To simplify the work of a lab designer,
We have standard refutation protocols to choose from.
For the same reason,
we offer standard binary games.

I am not sure about the name: binary game.
It is an important part of a binary game.

binary game 1
P is given claim c.

assertion T:
P asserts assertTrue(c).
Substantiation:
P defends c against O, P wins refute(c,P,O).

assertion F:
P asserts: assertFalse(c)
Substantiation:
P refutes c against O, i.e., P wins refute(c,O,P).

binary game 2
P is given a claim c.

assertion Lopt:
P asserts: c is locally optimum.
Substantiation:
P defends c against O, i.e., it wins refute(c,P,O).
P proposes a stronger c, called c', and refutes c' against O,
i.e., P wins refute(c',O,P).

assertion !Lopt:
P asserts: c is true but not locally optimum.
Substantiation:
P defends c against O, i.e., it wins refute(c,P,O).
P proposes a stronger c, called c', and defends c' against O,
i.e., P wins refute(c',P,O).

binary game 3
P is given a claim c.

assertion indet:
P asserts that claim c is indeterminate.
Substantiation:
If refute(c,O,P) is played n times, the probability that O
wins n times is 2^-n.

assertion T (see above)
assertion F (see above)

binary game 4
P chooses two claims c1 and c2 from claim set
and asserts c1 => c2.
Substantiation:
If P is given the oracle to win refute(c1,P,O) then 
P can win refute(c2,P,O).
Not constructive enough.
Use a transformation.