I was wondering if anyone has any insight into the nature of contradiction. I have encountered contradictions before, but it is difficult to define the concept. I might attempt to define it as the presence of incompatible statements which might mean one of them must necessarily be false. Please let me know if you have any thoughts on this topic because I would love to learn more about it.
Care to join me?
For every truth, there is unlimited potential failures and falsehoods. Like we can talk about a unicorn. But we can say there are two unicorns, or one has a black horn, or they have golden hoofs, etc. All that is false, but one thing is true = there is no unicorn.
That’s not true though, there are unicorns. They exist in many ways, for example as an idea within human minds. Also as drawings and in films. And who knows, maybe there is or was a horse with a horn on its head, so it could also be said that unicorns do or did exist as physically alive animals. We are in no position to say 100% that this is impossible.
The whole thing about logical contradiction is that it is rooted in LOGIC. Not in looking at human ideas about exotic or mythological animals. There is no logical contradiction in the idea of a unicorn, in fact the unicorn is quite logically consistent: a horse with a horn on its head. There is no contradiction implied here, only the fact that as far as we seem to know no such animal really does exist in the world. Humans seem to have made up the idea by combining the idea of horse with the idea of “horn on the head”. Such an idea does not example a logical contradiction, it examples an act of creative-logical combination.
Hi, welcome here 
The nature of logical contradiction is the same as the nature of logical non-contradiction: necessity and impossibility. These are definitional, for example the whole “all bachelors are unmarried men” thing. Once you define A as A, and B as B, and you can form a relation of necessity between them then you have arrived at logic. The logic of necessity and the impossibility of this necessity being violated.
Just like 1+1=2. Let’s say we add to that, another statement: 1+1=3. Well, 1+1=3 is a logical contradiction. Why? Because it violates the meaning of “1”, “+”, “=” and “3”. Each of those four terms have a necessary meaning. To say that 1+1=3 is to construct a contradiction that violates the necessary meaning of one or more of the terms in question.
Talking about it as incompatible statements isn’t wrong either, but we should be careful not to only attempt to understand this issue at the level of statements and language. Logic is really about the level of meaning itself, which is beyond and prior-to the level of statements and language-utterances. Logic can and should be viewed as necessity-as-such or as this necessity-as-such manifested in a particular meaningful way. Once this is realized and understood philosophically, the nature of logical contradiction follows quite easily as simply the exact inverse of that.