What is logical about the prisoner's dilemma? In the Wikipedia example of The Prisoner's Dilemma it states that "all purely rational self-interested prisoners will betray the other, meaning the only possible outcome for two purely rational prisoners is for them to betray each other."
https://en.wikipedia.org/wiki/Prisoner%27s_dilemma
My question is why would they betray each other if they are rational?
Why wouldn't being rational (logical) result in each concluding that there's no reason to think the other would choose differently...therefore the only choice is to stay silent to guarantee the minimum sentence as there is no possibility for the other options?
 A: Other answers and comments explain why betrayal is dominant, echoing  the standard argument (in the linked wikipedia page) that

the only possible outcome for two purely rational prisoners is for
  them to betray each other.

Your question suggests a different definition of "rational", one  which makes "do unto others" a rational strategy. I think that is a defensible philosophical position. 
It's this kind of argument that makes the prisoner's dilemma worth thinking about.
A: Confessing is a dominant strategy for both players in the Prisoner's Dilemma. 
A few more lines on why confessing is a dominant strategy. Consider yourself in the game. Consider both the possibilities of what the other guy would do (confess, stay silent) and convince yourself that you would pick confessing as it has a better payoff. Since the game is symmetric, the other guy would do the same to you and you'd both end up confessing.
A: I think you're right. Wikipedia says that "all purely rational self-interested prisoners will betray the other". However, in order to show that a prisoner will betray the other, we actually need three assumptions:


*

*The prisoner is purely rational. 

*The prisoner is self-interested. 

*The prisoner assumes that the other prisoner's decision is totally independent of her own. 


Betraying the other player is said to be a dominant strategy. 
You have noticed (correctly) that prisoners will instead cooperate if we make these three assumptions:


*

*The prisoner is purely rational. 

*The prisoner is self-interested. 

*The prisoner assumes that the other prisoner will always make the same decision that she herself makes (since, after all, the two prisoners are identical).


I don't know of a particular name for a strategy which is best according to these criteria. However, your reasoning reminds me of Eliezer Yudkowsky's "timeless decision theory", which states that if you're in a prisoner's dilemma, and you and the other prisoner are both using timeless decision theory, then you should cooperate (since the other prisoner will make the same decision that you will make). 
A: Betrayal nets you freedom if the other stays loyal, or two years if the other also betrays.
Loyalty nets you one year if the other is also loyal, or three years if the other betrays.

My question is why would they betray each other if they are rational?

It is in the "self-interested" part of "purely rational self interested."   That aims for freedom and avoids three years but accepts two years as the result of  neither party trusting nor caring about the other.

Only if one could trust the other and each cared about the collective best result , then would loyalty be the rational best choice. 
The very point of the experiment is that people tend to not be purely self interested or rational in such a scenario.  In practice people would stay silent trusting the other to do the same, even though it is in their self-interest to--as the experiment's wording deliberately frames it--'betray' the other.
A: I USED TO agree that betraying is a good strategy.
If Prisoner B thinks that he is better off staying silent, Prisoner A can be free while Prisoner B can stay in jail for 3 years.
Now, I agree with Vizag.
CONFESSING, HOWEVER, SEEMS LIKE A BETTER STRATEGY. Just consider the combinations and pick the choice to confess. Prisoner B would do the same thing as you.
