So what is a logical contradiction? When does something constitute as such? I've found the term is often used without any logical proof, which is necessary to prove a contradiction. Here I will attempt to do away with the ambiguity. After going through this I think you'll find that logical contradictions require much more than is typically thought. Prepare yourself, it may be a little on the technical side.

There are 3 kinds of contradictions I'll cover, namely *explicit*, *implicit*, and *formal*.

**EXPLICIT CONTRADICTIONS**

Explicit contradictions are ones of a certain sort and, as such, are rarely ever espoused. And by rarely, pretty much never. An explicit contradiction would be a conjunctive proposition where one conjunct is the negation of the other. For example:

(1) John is a good tennis player, and it is false that John is a good tennis player.

It should be obvious that this conjunctive proposition is explicitly contradictory. Not much more needs to be said here. The other two are much more involved.

**IMPLICIT CONTRADICTIONS**

Implicit contradictions are those that require *additional necessarily true propositions*, that when added to the original proposition create a formally contradictory set of propositions. Take the implicitly contradictory proposition "There exists a married bachelor". This is quite clearly contradictory but it is not *explicitly *contradictory, so defined. It requires an additional necessarily true proposition to create a formal contradiction (which eventually and inevitably explicates an explicit contradiction).

(2) There exists a married bachelor

(3) Bachelors are unmarried

**FORMAL CONTRADICTIONS**

In the above example, (3) is broadly necessarily true. Here we can see that the conjunct of (2) and (3) create a formal contradiction, but not yet explicit. Formal contradictions are those that, by using the laws of logic, can arrive at a conclusion that when combined with the original set derive an explicit propositional contradiction.

(4) If there exists a married bachelor and bachelors are unmarried, then there exists a married unmarried man.

(5) If there exists a married unmarried man, the law of non-contradiction is false.

(6) But the law of non-contradiction is strictly necessarily true.

(7) Therefore, no married bachelor exists.

The conjunct of (7) and (2) combine to create an explicit contradiction: "There exists a married bachelor, and no married bachelor exists."

I'd be happy to apply this to practical usages such as in the logical PoE, just let me know and we can discuss. Otherwise, feel free to comment!