For the definitions, see yesterday's post.
Horn 2: Truth is nomically coextensive with a Natural property.
We will now generate a problem for the naturalist from the following very plausible claims:
- If naturalism holds, any syntactic property of utterances is a Natural property.
- If naturalism holds, then Utterance is, of nomic necessity, a Natural kind.
- Nomically necessarily, if naturalism holds, and s is any sentence in an interpreted typed first-order language L such that (a) the types coincide with Natural kinds, (b) quantification is restricted to within a type and (c) all predication is of Natural properties, then s expresses a proposition which is either true or false.
Now, let L be a rich enough subset of technical English that satisfies the condition in (3). Let T be the Natural property that is of nomic necessity coextensive with Truth. Let s be any sentence-type of the form:
For all utterances u, if P(u), then u does not have T,where P is an explicit statement of a finitely-expressible Natural property of an utterance sufficient to nomically entail that all and only utterances of type s satisfy P. (I'll construct P in a moment. Technically, P(u) in the above should be in right-angle brackets.)
Now, if naturalism holds, then for any finite sentence type in L, there is a nomically possible world in which naturalism also holds and where that sentence type is uttered exactly once. Let w be a world where s is uttered exactly once. If u is the utterance of s in w, then at w, u satisfies P and only utterances of s satisfy P. But then u is true if and only if u is false, since T is coextensive with truth at w. And this is absurd.
To construct P, we presumably can use Goedel numbers as in the proof of the diagonal lemma in Goedel's theorem. Or perhaps more simply, we can use what I call "modified Goedel numbers". A "numeric expression" is a literal number in a sentence, e.g., "44.58" or "-1909". The modified Goedel number of a sentence with no numeric expressions is -1. If a sentence s constains a numeric expression, we let s* be the sentence with its first numeric expression replaced by 0, and let n(s) be the numeric value of that first numeric expression of s. If the Goedel number of s* is equal to n(s), then the modified Goedel number of s is n(s). Otherwise, the modified Geodel number of s is -1. Then, it's really easy to construct P. We let N be a numeric expression of the Goedel number of "For all utterances u, if 0 equals the modified Goedel number of u, then u does not have T", and then let P(u) be "N equals the modified Goedel number of u" (where here N is expanded out—it should be in right-angle brackets, I guess). Since P(u) expresses a syntactic property of u, it follows that it expresses a Natural property of u.
The argument can be modified by replacing (1) with the weaker claim that enough basic syntactic properties for computing the modified Goedel number of a sentence are of nomic necessity coextensive with Natural properties.
Hence, absurdity also follows from the second horn of the dilemma.
Therefore, naturalism is false.