I am learning about the principle of noncontradiction ~(p^~p). I can see that this would work if we assume that 'p' can only be true or false. Why should I make this assumption. I can see a lot instances where we need more than 2 truth values (how people feel about the temperature of a room, for instance could have an infinite number of responses, and all would be true because the proposition is based on subjective experiences).
What is this type of logic called? If this is a possible logic then can't someone argue that everything is this way?