*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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