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A priori and a posteriori
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A priori and a posteriori are two categories of obtaining knowledge (epistemology). Depending on who you ask, only one is valid and the other is bullshit, or both are useful. In philosophy, a priori knowledge is constrasted with a posteriori knowledge, a priori knowledge being the backbone of deduction and rationalism and a posteriori knowledge being gained through observation and forming the basis for empiricism.
 A priori
A priori (literally "from before") is a Latin term used in formal logic (and philosophy) to mean a fact that is assumed to be true prior to any empirical research. A priori conclusions are usually reached through reasoning rather than observation and are the centerpiece of (philosophical) rationalism.
At the most solipsistic level, the only a priori statement that a tough skeptic can be assured of is "I think, therefore I am" ("cogito ergo sum"), although it can be argued that even Descartes' thought experiment had the a priori assumption that a thought requires a thinker.
 A posteriori
A posteriori ("From the later") is a Latin phrase used in formal logic (and philosophy) to denote knowledge that is derived from empirical observation (experience). While it has nothing to do with one's posterior, some people manage to extract "knowledge" from their posteriors. As the logical basis for induction, it is extensively used in the scientific method and in most courtrooms.