Define Synthetic A-priori and Analytic A-priori.

Synthetic a priori and analytic a priori are key concepts in Immanuel Kant's epistemology, particularly outlined in his work "Critique of Pure Reason."

Analytic a priori:
Analytic judgments are propositions where the predicate concept is already contained within the subject concept. They are considered self-evident and do not require empirical verification. Analytic judgments are a priori because their truth can be known independently of experience. For example, in the statement "All bachelors are unmarried," the concept of being unmarried is already inherent in the concept of being a bachelor.

Synthetic a priori:
Synthetic judgments involve the combination of concepts that are not contained within each other. Unlike analytic judgments, synthetic judgments add new information and extend our knowledge. Synthetic judgments are a priori when their truth can be known independently of experience, but they go beyond the mere analysis of concepts. Kant argued that certain mathematical and metaphysical propositions, such as the concept of space or the principle of causality, are synthetic a priori judgments because they provide new knowledge not derived from the concepts alone.

Kant's distinction between analytic and synthetic, a priori and a posteriori, forms the basis for his transcendental philosophy, challenging traditional views on the nature of knowledge and the relationship between the mind and the external world.