Thanks for the comments.

Mike: You’re probably right that (Q) doesn’t have unique status with respect to this problem – other norms might also interfere with the coordination strategy involved in developing canonical modes of reference. I mentioned (Q) because philosophers take it – or something like it – to be a fundamental norm which governs communication. That there is a default presupposition that speakers typically obey (Q) defines the space in which the kind of rational interaction that conversation is can take place.

I like the example of junk DNA. It is like the converse of the claim I’m making. I’m claiming: if you want some way of identifying an object, you should isolate the procedures for generating such identification from potentially conflicting norms. Conversely: if some process is isolated from potentially conflicting norms (as the transmission of junk DNA is isolated from natural selection) it’s outputs can be used to identify objects with which they are associated (i.e. organisms)

Matt: you’re right to point out that the predicate view of names has some interesting consequences for the problem of negative existentials. Those consequences will be the subject of a latter chapter of the dissertation. The situation is not quite as simple as you state it, though. According to the predicate view, names are basically analogous to incomplete definite descriptions. Definite descriptions do have a presupposition of existence, so there is still a question about negative existentials. Under what conditions are we allowed to violate that presupposition? Under what conditions could I sensibly say something like ‘The president does not exist’ ? What exactly would I then be taken to assert?

Aidan

Here is a naive thought I had the other day: perhaps your view takes care of the problem of negative existentials. For there isn’t anything unusual about statements like “Unicorns don’t exist”–in Frege-ese that becomes something like “The concept ‘is a unicorn’ has an empty extension.” Only negative existentials which feature logically proper names (like “Santa Claus doesn’t exist”) pose a problem. But if “Santa Claus” is a predicate, then perhaps Santa Claus sentences are no different from unicorn sentences.

That is, to get an actual value of a provably total computable function, there is indeed a proof of P, and with provable consistency that yields a proof of NOT Provable (NOT P), and rules out a proof of NOT Provable(P). So, the search above indeed finds all of the values of the provably totally computable functions. It cannot find an incorrect value, because the values of the functions are provably unique, the correct value is provable, and consistency (just the fact of consistency here, I think, not its provability) rules out a contradictory proof of an incorrect value.

I notice that similar conflicts arise in many informational phenomena, and are often resolved in a similar way. “Junk” DNA is particularly useful for identifying an organism, since it is not constrained by a physiological function and is therefore not affected much by natural selection. The law on trade names (not at all the same as copyright) makes whimsical names much easier to defend than names that might have been used descriptively by a reasonable disinterested party (e.g., it’s easier to defend the name “Glunk” to identify a brand of chicken than the name “Poultry”.) The Domain Name System on the Internet is plagued by social, legal, criminal, and all sorts of struggle to control attractive names (“sex.com” was actually stolen through a criminal forgery—the thief made millions and skipped the country before being convicted of the forgery), which interfere with the use of domain names just as permanent identifiers. I have proposed adding an alternative set of names that look like random numbers for those who need a robust identifier and don’t care to defend a contentious name. In other areas of computers science, it is very common to introduce arbitrary numerical codes for identities behind the scenes to resolve conflicts between different intuitive uses of names (e.g., variable names in programming languages are usually mapped to invisible arbitrary integers by the compiler).

There is a nice general theorem that “subrecursive” programming systems—systems in which all programs always halt (plus some weak assumptions about composability of programs, which are hard to avoid satisfying) cannot program their own interpreters. This is a major reason why all widely used programming languages allow infinite loops—to avoid the possibility of infinite loops they would give up the ability to write interpreters.

The provably total Turing Machines in almost any sensible theory of arithmetic form such a subrecursive programming system.

If our theory could prove FOR ALL P either NOT Provable(P) or NOT Provable(NOT P), then we could prove the termination of a TM that simultaneously searches for a proof of P and a proof of NOT Provable(P) (I’m pretty sure that I got the positives and negatives right here, but it’s confusing). By applying that program to formulae defining the functions computed by provably total TMs, we get a provably total interpreter for the provably total functions.

I’m not totally confident that I haven’t missed a loophole, but I think this is right. Not sure how satisfying it is. It certainly does not characterize provability of PI_1 formulae, but it relates independence of consistency to the provably total computable functions, which are the same ones that determine PI_2 reducibility to PI_1.

I think of the form FOR ALL P either NOT Provable(P) or NOT Provable(NOT P) as a constructive form of the statement of consistency. In a constructive theory, I suppose that we would prove this form directly. In a classical theory, it follows from the usual NOT Provable(0=1). I don’t think it makes a difference to the connection with the provably total computable functions which way we do it.

I thought of this point partly because I recalled that the question of provably reducing a PI_2 formula to a PI_1 formula, which depends entirely on the growth rate of the Skolem/witness function, turns out to be the same for Classical and Constructive proofs.