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
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
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
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.