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In the literature of interactive epistemology, for a player, knowing an event obtains with probability one and knowing an event obtains with absolute certainty are different. Is there a nontrivial (say, it could be the case, an event is a proper subset of the event that player $i$ knows this event obtains with probability one, but never the case for knowledge operator with absolute certainty) proposition that could serve as an example to show they're not interchangeable in the language which consists of two corresponding knowledge operators, a set of states of world, set relations, set operations, and all logical connectives.

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If you take the standard Aumann model, but allow for states occuring with probability zero, you can taje some nonempty event $N$ such that for the agents prior $p$, one has $p(N)=0$. Let $K\cap N=\emptyset$ and $p(K)>0$. If the agent knows $K\cup N$ but not $K$, then she assigns probability $1$ to $K$, even though she does not know $K$.

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I have some difficulties parsing your question (especially the "this" in "it could be the case, an event is a proper subset of the event that player i knows this event obtains with probability one, but never the case for knowledge operator with absolute certainty") so maybe what I am proposing here is "trivial" (and therefore not what you are looking for). But I don't have enough reputation to comment so this ends up an "answer" anyway. See it as a question asking for clarification.

Let $X\sim Unif([0,5])$ be a random variable.

Let $E_1=\{X\in [0,5]\}$. The agent knows with certainty $E_1$, since the agent knows with certainty that $X\sim Unif([0,5])$.

Let $E_2 = E_1\setminus \{3.14\}$. That is, $E_2$ is the event that $X$ is not $3.14$. Now, $P(E_2)=1$ so the agent knows with probability $1$ that $E_2$. However, the agent does not know $E_2$ with absolute certainty since it is possible (but only with probability $0$) that $X$ is in fact $3.14$.

Is this what you have in mind?

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  • $\begingroup$ Thank you. What you are proposing is correct. But I found them quite straightforward in light of definitions. I want to know why in literature, these two operators are distinguished, i.e. why interchange one operator with the other could generate some interesting result. $\endgroup$ – Metta World Peace Feb 11 '15 at 8:11
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    $\begingroup$ @MettaWorldPeace I would convert this to a comment but most of it would get cut off. I'm not sure what to do. I could either leave it here, because it's useful clarification. Or you could incorporate it into the original question as clarification. $\endgroup$ – jmbejara Feb 12 '15 at 9:03
  • $\begingroup$ @jmbejara Thanks for attention. In that case, I prefer to leave it here, because it's Monkeynomics' contribution. $\endgroup$ – Metta World Peace Feb 12 '15 at 12:44
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An event with a probability $1$ happens almost surely. There is the same difference between that and absolute certainty that in English. One cannot definitively say that these outcomes will never occur, but can for most purposes assume this to be true. The distinction is important because if $A_n, n \in \mathbb{N} $, with for each $n$ $P(A_n)=1$, then you cannot assume $P(\cap_{n \in \mathbb{N}} A_n)=1 $, which you could if the events were absolutely certain, which is why one need to pay attention to the distinction, even though it has no impact on the immediate consequences of the statement.

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