[FartyMcFarter]

> For any single program, one of these two functions will correctly output whether it halts or not.

Saying "one of these functions is correct" is not a decision procedure. You actually need to decide which one of them is correct.

> This is why it's not very meaningful to talk about the decidability of particular Turing Machines

I disagree. There are particular Turing machines whose decidability is extremely meaningful. For example, there are specific Turing machines which encode mathematical problems of interest such as the Goldbach conjecture. Deciding whether they halt is equivalent to solving those mathematical problems, which is definitely meaningful.

[Maxatar]

>Saying "one of these functions is correct" is not a decision procedure. You actually need to decide which one of them is correct.

That's precisely what a decision procedure is, it's an algorithm that takes as input the description of a Turing machine, and an input, and returns true if and only if the Turing machine halts when given the input. You are using the term "decide" as if there were some kind of agency involved, like you have to actually "choose" what is correct or incorrect, but no such agency is involved, it is a purely mechanical process.

>There are particular Turing machines whose decidability is extremely meaningful.

You are mixing up the notion of decidability with the notion of halting, and there is a subtle difference. The Turing machine that encodes the Goldbach conjecture proves the conjecture if it halts, and disproves the conjecture if it doesn't halt. That is an interesting and meaningful property of such a Turing machine. What is not meaningful or interesting is whether that particular Turing machine is decidable.

As I said, there is a subtle difference between whether a Turing machine halts or not, and whether it's decidable whether it halts or not. The former is interesting, the latter is not particularly insightful.

[FartyMcFarter]

> You are using the term "decide" as if there were some kind of agency involved, like you have to actually "choose" what is correct or incorrect

Isn't this literally what a decision procedure is? It's an algorithm that always outputs the correct answer, not one which tells you "the correct answer is either true or false". I'm not talking about agency or consciousness.

[Maxatar]

One of the two algorithms I posted tells you the correct answer for any specific Turing machine, so for any given Turing machine one of them is a decision procedure for it.

[FartyMcFarter]

Sure, that is trivially true, but I don't see how that statement is useful in any way.

[Maxatar]

Exactly, so then we agree, there is nothing useful about an algorithm that can decide whether one specific Turing machine halts. Utility only emerges from an algorithm that can decide whether entire classes of other Turing machines halt, not from whether a particular machine does.

[FartyMcFarter]

> there is nothing useful about an algorithm that can decide whether one specific Turing machine halts.

Yes there is. If someone gives me a working algorithm (this means one algorithm, not two algorithms either of which may work or not as you've been doing) which can provably decide whether the machine encoding the Goldbach conjecture halts or not, that is a very useful algorithm.

I must insist that giving two algorithms and saying "one of them works" without telling me which one works is is not a valid answer, since that is strictly different from giving a working algorithm which I can run in finite time in order to find out one correct answer.

You said this yourself earlier:

> The Turing machine that encodes the Goldbach conjecture proves the conjecture if it halts, and disproves the conjecture if it doesn't halt. That is an interesting and meaningful property of such a Turing machine.