Appendix
1. A Dialectical Game
What
if two sentences refer to each other’s provability? Here is a table of the
results of such a dialog:
(A,B) B= prv A not
prv A prv not A poss A
A=
prv B (T,T) (S,D) (S,S) (S,D)
not prv B (D,S) (D,D) (P,T) (D,D)
prv not B (S,S) (T,P) (S,S) (S,D)
poss B (D,S) (D,D) (D,S) (P,P)
Consider
the row A = prv not B. Player A has accused player B of being outright
refutable. A said to B, “I am sure
you are wrong!” If you were player B, then what would be your best reply?
It
would be to say to player A, “You may be right!” For if A = prv not B, and B =
poss A, then A equals self-shame and B equals self-doubt. In this
confrontation, A committed the error of excessive certainty; whereas B’s
modesty yields Gödelian truth.
Thus
‘a soft answer turneth away wrath’.
Consider
this subgame: the “doubt-shame game”:
(A,B)
B= not prv A prv not
A
A=
not prv B (D,D) (P,T)
prv not B (T,P) (S,S)
If you rank the
outcomes thus:
P < S < D < T
then this game
is a non-zero-sum “Prisoner’s Dilemma” game.
Now
consider this subgame: the “trust-pride game”:
(A,B)
B= prv A poss
A
A=
prv B (T,T) (S,D)
poss B (D,S) (P,P)
This is a
dilemma game if you rank the outcomes thus:
S < P < T < D
In
a dilemma game, both parties can benefit if they cooperate, but both are
tempted to cheat, to their mutual loss. It is a social dilemma, arising here
from mathematical paradox.
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