Epistemology
A Priori vs A Posteriori Knowledge
Justification before experience vs after
A priori knowledge is justified independently of sensory experience: 2 + 3 = 5, the law of non-contradiction, “all triangles have three sides.” A posteriori knowledge is justified by experience: that water boils at 100°C, that the Battle of Hastings was in 1066, that the cat is on the mat. The distinction is about how a belief earns its credentials, not what is known.
- Latin“from the earlier” / “from the later”
- Sharpened byKant, Critique of Pure Reason (1781)
- Complicated byKripke, Naming and Necessity (1980)
- ConcernsJustification, not concept acquisition
- Linked toRationalism vs empiricism
- Contrast distinctionNecessary / contingent
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How the distinction works
Imagine two questions. (1) Is 17 a prime number? You answer by trying to divide it: 17 / 2, 17 / 3, 17 / 5 — none give an integer, so yes. The proof needs no observation of the world; you could carry it out in a sensory deprivation tank. (2) Are there 17 planets in our solar system? You answer by consulting astronomy. Sense perception, in some chain of testimony, is doing the work. The first is a priori, the second a posteriori.
The textbook formula: a belief is a priori justified if its justification does not rely on experience. It is a posteriori justified if its justification does rely on experience. Two clarifications immediately follow.
- Concept acquisition vs justification. You can't believe “2 + 3 = 5” without learning the words; learning them takes experience. But once you have the concepts, the truth is not checked against experience. Kant put it this way: a priori knowledge can begin with experience without arising from it.
- Defeasibility. A priori beliefs are not infallible. Frege's Basic Law V was thought a priori until Russell's paradox showed it inconsistent. The category is about how a belief is supported, not whether it cannot fail.
A priori vs a posteriori — at a glance
| A priori | A posteriori | |
|---|---|---|
| Justification | Independent of experience | Depends on experience |
| Latin meaning | From what is prior (to experience) | From what comes after (experience) |
| Paradigm examples | Math, logic, conceptual truths | Physics, history, perception |
| Methods | Proof, conceptual analysis, intuition | Observation, experiment, testimony |
| Defeasibility | Reflective revision (e.g. paradoxes) | New evidence, better instruments |
| Allies | Rationalism (Descartes, Leibniz, Kant) | Empiricism (Locke, Hume, Mill) |
| Modal alignment | Often necessary, but see Kripke | Often contingent, but see Kripke |
Worked example: the prime number test
Suppose you wonder whether 1,729 is the sum of two cubes in two different ways (the “Hardy-Ramanujan number”). You sit down with a pencil:
1729 = 1³ + 12³ = 1 + 1728. Check.
1729 = 9³ + 10³ = 729 + 1000. Check.
The conclusion — that 1729 has the property — is a priori justified. You used no microscope, ran no experiment, asked no expert. The only inputs were the meanings of “cube”, “sum”, the rules of arithmetic, and your own checking. By contrast, the claim “Ramanujan first noticed this in a London taxi” is a posteriori — you'd have to read Hardy's memoir.
Kant's four-cell grid
In the Critique of Pure Reason (1781), Kant crossed a priori/a posteriori with another distinction — analytic/synthetic — to produce a 2×2 table.
| Analytic (true by meaning) | Synthetic (true by world) | |
|---|---|---|
| A priori | “All bachelors are unmarried” — uncontroversial | “7 + 5 = 12”, “every event has a cause” — Kant's signature |
| A posteriori | Empty — Kant says impossible | “The cat is on the mat” — uncontroversial |
The synthetic a priori cell is the one that carried the weight of Kant's project. If pure mathematics, Euclidean geometry, and the principle of universal causation are all synthetic a priori, then we have substantive knowledge of the world that doesn't require experiment — exactly what Hume's empiricism denied.
Kripke's twist: the four cells fill up
Saul Kripke's Naming and Necessity lectures (1970, published 1980) re-shuffled the deck. Kripke argued that necessary (couldn't have been otherwise) and a priori (justified independently of experience) had been wrongly conflated.
- Necessary a posteriori. “Water is H₂O.” If water is H₂O, it's H₂O in every possible world (so necessary). But you needed 19th-century chemistry to find that out (so a posteriori). Same shape: “Hesperus is Phosphorus”, “heat is mean molecular kinetic energy.”
- Contingent a priori. “The standard metre stick in Paris is one metre long.” By stipulation, the rod fixes the meaning of “metre”, so anyone who understands the stipulation knows it without measuring (a priori). But the rod could have been a different length (contingent). Kripke's example is artificial; the philosophical point is that a priori does not entail necessary.
The result: a priori/a posteriori is an epistemic distinction (about how we know), and necessary/contingent is a metaphysical distinction (about how things must be). Pre-Kripke, philosophers used the two interchangeably. Post-Kripke, they don't.
Objections and responses
- Quine's holism. In “Two Dogmas of Empiricism” (1951) Quine argued no belief is immune to experiential revision; even logic could be revised for quantum mechanics. If true, the a priori category is empty. Most philosophers think Quine showed the boundary is porous, not that it's missing.
- Empiricist suspicion of intuition. Mill (1843) tried to make even arithmetic empirical, derived from generalisations over apples. Modern naturalists (Maddy, Penelope) revive this for set theory: axioms are justified by their fruits in mathematical practice, not by raw intuition.
- Experimental philosophy. Survey work shows people's intuitions about thought experiments vary by culture, gender, and order of presentation. If a priori justification leans on intuition, this is uncomfortable. Defenders reply that expert intuitions, properly trained, are what counts.
- Bealer's “modal reliabilism.” Bealer argued intuitions track modal facts well enough to ground a priori justification. The view formalises what Kantians have always assumed, and the debate continues.
Variants and refinements
- Strong a priori (Kant, BonJour). Justification is fully independent of experience.
- Weak a priori (Kitcher). Experience plays only a triggering role; the warrant is non-experiential.
- Defeasible a priori. A priori beliefs can be overturned by other a priori reasoning (paradoxes), even if not by experience.
- Relative a priori (Reichenbach, Friedman). Some principles are a priori within a framework (Newtonian space-time within classical mechanics) but framework-dependent and revisable when the framework shifts.
Why the distinction matters
- Demarcation of disciplines. Mathematics and logic are usually taken as a priori, the sciences a posteriori. The line is what makes “armchair” vs “laboratory” methodology coherent.
- Philosophical method. Most analytic philosophy proceeds a priori — argument, thought experiment, conceptual analysis. If a priori justification is bogus, philosophy needs a new toolkit.
- Religious and metaphysical arguments. Anselm's ontological argument and Aquinas's cosmological argument are paradigmatic a priori reasoning about non-trivial conclusions; their force depends on the legitimacy of the category.
- AI and theorem-proving. Computer-assisted proofs (Four Colour Theorem, 1976) blur the line: humans verify the algorithm a priori, but trust the silicon a posteriori.
Common confusions
- A priori ≠ innate. The distinction is about justification today, not whether you were born knowing it. Plato's Meno conflates the two; Kant tried to keep them apart.
- A priori ≠ certain. Frege's Basic Law V was a priori and false. Mathematicians make mistakes.
- A priori ≠ analytic. Kant's whole point was to find synthetic a priori truths.
- A priori ≠ necessary. Kripke showed a priori contingencies and a posteriori necessities exist.
- “I figured it out without help” isn't a priori. If you reasoned from past observations, the justification is still a posteriori.
Frequently asked questions
What's the simplest example of each?
A priori: 2 + 3 = 5 (you don't run an experiment to check). A posteriori: water boils at 100°C at sea level (you have to look it up or measure it). The line tracks how the belief is justified, not whether you happened to learn it from a book.
Doesn't all knowledge eventually depend on experience?
You need experience to acquire concepts (you can't know 2 + 3 = 5 without first learning what 2 means). But the justification of the truth doesn't depend on experience — once you have the concepts, no observation could refute it. A priori is about justification, not concept acquisition.
What did Kant add?
Kant cross-multiplied a priori/a posteriori with analytic/synthetic. Three of the four cells were familiar; the explosive one was synthetic a priori — claims that extend our knowledge yet are justified independently of experience. Mathematics, geometry, and the principle that every event has a cause were his stock examples.
What's a Kripkean a posteriori necessity?
Saul Kripke (1980) argued that some identities are metaphysically necessary but only knowable through empirical investigation. “Water is H₂O” is necessary (in any possible world, water just is H₂O) but a posteriori (we needed 19th-century chemistry to discover it). “Hesperus is Phosphorus” — that the morning and evening star are the same planet — is the same shape.
Are a priori truths certain?
Not necessarily. Frege thought Basic Law V (the comprehension principle for sets) was a priori; Russell's paradox showed it was contradictory. Mathematicians have made errors. A priori justification is justification without sensory input — that doesn't immunise it from being wrong.
Is moral knowledge a priori?
Disputed. Rationalists (Kant, Ross, modern intuitionists) say basic moral principles like “gratuitous cruelty is wrong” are knowable a priori. Empiricists and naturalists (Mill, evolutionary ethicists) say moral knowledge depends on facts about wellbeing, social cooperation, or evolved emotions — and is therefore a posteriori. Both positions have lively contemporary defenders.