Here's the promised conclusion to my last post. (Thanks to Robbie for a helpful discussion of this stuff over dinner last night).
The reason I'm uneasy about Field's project is simply that I need to hear more about why the notion of truth that we end up with is (the) one that we're interested in when we get worried about semantic paradox. We are to understand truth as the thing governed by Schema (T), and we are to understand (T) by understanding its logical constants, and we are to understand its logical constants by understanding which arguments are valid. (The notion of validity which is used to give us a grip on the logical constants in Schema (T) - and hence on the truth-predicate - will render the paradox-generating argument invalid - specifically, by rendering unrestricted LEM invalid).
But what reason is there to think the kind of truth we were concerned with when we started out is so definable? Maybe Field's project is just to show that a predicate obeying (T) can be used consistently. But even this claim presents difficulties: if how we understand (T) depends on how we understand the constants it contains (undeniable), and if how we understand of those concepts is governed by which arguments we take to be valid (Field's claim), then we have first to agree with Field about which arguments are valid in order even to accept that the (thing which looks like a) T-schema that he ends up preserving is the same as the one we wanted to preserve. Otherwise, he may have preserved the truth of the sentence 'T(< A >) iff A' but only at the expense of making it mean something else.
In addition, traditional worries about implicit definition seem relevant to the claim that we can understand logical constants by understanding which arguments are valid, and the claim that we can understand (T) once we understand the constants involved. (To illustrate with the first case, suppose you have a bunch of arguments involving '&' which you're told are valid. If you're meant to be able to tell from that what '&' means, you're presumably supposed to do this by noting that the intended interpretation of '&' is (the) one which will make all these arguments come out valid. But what guarantees that there is such an interpretation, and exactly one of them?)
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Nothing guarantees that at all. I think that if it's going to work, it's as a part of a larger program in which you try to tell just that story.
I was talking to Hartry at dinner and he seemed especially keen when I said that someone should give an independent, understandable "semantics" (here any sense of the world would be OK -- proof theory, model theory, whatever) for some logical system no stronger than the system given the consistency proof. If he can, say, give an independent account of what is going on when you use vocabulary in such a way as to render these arguments valid (and more importantly, for his project, to render those other arguments *invalid*) -- and, as you point out, give something approaching an argument convincing us that you *can* do that -- then he'll have started discharging that debt.
(I think Hartry seemed keen because he thought I was volunteering for the project.)
Hi Greg,
That seems right. Until we have a story about that, while the project's still formally interesting, philosophically I'm always going to be uneasy.
So: best of luck with your new project! :)
New project? You've got to be kidding me. I've got enough on my plate as it is...
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