The human mind is deceitful and desperately wicked.  You and I can be fooled by our own minds.  Self-refuting ideas are an example of this wickedness.  Here are some examples:

Post-Modern and Relativist ideas are self refuting.  They claim that there is no truth or error, no right or wrong.  They claim that all things are relative.  Here is how that refutes itself:  If all things are relative, then the statement that "all things are relative" is also relative and you can't depend on it.  If there is no truth, then the notion that "there is no truth" is not true.

Other people may say, "There may be truth, but you can't know what is true and what is false."  Then that statement refutes itself because you can't tell whether it is true or false.

Some Christians claim that there are no miracles, no apostles, no revelation, no spiritual gifts, or any such any more.  The problem is that this is a self-refuting idea since, if this were true (it is not true), then only by extra-biblical revelation could anyone come up with such an idea.  In fact, any interpretation of Scripture is either an example of speculation or revelation.  The assumptions of rationalism and Naturalism are so ingrained in us by the culture that it makes it easy for us to be tricked by naturalistic explanations, but the Christian walk is a supernatural walk, even in an unbelieving culture.

Naturalism is the notion that only natural things exist.  It is the claim than no angels or spirits do anything.  It is the claim that God does nothing.  It's the claim that God doesn't reveal Himself and that the only things that can be known must be known by natural means.  That is self-refuting because the ideas of Naturalism cannot be shown to be true by natural evidence.  It is either a divine revelation or an out-and-out lie.  And they claim that it is true, which claims supernatural revelation to know it, which violates the tenats of Naturalism.

Materialism is the claim that only material things exist.  In the same way that Naturalism is self-refuting, Materialism is self-refuting.

These are just a few examples.  There are many others.  These self-refuting ideas are constantly creeping into human thinking.