A necessary truth is a true statement whose negation must imply a contradiction in reality, such that the negation would be impossible.
So, if “One plus one equals two,” is a necessary truth, then the statement “One plus one does not equal two” will imply a contradiction. Given the meanings of “one” and “two,” we can immediately see that the addition of two “ones” (units) always does yield “two,” yet the statement “One plus one does not equal two,” contradicts this. It’s incomprehensible that one plus one should ever add to anything but two. So “One plus one equals two,” is commonly held to be a necessary truth, with its negation being impossible.
A contingent truth is a true statement whose negation does not imply a contradiction in reality, such that the negation could have been the case.
So, if “John married Jessica last Sunday,” is a contingent truth, then the statement “John did not marry Jessica last Sunday,” could have been true, without implying a contradiction in reality. Since John could have chosen not to marry Jessica, or to have married her on a different day, we can see that this is indeed a contingent truth.
The Objectivist View on the Necessary/Contingent Distinction
Causality (the Law of Cause and Effect) is the Law of Identity applied to action. This means that an entity’s actions follow from its nature. That is, the nature of the entity (its attributes, properties, etc.) causes the action it will take in any given situation. In any given context, there is only one action open to it: the one in accordance with its nature. Any other action would contradict its nature.