Concept

Closure Principle

Intro

The closure principle says that if you know something, and you know that something logically requires something else, then you know that second something too. Simple example: if you know you are in your kitchen, and you know that being in your kitchen means you are not on the moon, then you know you are not on the moon.

That sounds obvious. But it has a sharp edge when the skeptic gets hold of it. Take a brain-in-a-vat scenario: a brain on a lab bench getting fed simulated experiences. Suppose you do not know whether you are such a brain. (Most epistemologists agree you cannot rule that out by any test from inside the experience.) Now apply closure: if you knew you had hands, then you would know you are not a handless brain in a vat. But you do not know you are not a handless brain in a vat. So by closure, you do not actually know you have hands.

That conclusion bothers most people. So one option is to deny closure. Fred Dretske and Robert Nozick famously argued that closure fails: you can know ordinary things like I have hands while still failing to know the consequences that logically follow, like I am not a brain in a vat.

This sounds desperate at first. It looks like cheating. But Dretske and Nozick build it on a more general theory called tracking theory of knowledge, which gives them some traction. The page lays out the principle in its various strengths, walks the skeptical argument that depends on it, and surveys the responses (Nozick's tracking conditions, Dretske's relevant-alternatives framework, the contextualist objections, the Mooreans who keep closure and reject the skeptical premise instead).

In full

The principle that knowledge is closed under known logical entailment, if I know p, and I know that p entails q, then I am in a position to know q (or, on stronger versions, I do know q). The closure principle is the second premise of the standard formulation of the Cartesian skeptical argument (P2: if I know I have hands, then I know I am not a BIV). Denying closure, nonclosure, is one of the major anti-skeptical strategies in contemporary epistemology, principally associated with Fred Dretske and Robert Nozick. The denial is bold: it claims we can know the things ordinary perception tells us about (that I have hands) while failing to know things that follow logically from those things (that I am not a handless BIV). Whether this is a coherent position or a desperate ad-hoc maneuver to save ordinary knowledge from skepticism is one of the live disputes.

The closure principle stated

There are several versions of closure of varying strength. A standard formulation:

(Closure) If S knows that p, and S knows that p entails q, then S knows q.

Stronger and weaker variants:

Strong closure: If S knows p, then S knows everything that p logically entails. (Implausibly strong, entails I know all logical consequences of all my knowledge, including unrecognized ones.)
Single-premise closure: If S knows p and S competently deduces q from p, then S knows q. (Hawthorne; Williamson.)
Multi-premise closure: If S knows p, knows q, and knows that p ∧ q entails r, then S knows r. (Considered separately because of preface-paradox-style worries.)

The closure version most relevant to skepticism is something like: if S knows p, and S knows that p entails ¬h (where h is a skeptical hypothesis incompatible with p), then S knows ¬h.

The closure-skeptical argument

The standard form:

(1) I know I have hands → I know I have hands. [Trivial]
(2) I know that "I have hands" entails "I am not a handless BIV." [Logic]
(3) By closure: If I know I have hands, then I know I am not a handless BIV.
(4) I do not know I am not a handless BIV. [Skeptical premise]
(5) Therefore, I do not know I have hands. [3, 4 by modus tollens]

The closure principle is doing the work of moving from (3) to (5) via (4). If closure fails, the argument doesn't go through, even if I don't know I'm not a BIV, I might still know I have hands.

The Dretske-Nozick denial

Both Fred Dretske and Robert Nozick, in independent work in the 1970s, proposed that closure fails, and that this failure is how we should think about ordinary knowledge in light of skepticism.

Dretske: relevant alternatives + epistemic operators

In "Epistemic Operators" (1970) and Knowledge and the Flow of Information (1981), Dretske argues that knowing requires ruling out the relevant alternatives, but not all alternatives are relevant in all contexts. In the context of "knowing I have hands," the relevant alternative might be "having no hands" or "having prosthetics," not "being a handless BIV." So I can know I have hands without ruling out the BIV scenario, because the BIV scenario is not a relevant alternative.

This is structurally a denial of closure: knowing-A and knowing-that-A-entails-B does not require ruling out (and so knowing) ¬B if B's negation is not relevant.

Nozick: tracking / counterfactuals

In Philosophical Explanations (1981), Nozick proposes a counterfactual analysis of knowledge:

Tracking conditions for knowledge: S knows p iff:

p is true.

S believes p.

If p were false, S would not believe p. (Sensitivity / "variation" condition.)

If p were true (in nearby possible worlds), S would believe p. (Adherence / "adherence" condition.)

The third condition (sensitivity) is what does anti-skeptical work. My belief that "I have hands" tracks the truth: in the nearest possible world where I don't have hands (e.g., I lost them in an accident), I would not believe I have hands. So I know I have hands.

But my belief that "I am not a handless BIV" fails the sensitivity test: in the nearest possible world where I am a handless BIV, I would (by stipulation of the BIV scenario) still believe I am not. So I don't know I'm not a BIV.

Closure fails: I know I have hands (sensitivity passes); I don't know I'm not a handless BIV (sensitivity fails); but I have hands entails I am not a handless BIV.

Why closure denial is attractive

The denial is attractive because it preserves both of the intuitions the Cartesian skeptical argument exploits:

(a) Ordinary knowledge: I do know I have hands. Of course I do.
(b) Skeptical-hypothesis ignorance: I don't know I'm not a BIV. (Whatever evidence I might marshal would be exactly the same in the BIV scenario, so I can't distinguish.)

Both intuitions are strong. Mooreanism rejects (b); the skeptic rejects (a); only the closure-denier preserves both. This is the principled appeal.

Why closure denial is suspicious

The cost is denying a principle that seems obviously true and doing fundamental work in deductive reasoning. If I genuinely know p, and I genuinely deduce q from p, can I really fail to know q? Concrete cases:

I know my keys are in my pocket. From this it follows that my keys are not in the kitchen. Do I not know my keys are not in the kitchen? It seems I do.
I know that 2 + 2 = 4. From this it follows that the fourth Fibonacci number is not 5. Do I not know this?

Closure denial threatens to undermine the most basic feature of deductive reasoning: that valid arguments transmit knowledge from premises to conclusions. The cost of accepting closure denial is having to specify which entailments knowledge is closed under (the everyday ones?) and which it isn't (skeptical-hypothesis-related ones?), and the demarcation is hard to make non-arbitrary.

Williamson's defense of closure

Timothy Williamson, in Knowledge and Its Limits (2000), defends a form of closure: if you competently deduce q from p, knowing p, you know q. Williamson argues that closure denial conflates the bare logical entailment (p entails q regardless of anyone's beliefs) with the epistemically loaded entailment (S knows p, competently deduces q, retains the deduction). The latter does transmit knowledge; the skeptical scenarios involve a different failure mode (failure of S to actually perform the deduction in cases that matter).

Williamson's defense puts pressure on the Dretske-Nozick anti-skeptical strategy: if closure holds, then nonclosure-based anti-skepticism collapses, and the skeptic's argument goes through unless one of the other premises (the skeptic's P1 or some hidden assumption) is rejected.

The relation to Mooreanism

Mooreanism and closure denial are competitors among anti-skeptical strategies. The Moorean accepts closure but rejects the skeptic's antecedent (claims we do know we're not BIVs); the closure-denier accepts the skeptic's antecedent but rejects closure (claims we don't know we're not BIVs but still know ordinary things). They cannot both be right.

In recent literature, the consensus has shifted somewhat toward closure (and so toward Mooreanism, contextualism, or externalist approaches and away from Dretske-Nozick), largely because Williamson's arguments and the costs of denying closure in non-skeptical cases have outweighed the appeal of preserving both intuitions.

Closure in non-skeptical contexts

Closure is invoked across philosophy in many non-skeptical contexts:

Mathematical knowledge, knowing axioms, derivative theorems are knowable through valid proofs (Frege's Begriffsschrift program).
Knowledge of consequences in deliberation, moral / practical reasoning depends on closing knowledge under known entailments.
Logical omniscience in Bayesian and possible-worlds frameworks, sometimes idealized, but the principle is closure-shaped.
Knowledge of one's own beliefs and attitudes, KK principle (knowing implies knowing one knows) is a closure-style claim.

So closure denial is not free: it costs not just the anti-skeptical move but the principle's role in the rest of epistemology.

Christian-apologetic relevance

Closure is implicitly invoked in many Christian-apologetic arguments:

Cosmological arguments, knowing the universe began to exist; knowing this entails a cause; concluding to a first cause. Closure required.
Inference from miracles to God, knowing the resurrection happened; knowing this entails divine intervention. Closure required.
Logical-problem-of-evil response, knowing premises about omnipotence and goodness; deriving consistency results.
The transmission failure objection to Mooreanism, when applied to "knowing I have hands → knowing I'm not a BIV," critics argue justification cannot transmit. The same shape of objection (transmission failure / question-begging closure) appears in skeptical-theist responses to the problem of evil: from "premises I accept" we derive "God exists," but if "I accept these premises only because I already accept God exists," there's a kind of closure-failure / circularity.
Reformed Epistemology and closure, Plantinga's belief-in-God-as-properly-basic doctrine sidesteps closure-style skeptical arguments because the basic belief doesn't require inferential justification; closure-failure objections (which target inference-based justifications) don't apply.

The contextualist alternative

Epistemic Contextualism avoids the closure-denial problem differently: closure holds in any single context, but the skeptical argument and the ordinary-knowledge claim are made in different contexts, with different standards for "know." So we are not committed to denying closure or to denying the skeptic's antecedent; instead we recognize that "know" picks out different relations in different contexts.