
Door images from Woodstone [5] 
Welcome to the third of the Three Doors of Serendip [4]. If you haven't been through the first two, you should go there and check them out first.
The first two doors consider understanding in terms of experience and empirical observations, but suggest that there must be something more to a broader understanding. Is there a way to reach understandings other than through experiences and observations? Perhaps a way of doing so that is more convincing? Could we approach this problem (and others) logically instead of through experiences and observations?
Logical Understanding
Let's think about that for a minute. Below are two different explanations of what the best strategy is for playing the Three Doors game. Can you decide which one is "right"? Could you have decided which one is right without having gone through the first two doors, i.e. logically?
EXPLANATION 1  EXPLANATION 2 
After one sees the prize is not behind the first door opened, one knows there are only two places the prize can be... and nothing more. Therefore, the prize has the same chance of being behind either of two remaining doors. Pick either one; it doesn't matter.  There is one chance in three that the door you first picked is the correct one. This can't change just because you're shown there isn't a prize behind some other door. Since the prize has to be behind either the door you picked or the other closed door, and there is one chance in three that its the one you picked, there must be two chances in three its behind the other door. Switch, and you'll be right twice as often. 
When the Three Doors game was first described, people argued at great length about the best strategy to use [6], using stories like those above, without anyone successfully persuading other people that they were right. What this suggests is that logic is not by itself a good way to settle many questions. Logic is a process of deducing consequences given an agreed upon set of starting points and rules to reach further conclusions. Therefore there is no single "logical" way to approach a question. Both explanations above are "logical", they differ in their conclusions because they differ in their starting points. There is no logic which can tell one how to pick among the possible starting points (cf. Paths to Story Telling as Life [7]).
To put it differently and more generally, logics are interesting in their own right and sometimes useful but one needs some empirical experience to decide which logic is relavant in a given case. For science, logic is something that provides useful ways to make sense of observations and implications of those, as well as to communicate them, rather than a starting point for them making sense of observations. Logic is a tool for constructing knowledge, not a criterion for it. The tools develop out of the attempts to understand rather than existing prior to them.
But you may still be puzzled by the observation that explanation 2 works better than explanation 1 and want to know why. You have a kind of understanding, empirical understanding, that derives from observations and experiences. But that understanding is inconsistent with some other understanding that you also have (i.e., a strong sense that since there are only two doors and the prize is behind one of them, there must be a 50% chance of it being behind each door). If there is no "abstract" logic that allows you to figure out which of your current understandings is correct, how do you decide which to trust? What is "broad" understanding if it isn't logic?
Maybe "broad" understanding is nothing more, and nothing less, than not having conflicting understandings. And maybe one acheives a broader understanding by noticing conflicting understandings and using their differences to create new ways of understanding things that eliminate the current conflicts. From this perspective, broader understanding, is not something one "should have had" nor something which is absolute and never to be in turn challenged. Broader understanding is simply the new understanding that one creates to resolve current conflicts in understanding (either within oneself or with other people).
Now that's an interesting idea with lots of implications beyond the case at hand. Let's try it out by seeing if we can reach a broader understanding of the Three Doors problem. To do so, go here [8]. Or if you want to think about the broader implications right now, go here [9].
Hands on understanding [10] 
Experimental understanding [11] 
Broader understanding 
 complete exhibit index [12] 
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