, there is a proposition that says of that object that it has that property. So, for example, there is a proposition that the number two is identical with Humphrey Bogart. Neither criticism will use this assumption in its full generality: the first requires the existence of a paradoxical proposition, and the second requires only that there are at least as many propositions as objects.
A theory is good only if it can explain the phenomena. In our case, a good theory of truth will be able to explain all the relevant facts. Horwich says, “According to the minimalist thesis, all of the facts whose expression involves the truth predicate may be explained … by assuming no more about truth than the instances of the equivalence schema” (1998b, p. 23). This is usually called, following Gupta (1993), the Adequacy Thesis. Explanation, for an axiomatic theory, goes by way of logical inference; the explananda are logical consequences of the axioms, or of the axioms and some additional assumptions. With an inclusive enough definition of “theorem,” the explananda are theorems of the theory. This phrase “by assuming no more about truth” emphasizes that for many facts, the explanation will involve assumptions that are not about truth. Another way to say this is that a theory of truth is always an ancillary theory over a base theory. For the theory to be of truth, and not just a theory of truth-for-L for some system L, the base theory must include everything—everything but truth.
The Adequacy Thesis says that the theory can explain the truth of every fact about truth, and that in these explanations, the only assumptions involving truth that are needed will be axioms of the theory. Any worldly facts needed in the explanation will not involve truth. Any proposition that follows from the axioms of the theory and possibly some other facts not involving truth, we can call a theorem.5
Adequacy Thesis: Every proposition containing [true] is a theorem of TT.
Horwich’s formulation of the Adequacy Thesis talks of “facts,” a word that implies that the theorems are true. As I have phrased the Thesis, there is no such implication. There is some reason to think this looser formulation is sufficient. The purpose of an axiomatic theory of truth is not to tell us what is true, but to tell us what is true according to the base theory. It is no objection to a theory of truth that its axioms include [[ravens are red] is true] and [[Brazil is in Europe] is true]. If the base theory included such “facts” as [ravens are red] and [Brazil is in Europe], this is just what we should expect.
At the same time, since it is truth we are interested in, there is good reason to restrict our attention to a true base theory and true consequences. The Adequacy Thesis does not, by itself, guarantee that the theorems will be true when the propositions of the base theory are true—does not guarantee that these theorems will be the facts about truth. To guarantee this, we need to restrict the Thesis to truth. Let me introduce a term to stand in for these “facts about truth”: a supertheorem is a proposition that is a consequence of a consistent set Δ whose members are either axioms of TT or are true propositions not containing the truth predicate.
Super-Adequacy Thesis: Every true proposition containing [true] is a supertheorem of TT.
Super-Consistency Thesis: Every supertheorem of TT is true.
These Theses, Super-Adequacy and Super-Consistency, are troubling because they can be used in deriving contradictions. That is, analogs of Gödel’s theorems can be proved of TT using these Theses. A certain Gödelian proposition that says of itself that it is not a supertheorem can be proved to be equivalent to the Super-Consistency Thesis (hence the Second Incompleteness Theorem), and its negation is a consequence of the Super-Adequacy Thesis (hence the First Incompleteness Theorem).7 Hence, no axiomatic theory of truth succeeds by its own lights.
I said that there is a certain proposition that says of itself that it is not a supertheorem. This is common language where self-reference is concerned, but it is often rather metaphorical. A big part of Gödel’s genius lay in finding a way for numbers to code formulas, to achieve self-reference. A big part of the genius of Horwichian axiomatic theories is that removes the distinction between the language of the theory and the language that we use to talk about the theory. We say that ConPA says that PA is consistent, but really it does not do this. It really says nothing at all. It is a formula in the language of PA—really, given Gödel coding, just a number—that may be interpreted as asserting that PA is consistent. The Super-Consistency Thesis, in contrast, is a proposition, and propositions need no interpretation.
In this case, the erasing of the distinction between the language of the theory and the language in which we talk about the theory may seem puzzling. Many papers have been written over the question of whether Horwich’s theory of truth is adequate; it turns out this question is a theoretic question. Perhaps the feeling of unease comes from our mistrust (since Gödel) over what a theory says about its own completeness and consistency. It is, at best, irrelevant whether a theory asserts its own completeness. This is something that can be known, if at all, only outside the theory. Perhaps the feeling of unease comes because the circle just feels too tight. How do I know the theory gets all truth? Because it says so! But just the fact of the theory’s being able to talk about itself is not really problematic. The real problems come when, as here, the theory is self-defeating.
Not every theory of truth, meaning “theory” in the broad sense, has these problems of incompleteness. Gödel’s theorems apply only to axiomatic theories that meet certain criteria, and because most theories of truth are not theories in the appropriate sense, most theories of truth do not have these problems. They do, however, have similar problems. My proof of the theorem (in footnote 6 above) rests on a paradoxical proposition, and the semantic paradoxes are problematic for almost every theory of truth. One might, then, say that my complaint is dialectically inappropriate. To complain against a theory of truth that it cannot account for the paradoxes would be dialectically appropriate only if it feels these problems in some special way, only if the semantic paradoxes are particularly problematic. This is the situation that I find Horwichian axiomatic theories in. I will not present here a sustained attack on these theories’ ability to deal with the paradoxes.8 But because these theories are logical, the most appropriate criticisms are logical. Other theories of truth with different aspirations need not be particularly concerned with the paradoxes. If, for example, a theory intends only to explain our use of the truth predicate, a Wittgensteinian approach is appropriate.
The upshot of these concerns is this: There is tension in the methods of an axiomatic theory of the property of truth, between the strictness needed for the theory’s brand of explanation and the looseness needed to avoid paradox. It is a tension that is, I think, insoluble. A philosopher’s solution to the liar may make use of more than truth’s logic alone, and the axiomatic truth-theories of the mathematicians scrupulously avoid paradox, at the cost of making their theories only “like enough” to truth for the intended application. If a philosopher wants to use an axiomatic theory to explain truth itself, in all its wildness and transcendence, he finds himself stuck with this tension.9
5 Infinity and Generalizations
I have argued that Horwichian theories are self-defeating, i.e. that they entail that they are inadequate. In this section, I offer independent grounds for believing that the theories are inadequate. The grounds are independent, but in many ways similar. They both rely on certain assumptions about propositions. Here, the assumption is their innumerability. If we assume that everything—including, say, the real numbers—is the subject of some proposition, there are uncountably many propositions. Indeed, as Grim (1991) notes, the problem is worse, since a diagonal argument shows that the propositions cannot be gathered into a set.
This result is problematic for Horwich’s theory because the theory requires one axiom per proposition, and so, since there is no set of propositions, there is likewise no set of the axioms of MT. Horwich attempts to solve that by saying that the theorems of MT do not follow from “a certain entity, the minimal theory; but rather … [from] some part of the minimal theory” (1998a, b, p. 21n4, italics in original). There is no one Minimal Theory of truth, only countless partial theories. This is already a setback in the search for an axiomatic theory of truth, and I think, an insurmountable one. It turns out there is no such thing as the Minimal Theory of truth.
But it is not only Horwich’s theory that has this problem, since it is no help to have any principled restriction on the propositions. Many axiomatizations of truth are compositional; that is, they follow Tarski in treating atomic sentences differently from molecular sentences. The Kripke–Feferman system KF, for example, has as one of its axioms a T biconditional for atomic sentences; other axioms deal with logical connectives and with quantifiers. For formal languages, this is fine, since under their intended interpretation the models of the theories are of respectable size. But the atomic propositions already have the problems of inordinate infinity—there is no set of atomic propositions just as there is no set of all propositions—so it does no good to have as axioms only the T biconditionals for atomic formulas. Thus, there can be no Horwichian theory which includes among its axioms the T biconditionals for the atomic formulas. And similarly, for a restriction on the form of the propositions: given any form M, there are infinitely many propositions of that form.
Non-Existence: There is no set of true propositions of type TT.
Related to this objection is the problem of generalizations. Tarski rejected his sentential version of MT because it does not imply generalizations like “For all propositions p, either p is true or p is not true.” This is a criticism frequently leveled against Horwich’s theory.10 Horwich responds by saying that some propositions about truth follow from nothing less than the entire Minimal Theory, that there is a kind of propositional equivalent of the ω rule that allows us to infer from “p is F” and “q is F” and … to “all propositions are F.”11 David (2002) objects that this disregards compactness, but I am not convinced that the logic of propositions is compact. Compactness seems rather to be a counterintuitive consequence of classical first-order logic. I think it a worse problem that this solution works only if there is a complete Minimal Theory, and there is not.
All truths have property F,
Only things with property F are true.
There are some theories of truth that imply these propositions. For example, theories that include them as axioms will trivially imply them; likewise, a theory whose premises are generalizations may well imply some of them. MT, however, is a collection of nongeneral truths about truth, and so, it is hard to see how these propositions could follow from MT. Indeed, Horwich admits in response to Gupta that Monogamy does not follow from MT, and must be added as an axiom (1998b, p. 23n7). But if this is so, Monogamy is not the only axiom that must be added. Since Monogamy follows from them, if MT does not imply Monogamy, it also does not imply the Pragmatist’s Insight or any other interesting generalization about truth.
Assuming the propositional ω rule, Monogamy follows from a theory of propositions, which is presumably part of the base theory of MT. For every proposition p, that theory presumably implies [p is a proposition], and hence [if p is true, then p is a proposition]—i.e., Monogamy. Why then doesn’t it follow from MT? It would follow only if we have already established that all true things are propositions, that the theory of truth hasn’t added any non-propositions to the list of truths. The guarantee that this has not happened is Monogamy, and thus, any deduction from MT would be circular.
MT fails, then, because it has too many axioms and too few theorems.
6 Beyond the Theory
All three of these propositions—Monogamy, Non-Existence, and the Strengthened Adequacy Thesis—are all, in some sense, metatheoretic statements, statements about the theory of truth. It is surprising, then, that they should also be theoretic statements, statements within the theory. It may seem that this leads to the possibility that, for example, the Super-Adequacy Thesis can be demonstrably untrue about a certain axiomatic theory and yet be a theorem of that theory. A tempting response to this difficulty is to claim that the Thesis as a theorem does not say what it appears to say, that its symbols cannot be interpreted straightforwardly into natural language. Indeed, the response might run, since “true” is just defined by what is a theorem, the Super-Adequacy Thesis, as a theorem, says only that its theorems are theorems. But herein lies a difference between a theory of the truth predicate for some formal language and a Horwichian theory of truth itself. The latter is not in an uninterpreted language, and hence, there is no meta-language/object-language split. If the Thesis is a theorem of a theory, it must be taken at face value, and likewise for Non-Existence and Monogamy. To be sure, there are many propositions about the theory that are not theorems of the theory: [TT is conceptually inflationist], for example, or [TT is intuitively satisfying]. Super-Adequacy is so problematic for a theory because it must be both.
This erasure of the formal-language/interpretation divide is an essential part of Horwichian axiomatic systems. Because the axioms of the theories are not sentences of a formal language that is interpreted in a different language, the theories behave very differently from formal theories. In one sense that is their appeal. They are not explications of “true”; they are not analyses of the concept of truth; they are theories of truth itself. They get at reality unmediated by language. This erasure may not pose a problem for Horwichian theories—theories on the model of Horwich’s MT—of knowledge or causation, but it does pose a problem for truth. The Super-Adequacy Thesis was problematic not just because it was simultaneously theoretic and metatheoretic, but also because it is about truth in such a way that paradox was the consequence. It is not likely that there will be problematic metatheoretic propositions about the causal powers of a theory of causation, so problems of the sort I have pointed out in this paper will be unlikely to derail an axiomatic theory of causation.
Theories work well for Abelian groups and formal-language predicates. And the Horwichian method is clever and may well prove useful in formulating some metaphysical theories. But in the end, no theory can be a theory of truth.
Given the results of Cantor, it seems impossible for a formal theory to have a truly comprehensive ontology. And given the results of Löwenheim and Skolem, there are difficulties in modeling an uncountable ontology.
I am assuming for simplicity that propositions have constituents, and using brackets to name these constituents, just as I use brackets to name propositions. My arguments could be made, at greater length, without this assumption.
Horwich sometimes calls (E*) a “structure,” which sounds like a kind of propositional schema. This would make (*) an abbreviation of ‘∀x (x is an axiom of MT ↔ ∃y(x = [[y] is true iff y])). Propositional schemata are metaphysically dubious entities, and this version uses higher-order quantification, which Horwich wishes to avoid. If (E*) is taken as a function, the formula (*) is scrupulously first-order, even though the quantifiers range over propositions.
Horwich seems to consider this to be a major criticism, and addresses it in (1998b; 1999; 2002; 2006a, and b). One feature of his response that puzzles me is the change to the footnote that outlines this presentation of the axioms (1998b, p. 18, n. 3), in which the axioms are taken to be propositions expressed by a certain form of sentences. In the first edition, this method was presented as a salve to those who do not believe in structured propositions, showing that his theory can do without that assumption. In the second edition, the footnote remains largely the same but is introduced as a solution to Davidson’s criticism. Since Davidson’s criticism is semantic, and the primary method of generating the axioms is not, it appears that the criticism holds only when the solution is applied.
This definition differs from the standard definition in two crucial respects: (1) these theorems, like the axioms, are not sentences, but propositions; (2) these theorems follow from the axioms in addition to a set of propositions not containing [true].
The Thesis is not the sentence displayed, but the proposition this sentence expresses. I have asserted that this proposition contains [true]; the clearest evidence for this is that the sentence contains “true.” Deflationists—in a sense in which Horwich himself is not a deflationist—might claim that Super-Adequacy proposition might be expressed thus: “for all p, if p, and if p contains [true], then p is a supertheorem of TT.” That is, the “true” may be dispensed with given the right quantification, and there are no propositions containing [true]. If this is right, TT is vacuous. TT has nothing to explain if deflationism has succeeded in explaining truth away.
- 7.Proof that the Super-Adequacy Thesis is not true of any consistent axiom system closed under consequence. Consider this proposition:
G: [G is a supertheorem] is not true.
If the Super-Adequacy Thesis is true, G is not. To see this, assume G is true, and hence (because of what it says) not a supertheorem. But then we have a true proposition (G itself) that contains [true] but is not a supertheorem, so the Thesis is false. By transposition, the Super-Adequacy Thesis entails ∼G. And the Thesis itself, if true of some theory, must be a supertheorem of that theory (since it, unlike the original Adequacy Thesis, contains the truth predicate). Since the theory under consideration is closed under consequence, ∼G is also a supertheorem. But since ∼G implies that G is a supertheorem, the theory has both G and ∼G among its supertheorems. So, since the theory is consistent, the Super-Adequacy Thesis cannot be true. QED. The proof corresponding to the second incompleteness theorem is similar; the crucial step is showing that G is logically identical to the Super-Consistency Thesis (i.e., that we can deduce the Thesis from G and vice versa).
I do, however, want to make two brief points. First, many solutions to the paradoxes weaken classical logic to some degree. For the kind of theory under discussion, unlike for theories of the truth predicate, there can be no inner-logic/outer-logic distinction. As I said in Section 3 above, any proposal about the logic of the theory is a metaphysical claim about the consequence relation that holds among propositions. And second, many solutions, including dialethism and Kripkean ungroundedness, produce what are in effect truth gaps or gluts. If a proposition is gappy or glutted, this is a fact about truth the theory ought to explain. Given revenge problems, it seems that it is not possible.
This opinion has been expressed several times. The first I can find it is in Kirkham (1992), quoting unpublished work by Adam Morton to this purpose, speaking specifically about Horwich’s Minimal Theory: “If a non-minimal theory could motivate a plausible line on semantical paradoxes it might claim to have uncovered an essential feature of truth about which the [MT] is silent” (p. 348, addition Kirkham’s).
See, e.g., Gupta (1993), Soames (1999), Halbach (1999), David (2002), Raatikainen (2005). Most of the literature discusses the problem for formal disquotational theories; the difficulty is different for a propositional theory like Horwich’s, and David (2002) is perhaps the best discussion of the specific problem.
Of course, since there are uncountably many propositions, the term “ω rule” is incorrect.
This article has been improved by comments on earlier drafts from many people. I want to particularly thank Paul Horwich, Mark Crimmins, John Etchemendy, and John Perry.
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