Abstract
Vann McGee has argued against solutions to the liar paradox that simply restrict the scope of the T sentences as little as possible. This argument is often taken to disprove Paul Horwich’s preferred solution to the liar paradox for his Minimal Theory of truth (MT). I argue that Horwich’s theory is different enough from the theory McGee criticized that these criticisms do not apply to Horwich’s theory. On the basis of this, I argue that propositional theories, like MT, cannot be evaluated using the same methods as sentential theories.
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Notes
For example, [[ravens are red] is true iff ravens are red]. I follow Quine’s convention of using of square brackets to refer to propositions. More precisely, square brackets indicate mention: they surround a sentence to indicate the proposition expressed by that sentence is being referred to. This distinction holds also when propositions are embedded in other propositions, as in the above example. It is a tricky question what the ontological difference is between a proposition’s being contained in another as use and being contained as mention. It is likely that no criticism of propositional theories like McGee’s can have the same rigor without better knowledge of the metaphysics of propositions. I will also occasionally use square brackets to refer to a propositional form, as in ‘every proposition of the form [if p then p]’.
For example, Simmons (1999, p. 459) claims that Horwich’s “minimal theory of truth for utterances” takes as its axioms the T sentences. He goes on to say, in footnote 30, that McGee’s argument applies to this theory. Simmons is merely one example of a tendency I have seen often in print and conversation.
For more on the distinction between Horwich’s two accounts of truth, see Horwich (1998, p. 135ff).
On some theories of propositions, [[ravens are red] is true] and [what Smith said is true] are both names of a single proposition in the case where Smith said that ravens are red. On these theories, oblique reference does not represent a problematic case, but quantification still does. [Everything Smith said is true] is a different proposition than [[ravens are red] is true], even if [ravens are red] is the only thing Smith said. (The first is a general proposition, the second a singular proposition.) The argument of this section would also work with a generalization.
That is, if R and SR are in the base theory, S will be a theorem of the theory that is the union of this base theory and MT.
This is in contrast with another reading. I may not have heard Smith very clearly, but I’m confident that he at least sometimes speaks the truth. To avoid committing myself to the truth of his statement, I say only that it is possibly true. My statement is an abbreviation for ‘If Smith said that ravens are black, then what Smith said is true, and if Smith said that ravens are red, then what Smith said is false, and if Smith said that...’.
This derivation requires the necessitation only of the right-to-left direction. The necessity of the left-to-right direction is required from the premise that it’s possibly true that ravens are red to the conclusion that it’s possible that ravens are red.
Because of the need to make MT as conceptually simple as possible, Horwich tries to avoid this, claiming that the necessity need not be built in Horwich (1998, p. 21). He is wrong. If the axioms are not modal, the theory will not be adequate.
Raatikainen (2005) has a cluster of worries about Horwich’s appeal to the “\(\omega \)” rule—which is not just infinite and not just uncountable, but larger than any set. Most of these worries are related to the claim that “even if the rule would in theory entail the desired generalizations about truth, we human beings would never reach any of these generalizations.” As I argue in the text, Horwich accepts this conclusion for MT; it does not, I think, damage that project.
More precisely, what he shows is that the complexity of the set will be either \(\varDelta ^0_2\) or \(\varDelta ^1_2\), depending on which notion of consistency we’re assuming.
In a footnote (footnote 5), McGee mentions that Horwich’s account is for propositions rather than sentences, but his paper ignores this distinction, continuing to press his objection as if it were an objection to Horwich’s theory.
In McGee’s proof, (1), and hence (2), are theorems of R. Given the difference in the mechanics of the proof for propositions, I see no need to couch this in terms of an arithmetic base theory, even if there were such a thing as propositional Robinson Arithmetic.
The Gödel self-referential lemma allows us to find, for any formula f, a sentence S such that f(S) iff S. The propositional analogue would allow us to find, for any propositional function f, a proposition P such that f(P) iff P. I doubt the propositional analogue of the lemma is true in general, but McGee doesn’t need the full strength. He needs only for there to be some proposition [\(\hbox {B}_\phi \)] that corresponds to the propositional function [x is true iff \(\psi \)]. There is such a proposition: [[\(\hbox {B}_\phi \)] is true iff \(\phi \)].
It might be asked: Why is the corresponding step not available for the sentential version? The answer is that the propositional task is greatly simplified by not getting into the mechanics of self-reference. With formal sentences, self-reference cannot be taken for granted. This is a case, as mentioned in footnote 1, where a propositional analogue of McGee’s proof cannot be pressed rigorously without taking into account the metaphysics of propositions.
To be two-way possible is to be possible but not necessary.
This counterexample works in general, proving that \(\phi \leftrightarrow (\psi \leftrightarrow \chi )\) does not imply \(\chi \leftrightarrow (\psi \leftrightarrow \phi )\). But it also respects the semantics of this particular case. It takes [\(\hbox {B}_\phi \)] and [[\(\hbox {B}_\phi \)] is true] to be necessarily equivalent, as we would expect for truth, and since it takes [\(\hbox {B}_\phi \)] to be false, it takes [[\(\hbox {B}_\phi \)] is true] to be nonequivalent to [\(\phi \)], as we would expect given the meaning of [\(\hbox {B}_\phi \)].
It may be, as Horwich says, that every proposition is expressible in some extension of English; but they cannot all be simultaneously named. We can name any real number we choose to name, but we cannot in a finite language name all of them.
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Christensen, R. McGee on Horwich. Synthese 193, 205–218 (2016). https://doi.org/10.1007/s11229-015-0753-x
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DOI: https://doi.org/10.1007/s11229-015-0753-x