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I didn't really understand what the examples are for. The essence of these philosophical positions is quite definite.

Empiricism-believes that any true knowledge is always based on sensory experience, without options.�

Rationalism-claims that in addition to this, there is also knowledge that is not derived through experience, but relies only on mental reasoning and logic (for example, mathematics).

The consistent empiricist knows the world through incomplete induction. �Let's take a trivial example: � 2*n > n+2, n is a natural number greater than 2 (statement).

The empiricist checks the statement for the beginning of a natural series. �For 3, 6 is greater than 5.

For 4, 8 is greater than 6. �For 5, 10 is greater than 7. �Works! �But the empiricist cannot be completely sure. �After all, for the entire infinite series – you can't check it!

A rationalist works by proving rationally. �For example, by the method of full induction. �In the previous example, this means:

Base of induction. �For 3, 6 is greater than 5.

The induction step. �If the statement holds for n: �2*n > n+2, you must prove that it holds for n+1 as well

n+1: � �2*(n+1)>(n+1) +2

2n +2 > n+3

2n > n+1

And if 2*n > n+2, then 2*n >> n+1-even more so. �Step of induction-proved!

Now, the rationalist knows for sure that the statement is true for the entire natural series, starting with three. He does not need to test every case in practice, like an empirical one.