Horwich's minimalism is a theory of truth generated by the axiom:
- If p is true, then p is a proposition
- <s> is true if and only if s
Here are some issues. None of them are fatal. But they all mean that minimalism isn't quite as simple as it initially seems.
Issue 1: What is an extension of English? We need to include sentences of extensions of English because no doubt there are propositions that no sentence of English can express. Now, an extension of English had better not change the meanings of "is true" or "if and only if"—for if that is allowed to change, then some instances of (2) will become false. Presumably, then, what makes L an extension of English is that for any linguistic element e of English, e is also a linguistic element of L, and it has the same meaning (semantic value, etc.) in L as it does in English. Thus, Horwich's minimalism in its description of the axiom schema presupposes the concept of meaning (semantic value, etc.). To avoid circularity, the concept of meaning had better not depend on that of truth.
Issue 2: Nitpicky stuff. Strictly speaking, (2) generates bad orthography. Suppose s is "Snow is white." Then we are told that <Snow is white.> is true if and only if Snow is white. But the last "Snow" should not be capitalized. This can be easily handled—we specify that when substituting s in, we adjust the first letter's case as needed. There is also that odd looking period after the first "white"; again, we can specify that it is to be omitted. A slightly less easy case is where s is "This sentence is short." Suppose s is true. But now consider:
- <This sentence is short> is true if and only if this sentence is short.
- <u if and only if v> if and only if u if and only if v.
- <s> is true if and only if (s).
Issue 3: Contingent liar. Axioms normally are supposed to not vary between worlds. But there are contingent liar sentences, like "The sentence on Alex's board is false", which is paradoxical when it is the unique sentence on Alex's board but need not be paradoxical when written on Jon's board (unless we have something like "The sentence on Jon's board is false" as the only sentence on Alex's board). This means that dropping those instances of (2) (or of (5)) that generates the paradox requires dropping different instances in different worlds, thereby making the axioms of truth differ from world to world.
There are two ways for axioms to differ between worlds. In the weak sense, whether p is an axiom varies between worlds, but p is true at all worlds. In the strong sense, the truth value varies between worlds, too. This is the kind of variation that we'd need to get out of the contingent liar. And this just doesn't fit with what we understand by "axiom", I think.