From the distinction established by Hume, a group of philosophers in the sciences and mathematics would adopt his idea. This would lead to the traditional distinction of two general types of knowledge as formal and empirical knowledge. This demarcation has become the basis for the distinction of the disciplines in the sciences as the formal sciences of mathematics, logic, geometry, etc. and the empirical sciences of biology, chemistry, physics, the social sciences, and others.

Formal knowledge corresponds to knowledge in the formal sciences whose main concern is the validation of their knowledge claims within the formal system in their respective disciplines. This could be logical, mathematical, linguistic or any formal system whose method of validation depends entirely on the particular system being used. The formal sciences is characterized by the consistency of the system being used. Therefore, they are not dependent on the accumulation of empirical data but they are concerned with the structure or validity of their sciences within the formal deductive framework that governs them. For example, a mathematical calculation can be done precisely without having to rely on actual empirical data. In other words, you can perform the mathematical operation of adding 2 and 3 apples as equivalent to 5 without necessarily having the actual empirical objects at hand, in this case, the apples.

Empirical knowledge, on the other hand, is the general term used to describe the different disciplines in the empirical sciences, ranging from the hard sciences of physics, chemistry, biology, and others to the soft sciences of sociology, political science, psychology, and others. In as much as empirical statements use the faculty of experience and sense perception in order to establish their knowledge claims, empirical knowledge also takes emphasis and makes use of the data or the content from experience and its correspondence with the state of affairs to establish the truth or falsity of their knowledge claims from these empirical sciences. The empirical sciences give information about what the world is.

A claim being made under formal knowledge can be rejected right away once contra-dictions are seen. This is not the case with empirical knowledge. Instead of looking for contradictions, inconsistencies or invalid arguments in logic, the empirical sciences deal more with probabilities using the framework of induction. These probabilities are results of data from experience. Thus, empirical knowledge is working within the assumption that it is always probable that experience could prove you wrong later on.

Using correspondence with experience and induction, the empirical sciences try to accumulate data and information coming from experience in order to establish more reliability or probability for their hypothesis. They are not governed by a formal system or structure for the validity of their knowledge claims. They are concerned, instead with the process of verification from experience, to establish whether or not the hypothesis have been confirmed by as many instances as possible using data coming from experience. Thus, in the empirical sciences, the reliability of information depends on their correspondence with the state of affairs, it is not a matter of coherence within a given system. They use the method of induction to arrive at their conclusion. In induction, the grounds or premises support the conclusion only with probability and not with logical certainty.

For example, you went on a pond and saw swan no. 1 which is white. Then, another pond and swan no. 2 is again white, and another, swan no. 3 is again white, and others. Then you would conclude, based on the observed cases, that all swans are white. It is only with warranted assertability that you are making the conclusion all swans are white. That is as far as your experience could prove and what your evidence could warrant. Unfortunately, it is not possible for you to observe all the swans in the world. Thus, you are not completely certain about your knowledge claim. As far as you are concerned, this is what your evidence could warrant, thus it is possible that experience could prove you wrong later on. In fact, when everybody thought that all swans were white, scientists discovered this black species of swans in Australia.

To summarize, the two general classifications of knowledge in the sciences belong to two different paradigms, empirical knowledge gives emphasis on the criterion of verification while formal knowledge gives emphasis on the validity or coherence within the system being employed.