1

According to the Cambridge dictionary,

Counterintuitive: Something that is counter-intuitive does not happen in the way you would expect it to

and

Contradiction: a fact or statement that is the opposite of what someone has said or that is so different from another fact or statement that one of them must be wrong

How something was counterintuitive? Someone must have read or proved facts in the past and that is why one will not expect opposite things to happen in future.

If both the words are providing a sense of opposition, then what is the exact difference between them?

Also, what are the appropriate contexts to use them?

2

If something is counterintuitive, it does not work in the way that you instinctively expect - for example, a swing door that you need to push from one side sometimes has a handle on that side as well, which tempts people to pull on the handle instead of pushing.

It's not about having 'read or proved facts'; it's the opposite of what your intuition tells you.

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