On Aug 4, 2013, at 7:07 PM, Jonathan Rouillard wrote:
>
> ============================== CFJ 3381 ==============================
>
> I am a player.
>
> ========================================================================
>
> Caller: omd
Arguments:
Recently, Fool purported to deregister all first-class players other than
emself, by means of a logical deduction from a certain set of Promises. This
CFJ asks whether Fool's attempt succeeded.
At first glance, it seems like the unavoidable conclusion is that Fool's
attempt did indeed succeed. Fool submitted two promises, titled
"Paraconsistency is overrated, part 1" and "Paraconsistency is overrated, part
2" (which we will abbreviate as P1 and P2), with (essentially) the following
conditions for destruction by author:
P1: P2 CAN be destroyed with notice.
P2: If P1 CAN be destroyed with notice, then Fool CAN deregister all other
first-class players.
Rule 2337 "Promises" (along with the other clause in the rules stating that if
one rule says an action CAN be taken under certain circumstances, then the
action CANNOT be taken outside those circumstances) states (again, essentially)
that a promise can be destroyed with notice if and only if its destruction
condition is satisfied. So we apparently have the following two axioms:
(D-P1) P1 can be destroyed if and only if P2 can be destroyed.
(D-P2) P2 can be destroyed if and only if (if P1 can be destroyed, then Fool
can deregister all other first-class players).
Under classical logic, it can be proven from these axioms that Fool can
deregister all other first-class players. The proof is as follows:
Suppose that P2 cannot be destroyed. Then, by D-P1, P1 cannot be destroyed,
either. This means that the statement "if P1 can be destroyed, then Fool can
deregister all other first-class players" is vacuously true. But this means
that the left-hand side of the biconditional of D-P2 is false, whereas the
right-hand side is true; this is a contradiction. So we can conclude that P2
can be destroyed.
Since P2 can be destroyed, by D-P1, P1 can be destroyed, too. Thus, by D-P2,
Fool can deregister all other first-class players.
It has been suggested that intuitionistic logic ought to be used to interpret
the rules, instead of classical logic. Unless I have made a mistake in querying
lambdabot, intuitionistic logic does not allow us to conclude that Fool can
deregister all other first-class players:
<tswett> @djinn (p2 -> p1) -> (Not p2 -> Not p1) -> ((p1 -> fool) -> p2) ->
(Not (p1 -> fool) -> Not p2) -> fool
<lambdabot> -- f cannot be realized.
However, intuitionistic logic does allow us to conclude that it is not
IMPOSSIBLE for Fool to deregister all other first-class players. This
conclusion seems no better than the conclusion that it is POSSIBLE for em to do
so.
Agoran tradition seems to be to use a sort of vague paraconsistent rule of
thumb when dealing with paradoxes, namely, something like this: "when some part
of a rule contradicts itself, declare the truth value of the contradictory
statements to be 'paradoxical', and do not let this declaration lead to any
unreasonable consequences". But this is irrelevant here, because there is no
contradiction; Agora has no tradition (and probably shouldn't have a tradition)
of using any form of paraconsistent or otherwise non-classical logic in the
absence of contradictions.
So, to recap, given the statements in the rules, it seems to be an unavoidable
logical conclusion that Fool's attempts to deregister all other first-class
players succeeded. However, I think there is a reasonable nomic-philosophical
(nomicological?) viewpoint according to which Fool's attempts to deregister
failed.
Agora is a game that is played according to its rules. But what does it mean to
play according to a set of rules? One interpretation, perhaps the traditional
interpretation of Agora, is that the rules should be treated as axioms in a
logical system, and then the state of the game is whatever can be concluded
from these axioms. But the axiom interpretation is not without its problems;
indeed, one significant failing of this interpretation seems to be the fact
that it can produce paradoxes.
I would like to suggest an alternative interpretation of the rules: namely,
that the rules are a complete and comprehensive set of mechanisms for
interacting with the game. Thus, even if it is possible to prove, using
classical logic, that some action CAN be taken, this proof is irrelevant to the
possibility of the action; the action can still only be taken if there is in
fact a mechanism for taking that action.
Let us take another look at Rule 2337 "Promises" using the mechanism
interpretation. The relevant paragraph says:
If a promise has one or more conditions under which the author
of the promise can destroy it, and they are all satisfied, then
the author CAN destroy that promise with notice.
Under the mechanism interpretation, this paragraph provides a conditional
mechanism for destroying a promise, and no other mechanisms. Even though the
paragraph logically entails that Fool CAN deregister all other first-class
players, e in fact CANNOT do so, because there is no mechanism for doing so.
So is the mechanism interpretation acceptable? I dunno. I feel like there are a
few things going against it.
One objection to the mechanism interpretation is that you just aren't actually
playing the game if you're using the mechanism interpretation; you're only
actually playing the game if you're treating the rules as axioms. Though a
similar argument would claim that you're only actually playing the game if
you're treating the rules as axioms *under classical logic*, in which case
Agora became unplayable as soon as its first paradox was introduced, or perhaps
even as soon as two rules contradicted each other.
Another objection is that some clauses consist of definitions, and it doesn't
make sense to interpret a definition as a mechanism. But this could perfectly
well be fixed just by saying that some clauses are mechanisms, and some clauses
are definitions instead.
Another objection is that a statement of the form "if X then Y" can't really be
interpreted as a mechanism, since the "if" and "then" form a logical connective
that's meaningless as a mechanism. I don't think this objection is right at
all, because you could perfectly well interpret that as a mechanism for doing Y
that can only be used under circumstance X.
I definitely feel like there are more possible objections I've missed, but I
can't actually think of any.
Consideration of the best interests of the game seems to prefer the mechanism
interpretation over the axiom interpretation: after all, under the axiom
interpretation, everyone but Fool has been deregistered, whereas under the
mechanism interpretation, we're all still players.
TRUE appears to be appropriate.
—Machiavelli
