I don't really like the term 'non-empirical sciences' because it sounds as if it is opening the door to lots of snake oil, as well as formal systems. Formal science is a better term in my view, but that's a bit too restrictive because it limits the topic to science, whereas they can be used for other things, like linguistic analysis.

A formal system is a set of distinct symbols and a set of pairs of patterns, with the rule that given a sentence which is an ordered sequence of the symbols any occurance of the 'left hand' pattern in a pair can be replaced by the corresponding right hand pattern.


I hope that helps!

The reason is it 'non-empirical' is that observations and measurements play no part in the formal system. The sentence might include a measurement, but as far as the system is concerned it has no meaning. Even 'true' and 'false' are merely symbols without a meaning.

This may sound so abstract it is without value, but actually it powerful in the same way as algebra is powerful: circumference = 2 pi * radius gets it power from the fact it is true for all radii. Similarly this approach allows you to come up with relationships that are valid whatever the symbols mean. And you can see why logic is non-empirical: For a given a,b. 'a and b' has a valid interpretation what the 'truthfulness of a' is meant to represent.