Abstract
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.
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Notes
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.
Proof that the Super-Adequacy Thesis is not true of any consistent axiom system closed under consequence. Consider this proposition:
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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).
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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.
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Acknowledgements
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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Christensen, R. Theories and Theories of Truth. Int Ontology Metaphysics 12, 31–43 (2011). https://doi.org/10.1007/s12133-011-0075-5
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DOI: https://doi.org/10.1007/s12133-011-0075-5