Paradox or Contradiction? Which is which?

Contradiction.  Paradox.

I see these two terms a lot, usually in arguments over the legitimacy of some controversial conclusion.  Despite the evidence, an apparent logical consistency, or appearances to the contrary, something is asserted to be true.  When it is pointed out that the evidence does not support that conclusion, supporters of the conclusion will generally call the equation a “paradox” while detractors will call it a “contradiction”.  And the conversation can go no further.   

It got me thinking, how DO we actually define these words?  How do we determine whether something is a paradox or a contradiction?

Paradox:

-a statement or proposition that, despite sound (or apparently sound) reasoning from acceptable premises, leads to a conclusion that seems senseless, logically unacceptable, or self-contradictory

-a statement or proposition that seems self-contradictory or absurd but in reality expresses a possible truth

-a statement that seems to contradict itself but may nonetheless be true: the paradox that standing is more tiring that walking

Contradiction:

-a combination of statements, ideas, or features of a situation that are opposed to one another

-a difference in two or more statements, ideas, stories, etc. that makes it impossible for both or all of them to be true

-someone or something with qualities or features that seem to conflict with one another <a loving father as well as a ruthless killer, the gangster is a living contradiction>

-a statement or phrase whose parts contradict each other <a round square is a contradiction in terms>

This doesn’t clear it up at all.  It just highlights the whole dilemma all over again.  Both “paradox” and “contradiction” acknowledge that a given conclusion doesn’t make sense in light of the evidence.  A paradox says it is true.  A contradiction says it isn’t.  The word choice, then, is based on the preference of the speaker. 

So when is a conclusion wrong, and when is it right and it’s our own harmonization of the evidence that is lacking?  Forget the purely mathematical for a moment.

Take the example of “a loving father as well as a ruthless killer, the gangster is a living contradiction”. 

A person might argue that this is a paradox, not a contradiction.  Both personas exist – the loving father and the ruthless killer.  That they seem contradictory does not seem to negate the sheer fact of their existence. 

A contradictory take on this individual would assert that one of these personas is not true.  For example, if the gangster is truly a loving father then he must not really be ruthless killer.  What do we mean by “ruthless” and “loving” after all?  These are not binary terms, and the thought experiment is thus based on categorical errors.

So who’s to say?

My point is this.  We often don’t know how or why we come to the conclusions that we come to.  Very often we find ourselves believing things and we aren’t sure exactly how or why we came to believe them.  That isn’t to say that there isn’t a deliberative process, or that we are fully irrational.  It’s to say that there’s a lot that happens in our depths, below the levels of conscious choice. 

That's not to say that a we let any old thing go under the banner of "paradox".  It's just to say that we’re not just mathematical models running on as-yet undetermined computer code.

We are blurry and, despite the importance and appeal of "choice", maybe we are not determined by our own sheer will power and rationality.  

Our complexity demands humility.  

We ourselves are walking contradictions.  Or paradoxes.  Which is it again?