Empirical or a posteriori knowledge is propositional knowledge obtained by experience or sensorial information. It is contrasted with a priori knowledge, or knowledge that is gained through the apprehension of innate ideas, "intuition," "pure reason," or other non-experiential sources.
For example, "all things fall down" would be an empirical proposition about gravity that many of us believe we know; therefore we would regard it as an example of empirical knowledge. It is "empirical" because we have generally observed that things fall down, so there is no reason to believe this will change. This example also shows the difficulty of formulating knowledge claims. Outside of the Earth's gravitational field, for example, things do not "fall down", as there is no "down".
The vast bulk of the empirical knowledge that ordinary people possess is gained via a mixture of direct experience and the testimony of others about what they have experienced—iterated in an interesting way that is studied in the field of social epistemology as well as other fields. More complicated and organized methods of gaining empirical knowledge are the methods of science—see scientific method—which result in perhaps the best examples of rigorously codified, scientific empirical knowledge, namely, physics.
The modern perusal of a posteriori thought began with Immanuel Kant in a reactionary movement to Hume's sceptical approach to knowledge in his Enquiries Concerning Human Understanding. Kant, in adding the distinction between synthetic and analytic truths to the distinction between a priori and a posteriori knowledge, created four categories of knowledge (one of which, the analytic a posteriori, is never possible). Thus, for Kant, the only type of a posteriori knowledge is the synthetic a posteriori. Because of this, Kant proposes that a posteriori propositions are, as a set, contingent, because a posteriori propositions all depend on external conditions, which may change in time, making the proposition false (e.g. "My dog is a puppy" has a truth value only ascertained by external verification).
Saul Kripke contends that the category of analytic a posteriori truths is nonempty, including, among other things, identity claims such as "Water is H2O" and "Hesperus is Phosphorus."