Four valued logic

Given a preordering relation there are four possible results of a comparison between two elements: they are incomparable, the first value is less then the second, the first value is greater then the second, and they are both equal. Given that the comparison is less then there are four possible logical values corresponding to these comparison results: none, false, true, and both.

There is a knowledge order corresponding to these four truth values [#{none}, #{false true} #{both}]. We can intersect the comparison results of preorders under the knowledge ordering to get another preorder. When we cannot produce an exact comparison like with irrational numbers the none value can be used to represent our uncertainity.

In general the values of none, false, true, and both can be used to deal with uncertain and/or non-monotonic reasoning. One advantage of four valued logic over three valued logic is that it is still binary so each truth value can be represented as two bits in this system.