A Priori and A Posteriori Knowledge

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a-priori a-posteriori necessity analyticity Kant

Core Idea

A priori knowledge is justified independently of sensory experience — mathematics and logic provide the canonical examples, since we can establish that 2+2=4 or that all bachelors are unmarried through reasoning alone. A posteriori (empirical) knowledge depends essentially on sensory experience for its justification. Kant famously complicated this picture by arguing that some a priori knowledge is synthetic (genuinely informative about the world, not merely definitional), generating the puzzle of how pure reason can extend our knowledge beyond conceptual truths.

How It's Best Learned

Contrast clear cases: 'All triangles have three sides' (a priori) vs. 'Water boils at 100°C at sea level' (a posteriori). Then examine boundary cases — mathematical knowledge, knowledge of modal truths, moral knowledge — to see where intuitions about priority become contested.

Common Misconceptions

Explainer

One of epistemology's most useful distinctions cuts across every field of knowledge: some things we can figure out by thinking alone, and other things we can only know by going out and looking. This is the a priori / a posteriori distinction. "A priori" (Latin: "from the earlier") means knowable before or independently of experience; "a posteriori" (Latin: "from the later") means knowable only through experience. The distinction is about justification — what entitles us to believe something — not about how we first encountered the claim.

The clearest cases are easy. "All triangles have three sides" is a priori: you can verify it by analyzing what a triangle is, without measuring any physical object. "The boiling point of water is 100°C at sea level" is a posteriori: you must run an experiment (or trust someone who did). Mathematics and logic supply the canonical body of a priori knowledge; natural science supplies the canonical body of a posteriori knowledge. Most philosophical work involves the boundary cases — moral claims, modal claims about what is possible or necessary, introspective reports — where it is genuinely contested which side applies.

Kant added a second dimension to this picture by crossing the a priori/a posteriori distinction with the analytic/synthetic distinction. An analytic statement is one where the predicate is already contained in the subject ("All bachelors are unmarried"). A synthetic statement genuinely extends our knowledge ("The cat is on the mat"). Before Kant, the dominant assumption was that all a priori knowledge was analytic — all you could know without experience was what was already packed into your concepts. Kant challenged this by arguing for synthetic a priori knowledge: claims that are both genuinely informative and knowable through pure reason. His examples included arithmetic, geometry, and fundamental causal principles. This claim is still debated today.

A third important wrinkle comes from Saul Kripke's 20th-century work, which showed that the a priori/a posteriori distinction is logically independent from the necessary/contingent distinction (about whether a truth could have been otherwise). We might assume: necessary truths are a priori, contingent truths are a posteriori. Kripke dismantled this assumption. "Water is H₂O" is a necessary truth — in any possible world, water just is H₂O — but we could only discover this empirically, through chemistry. So it is necessary a posteriori. Conversely, "the meter is the length of this particular rod in Paris" was once contingently true but was stipulated to be true by definition — making it contingent a priori in some sense. These cases reveal that what's necessary about a thing and how we come to know it are different questions.

For epistemology, the a priori/a posteriori distinction matters because it shapes debates about the limits of reason. Empiricists (Hume, Locke, the Logical Positivists) tend to restrict a priori knowledge to analytic truths — definitional or logical — and insist that anything substantive about the world requires experience. Rationalists (Descartes, Leibniz, Kant) argue that reason can reach further. Knowing which side of the line a given claim falls on is often itself the philosophical question under investigation.

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Prerequisite Chain

Counting to 10Counting to 20Understanding ZeroThe Number ZeroCounting to FiveOne-to-One CorrespondenceCombining Small Groups Within 5Addition Within 10Addition Within 20Two-Digit Addition Without RegroupingTwo-Digit Addition with RegroupingAddition Within 100Repeated Addition as MultiplicationMultiplication Facts Within 100Division as Equal SharingDivision as Grouping (Measurement Division)Division: Grouping (Repeated Subtraction) ModelDivision: Fair Sharing ModelDivision as Equal SharingDivision as GroupingBasic Division FactsDivision Facts Within 100Two-Digit by One-Digit DivisionDivision with RemaindersRemainders and Quotients in DivisionDivision Word ProblemsIntroduction to Long DivisionFactors and MultiplesPrime and Composite NumbersEquivalent FractionsRelating Fractions and DecimalsDecimal Place ValueReading and Writing DecimalsComparing and Ordering DecimalsAdding and Subtracting DecimalsMultiplying DecimalsDividing DecimalsDividing FractionsMixed Number ArithmeticOrder of OperationsInteger Order of OperationsVariable ExpressionsThe Distributive PropertyVariables and Expressions ReviewIntroduction to PolynomialsAdding and Subtracting PolynomialsMultiplying PolynomialsFactorialPermutationsCombinationsCounting Principles: Addition and Multiplication RulesIntroduction to Graph TheoryPropositional Logic FoundationsLogical Inference and Proof RulesProof Strategies in Discrete MathematicsSoundness and Completeness of Propositional LogicSoundness and Completeness of First-Order LogicCompactness Theorem for First-Order LogicBasic Model TheoryLöwenheim-Skolem TheoremsGödel's Incompleteness TheoremsIntroduction to Intuitionistic LogicIntroduction to Modal LogicA Priori and A Posteriori Knowledge

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