Principle of contradiction denies the existence of contradictions, that is, the existence of combinations of contradictories. What then are these contradictories one might ask? Contradictories themselves are a kind of opposites, Wolff answers. Opposites, on the other hand, are such things that cannot exist at the same time, in the same situation (for instance, complete blackness and complete whiteness cannot exist in the same surface). Contradictories are then opposites, one of which must exist in a situation.
It is a well-known fact that when one modal notion (e.g. possibility) is defined, the rest of the modalities can be defined from that beginning. Thus, impossibility is contradictory of possibility: what is not possible, is impossible, and vice versa, and things must be either possible or impossible.
More importantly, when the opposite of something is impossible, this something itself must be necessary. That is, when some situation or thing has no capacity of ever becoming actual, it's opposite must undoubtedly have the power to actualise itself in every situation. For instance, a figure with three sides, but not three angles would be something impossible and could not ever be actualised, thus, if we do have an actual figure with three sides, it must be actualised with three angles. Generally, all such combinations or propositions describing them are necessary, if the predicate could be deduced from the definition of the subject.
Now, there is a special case of necessary propositions, that is, propositions describing the existence of something necessary. Here it is not any feature of the thing that is necessary, but the very entity is supposed to be such that its non-existence would be impossible. In other words, the actualisation of the haecceitas of such an entity would be necessary. Because this haecceitas or individual essence would contain at least implicitly all the predicates of the thing, it could not really have any other predicates. In other words, it could not change into anything else, but would eternally be what it is.
Wolff leaves it open for now, whether there are any concrete necessary individuals – this is a task left for other branches of metaphysics. Then again, Wolff does find examples of more abstract necessary entities. In Wolffian ontological scheme, what is absolutely possible is defined by its non-contradictoriness, and thus, one cannot change what is possible. If something is then possible, it is necessary possible. Then again, essences or coherent combinations of essential predicates correspond to certain possibilities. These abstract combinations or lists of predicates are then necessary, which means merely that it must be possible that some things satisfy these combinations of predicates.
Wolff also points out that there are actually two different concepts of necessity. Firstly, one can speak of necessity plain and simple or absolute necessity – this is essentially what we have considered now. Then again, there is also hypothetical necessity, that is, necessity under some assumption. Wolff's example of hypothetical necessity is the relationship between a feature of a thing determining what other features the thing has: for instance, if a certain figure has three straight lines as its sides, then it is necessary on this condition of its trianglehood that it also has three angles.
This mathematical example is a good point to move to consider how mathematics is presented in Wolff's ontology, which will be the topic of my next post.