Consider:

- The first full sentence in this post is not true.
- The second is true.
- The third does not express a proposition.
- The fourth is true or false.
- The fifth expresses a proposition.

I am quite sure that: The token (1) does not express a proposition that is true. Therefore, the first full sentence (here I am stipulatively taking "sentence" in the usual grammatical sense, rather than in a more beefy sense that I actually prefer) in this post is not true. Note that there is no contradiction between the two preceding sentences. In fact, I think the token (1) does not express a proposition

I am pretty sure that the token (2) does not express a proposition that is true. Therefore, the second full sentence in this post is not true. It does not follow from this that the token (2) expresses a proposition that is false. I suspect the token (2) does not express a proposition.

I am strongly inclined to think that the tokens (3)-(5) likewise fail to express propositions.

Now, if only I had good arguments for my judgments about (2)-(5), I'd be happy.

## 6 comments:

George Englebretsen has proposed that we can assign to each sentence a 'propositional depth'; all five of the sentences you list would have an indeterminate propositional depth. But sentences with indeterminate propositional depth are such that they can't be used to assert anything determinately (because that's what propositional depth measures). &c.

There are a few things that might be problems for the account, but I've always thought that it's a very promising account, in part because if it could be motivated properly and the objections properly met it could serve as a principled basis for the sort of judgments you discuss here.

Is there a difference between infinite propositional depth, and indefinite propositional depth?

Say that a proposition p is nth order true iff it is true that it is true ... that p, where there are n "it is true that"s. Then, consider:

(*) That snow is white is nth order true for all n.

This seems to have infinite propositional depth. (For all n, it has at least propositional depth n.) Or for a slightly less trivial example, say that p is nth order possible if MMM...Mp, with n M's. Then:

(**) That snow is white is nth order possible for all n.

This seems unproblematically true, but has no finite propositional depth.

If (*) and (**) are to be analyzed conjunctively (as a conjunction of propositions of finite depth), the following examples will not allow that:

M(For all n, it is nth order possible that snow is white)

L(For some n, it is nth order possible that snow is white)

Another difficulty is that one is committed to the implausible fact that in the following case, neither person has said anything true:

Fred: 2+2=4 or what Jim is saying is impossible.

Jim: 2+2=4 or what Fred is saying is impossible.

I think this is not such a problem myself.

Here are more fun sentences:

6. This claim is not theological.

7. This claim is semantic.

Well, I think there are two ways you could go. Route One would be to regard "It is true that" as not really depth-increasing: it's verbally collapsible, so saying "It is true that snow is white" is just an expanded way of saying "Snow is white", with a propositional depth of 0. Then nth-order true would be 0. (The same could be said for 'It is possible' in systems where MMp can be simplified to Mp, etc. That is, propositional depth is measured by the simplest form that preserves the meaning.)

Route Two would be to say, given the quantification over n, that what * actually says is:

"All n is such-that-snow-is-white-is-nth-order-true."

Which has propositional depth 0. Englebretsen is big on term logic, so he would be inclined already to read it as predicating a complex predicate to the subject n.

The collapse won't work if we are clever. We just don't use M but some related operator. It's easy to come up with half a dozen.

The second approach will run into trouble with this paradox:

(*) For all n (this sentence is false or n = n + 1).

If "All n is such-that-snow-is-white-is-nth-order-true" has depth zero, so does (*).

You're right that the collapse wouldn't be a resolution to the problem; rather, it's a constraint on what the problem can be. That collapse blocks the problem shows that any propositional argument for infinite propositional depth depends crucially on the fact that the operators in question are not idempotent. A question: Can 'nth order X' be univocal given that X^n is not the same for every n? In such a case 2nd order X means something different from 3rd order X, the only commonality being that they are both built out of X. If it's not univocal, though, it seems to me that one could argue that your original (*) at Aug 13 2:59 is not sufficiently coherent to express a proposition, nor is anything of similar form. Where 'It is X that it is X that p' does not collapse to 'It is X that p', the first is a different proposition entirely from the latter. And therefore in 'p is nth order X for every n' p can only be a particular proposition. But particular propositions can't be nth order X for every n, because they are only whatever particular order X they are. It's as if you said, "This apple is n-many fruits for every number n" or "This train car is n-many cars behind the engine for every number n."

(*) in your Aug 13 6:47 comment would certainly count as having propositional depth of zero if you move from predicate logic to term logic. It's a claim about every n:

Every n is a thing such that this sentence is false or n=n+1.

Which has interesting features, but, of course, what to do with claims in which kooky predicates are predicated of arbitrary subjects is a somewhat different from the question of whether infinite propositional depth is possible.

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