Bayesian Programming and Learning for Multi-Player Video Games ...

• modus ponens, A entails B and A = true gives us B = false:
translates to
[A ⇒ B] ∧ [A = true]
B = true
P(B|A, C) =
As P(A, B|C) = P(A|C) we have P(B|A, C) = 1
P(A, B|C)
P(A|C)
• modus tollens, A entails B and B = false gives us A = false (otherwise B = true):
translates to
[A ⇒ B] ∧ [B = false]
A = false
P(A|C¬B) = P(A¬B|C)
P(¬B|C)
As P(A¬B|C) = 0 we have P(A|¬B, C) = 0, so P(¬A|¬B, C) = 1
Also, additionally to the two strong logic syllogisms above, plausible reasoning gets two weak
syllogisms, from:
•
•
we get
P(B|A, C)
P(A|B, C) = P(A|C)
P(B|C)
[A ⇒ B] ∧ [B = true]
A becomes more plausible
If A stands for “the enemy has done action a” and B for “the enemy is doing action b”:
“the enemy is doing b so it is more probable that she did a beforehand (than if we knew
nothing)”. Because there are only a (not strict) subset of all the possible states (possibly
all possible states) which may lead in new states, when we are in these new states, the
probability of their origin states goes up.
we get
P(¬A|B, C)
P(B|C¬A) = P(B|C)
P(¬A|C)
[A ⇒ B] ∧ [A = false]
B becomes less plausible
Using the same meanings for A and B as above: “the enemy has not done a so it is less
probable that she does b (than if we knew nothing)”. Because there are only a (not strict)
subset of all the possible states (possibly all possible states) which may lead in new states,
when we are not in one of these origin states, there are less ways to go to new states.
A reasoning mechanism needs to be consistent (one cannot prove A and ¬A at the same time).
For plausible reasoning, consistency means: a) all the possible ways to reach a conclusion leads
to the same result, b) information cannot be ignored, c) two equal states of knowledge have the
same plausibilities. Adding consistency to plausible reasoning leads to Cox’s theorem [Cox, 1946],
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