Curry's paradox hasn't gotten much attention in Agora. It came up in
discussion a couple of times, and in terms of usage in-game, all I found
was someone CFJing a free-floating sentence "If this sentence is true,
then I win." That was about 10 years ago.
Well, this isn't a free-floating sentence. I'm not claiming that I win
merely because I uttered "If this sentence is true, then I win", or the
mutually-referential version of that.
The sentences in question are not directly self-referential or even
mutually-referential. This is more of a Curry-flavoured confused deputy,
with rule 2337 as the deputy. It says that the author can destroy a
promise with notice IFF the sentence in its "destruction by author
condition" slot is true. So:
- Sentence A: I can do Y.
- Sentence B: IF (I can do X), THEN (Z is true).
- Rule 2337 says that (I can do X) IFF (sentence A is true)
- Rule 2337 says that (I can do Y) IFF (sentence B is true)
As a result, R2337 says that (I can do X) IFF (IF (I can do X) THEN (Z
is true)). So, R2337 says that Z is true. And Z in this case is that I
can de-register everyone else, and that nobody else can register. This
is consistent and within the power of R2337: R869 secures
de-registration, and it has Power 2. And it does not secure the
prevention of people from registering. R2337 has Power 3.
My announced intent to destroy the promises is a red herring, by the
way. The logical deduction only refers to my ability to destroy a
promise with notice. So as long as the promises exist, R2337 asserts Z.
It is sufficient here to assume classical logic (i.e. boolean,
true-or-false logic) works inside the "R2337 says that" prefix. And, it
is not even necessary to assume that. The conclusion requires only a few
principles of natural (relevant, constructive, paraconsistent)
deduction. I can elaborate on this, if someone wants to argue that some
type of formal logic other than classical is what works inside "R2337
says that".
In particular if you think that classical logic should work within
"R2337 says that", but tentatively, subject to revision if it results in
a contradiction, then you're out of luck, because there is no
contradiction. Z is consistent.
It is true that this same mechanism could easily be used to create
contradiction in the rules, by doing the same thing with NOT Z, for
example. But that's not a reason to declare the mechanism invalid. After
all, standard rule-amending processes could also be used to create
contradiction in the rules, if players wanted to. The only real
difference is that this can be done unilaterally and immediately. That
is merely a "political" rather than a logical difference. I've only used
it to cause the rule to assert Z. I have no plans to cause it to also
assert NOT Z or any other sort of contradiction. And nobody else can use
this mechanism to cause it to assert a contradiction, because Z closes
the door behind me, and locks it.
