Kripke Structure Intuition I was studying about Intuitionistic Logic and I saw that instead of assignments, there we use Kripke Structures. My Question is: Why can't we use boolean assignments to define Intuitionistic Logic aswell ? Why do we have to define Kripke  Structures ?
Sorry if this question is a bit unclear. 
 A: If you assign Boolean values to your propositions, then you just have classical logic again, which sort of defeats the point. What you want for intuitinistic logic is for your propositions to take values in a Heyting algebra, where excluded middle/double negation elimination doesn't necessarily hold.
So where do Kripke frames come in? Note that one of the things that has to be true about an intuitionistic Kripke frame $(P,\leq)$ is that we have $$p\leq q,\;p\Vdash\varphi\implies q\Vdash\varphi$$ for all $p,q\in P$ and propositions $\varphi$; that is, the set of conditions that force a proposition are closed upwards. In any partial order, the set of upwards-closed sets form a complete Heyting algebra under the appropriate operations, so that what we're really doing with a Kripke frame is assigning to $\varphi$ the Heyting truth value $\{p\;|\;p\in P\wedge p\Vdash\varphi\}$.
One of the reasons it's nice to work via Kripke frames, as opposed to dealing directly with Heyting algebras of various sorts, is that the way we define $\Vdash$ (and therefore, the way we define the Heyting truth value of each proposition in the model) is by giving a classical valuation associated with each $p$, which can be easier to think about. Our Heyting-valued assignment then comes about from the way these classical valuations are related to each other by the frame $(P,\leq)$.
