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I am reading about probability and beliefs in artificial intelligent agents, and came across the following passage:

Why the axioms of probability are reasonable

The axioms of probability can be seen as restricting the set of probabilistic beliefs that an agent can hold. This is somewhat analogous to the logical case, where a logical agent cannot simultaneously believe $A$, $B$, and $\neg(A \wedge B)$, for example. There is, however, an additional complication. In the logical case, the semantic definition of conjunction means that at least one of the three beliefs just mentioned must be false in the world, so it is unreasonable for an agent to believe all three. With probabilities, on the other hand, statements refer not to the world directly, but to the agent's own state of knowledge. Why, then, can an agent not hold the following set of beliefs, given that these probability assignments clearly violate the third axiom? $$\begin{align*} P(A) &= 0.4 \\ P(B) &= 0.3 \\ P(A \wedge B) &= 0.0 \\ P(A \vee B) &= 0.8\end{align*}$$

Source: Artificial Intelligence A Modern Approach

I do not understand the meaning of this paragraph. Previously, the book restated some axioms of probability, such that $P(A \vee B) = P(A) + P(B)$ for mutually exclusive events, and $P(A \wedge B) = P(A) \times P(B)$ for independent events.

From my understanding, the statement above says the following:

In a statement using purely logic, information represents the world itself. As such, laws of logic forbid maintaining simultaneously that $A$, $B$ and $\neg(A \wedge B)$ hold. As logic however is an individual's understanding of the world, then the set of probabilities above can be maintained even though they defy the axioms of logic.

My question is, wouldn't an agent have a built-in engine which signals when probabilities are not coherent with each other? How can an agent still maintain them?

Hope this makes sense! :)

Cheers!

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  • $\begingroup$ It might be useful to distinguish a few questions: how do real agents manage their beliefs, how should artificial agents manage their beliefs, and do you want your artificial agents to manage beliefs like real agents or like ideal agents? In any case, you might want to look at the recent dissertation Probabilistic Reasoning with Incomplete and Inconsistent Beliefs by M. Thimm. $\endgroup$ – Alan Nov 21 '14 at 13:58
  • $\begingroup$ @Alan Fixed -- thanks for pointing it out. You can also edit the question yourself by clicking the "edit" link below it. $\endgroup$ – David Richerby Nov 21 '14 at 14:16

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