The terms a priori (Latin; “from former”) and a posteriori (Latin; “from later”) refer primarily to species of propositional knowledge. A priori knowledge refers to knowledge that is justified independently of experience, i.e., knowledge that does not depend on experiential evidence or warrant. In contrast, a posteriori knowledge is justified by means of experience, and depends therefore on experiential evidence or warrant. The distinction between a priori and a posteriori knowledge may be understood as corresponding to the distinction between non-empirical and empirical knowledge. Mathematical knowledge is a paradigmatically a priori, whereas, the truths of physics, chemistry, and biology are instances of a posteriori knowledge. This a priori / a posteriori distinction has been blurred by Catholic theologians such as Karl Rahner who have constructively adopted Immanuel Kant's understanding of a priori in anthropology and theology.
The historical source for contemporary understanding of the a priori / a posteriori distinction is Kant’s Critique of Pure Reason. Kant articulates the distinction as epistemological in its nature, i.e., pertaining to knowledge. Since knowledge is understood as ranging over propositions the a priori / a posteriori distinction refers to a division within the class of propositions known or capable of being known. If a proposition is capable of being known a priori, then it may be known independently of experience. For example, your knowledge that bachelors are unmarried, that 5 + 2 = 7 and that the square on the hypotenuse of a right angled triangle equals the sum of the squares on the other two sides counts as a priori knowledge. By contrast, if a proposition is known or is capable of being known a posteriori, then it is known as a result of experiential evidence. For example, your knowledge that there is a computer in front of you, that you ate breakfast this morning, that snow is white, that Indian elephants have smaller ears than African elephants, all count as a posteriori knowledge. The distinction between a priori and a posteriori corresponds to the distinction between empirical and non-empirical knowledge.
It is important to distinguish  the claim that a proposition is knowable without any experience from  that claim that experience is not necessary for the proposition to be known. The proposition that ‘all bachelors are unmarried’ is something known a priori, but this is not to say that you could know this without any experience at all. Clearly this knowledge requires the conceptual and linguistic capacities involved in an understanding of English. Crucially, then, to say that a proposition is known a priori is not to endorse , but only to endorse . A proposition is known a priori only if, in addition to any experience needed to have beliefs at all, or to grasp the proposition that p, your justification for believing that p does not depend on experience. So the claim that ‘all bachelors are unmarried’ does not depend on conducting a survey of all bachelors, although exposure to English is necessary for knowing it. Similarly, your knowledge that women are female human beings presupposes, but is not based on, experience, and counts as a priori knowledge.
Although the primary usage of the terms a priori and a posteriori is with reference to knowledge and justification, philosophers sometimes also speak of a priori or a posteriori concepts. It is reasonable to think that concepts are constituents of propositions, and are therefore neither true nor false, and so are not capable of being known. Reference to a priori concepts may then be naturally understood as those that have significance or meaning independently of experience and do not require experience for legitimatization. Similarly, a posteriori concepts are those that cannot be understood independently of particular experiences.
The distinction between a priori and a posteriori knowledge must be separated from two other distinctions with which it is closely connected and sometimes confused. These are the metaphysical distinction between necessary and contingent truths and the semantic distinction between analytic and synthetic propositions.
Historically, most philosophers have maintained that all a priori knowledge corresponds to knowledge of necessary truths. A necessary truth is a proposition that cannot be false; it is true in all possible worlds. Mathematical truths such as ‘3 + 5 = 8’ are paradigmatic examples of necessary truths. By contrast, a contingent truth is a proposition that is true, as things are, but is conceivably false. For example, it seems contingently true that the population of New York is greater than five million. This proposition is said to be contingent because we can easily imagine it to be false. Whatever the initial plausibility of the claim that a priori knowledge is restricted to knowledge of necessary truths, this view has been challenged by some eminent contemporary philosophers.
Saul Kripke (1972) argues that some propositions known a priori are contingently true, while some propositions known a posteriori are necessarily true. As an example of the former, Kripke maintains that the proposition ‘S is one meter long’ is known a priori, when S refers to the standard meter bar. Kripke argues that although this proposition is known a priori it is contingently true since the length of S might not have been one meter long. Kripke’s main examples of a posteriori necessary truths involve identity statements such as ‘Hesperus is Phosphorus.’ These issues are controversial, and continue to provoke widespread debate.
The a priori / a posteriori distinction is also sometimes aligned with the semantic distinction between analytic and synthetic truths. A number of philosophers have held that a priori knowledge is restricted to knowledge of analytic propositions, and a posteriori knowledge to synthetic propositions (see the entry on the analytic-synthetic distinction). An analytic proposition is roughly, a proposition true by meaning alone, whereas, generally, the truth or falsity of a synthetic proposition does not depend on meaning. Kant (1781) famously challenged the alignment of a priori with analytic and a posteriori with synthetic, arguing that truths of arithmetic and geometry are synthetic propositions, which are capable of being known a priori.
Lastly, it is important to note that the distinction between a priori and a posteriori knowledge does not correspond to the distinction between innate and acquired knowledge. Innateness focuses on the genetic question of how a belief is acquired, whereas the a priori / a posteriori distinction concerns the nature of the epistemic warrant in support of a proposition. It seems possible for a belief to be innate and yet be justified a posteriori; and conversely, for a belief to be acquired by means of learning whilst being justified a priori. The truth of Fermat’s last theorem, for example, is something known a priori, but is not innate knowledge.
Some Catholic theologians such as Karl Rahner have gone beyond the Kantian distinction of a priori and a posteriori. According to the epistemology of Kant, when a posteriori "impressions" from objects are processed by a subject's a priori "forms of intuition" and "forms of the understanding," the subject's knowledge about the objects is established. But, this epistemology cannot let the subject know God, immortality, freedom, and "things-in-themselves," given the limited nature of the a priori "forms" or structures of the subject's capacity to know. Hence Kant's basic denial of natural theology and the initially negative Catholic reaction to Kant. But Karl Rahner and others in Catholicism in the twentieth century have taken Kant's understanding of a priori as an opportunity for a renewal of natural theology. According to Rahner, the elements that are a priori are God-given and therefore far broader than the Kantian "forms," and they are not only in subjects but also in objects. So, knowledge of a knowing subject is always at the same time a knowledge about objects including God. This way, the a priori / a posteriori distinction has been blurred.
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