The epistemic principle that knowledge is closed under known logical implication: if a subject knows proposition p, and the subject knows that p logically entails q (and is able to deduce q competently), then the subject is in a position to know q, absent defeating factors.

Principle

Knowledge can transmit across recognized valid entailments: possession of justified belief that p plus justified recognition that p entails q confers prima facie entitlement to q, subject to defeaters such as lack of awareness of the entailment, relevant defeat, or noncompetence in reasoning.

Demonstration

Illustrative scenario → Recognition → Action → Consequence: A driver knows that the key is in her pocket (Recognition) and knows that if the key is in her pocket then her car can be started (recognized entailment). She therefore is justified in believing she can start the car this morning (Action). If no defeaters exist (e.g., she has reason to think the car is broken), she is entitled to the inferred belief (Consequence).

Misapplication

Using closure to derive skeptical conclusions by chaining ordinary knowledge to remote skeptical hypotheses (e.g., from knowing sensory appearances to knowing that one is not a brain‑in‑a‑vat) overlooks that one of the premises—recognized entailment or the absence of defeaters—may fail. The error is treating formal entailment as sufficient for knowledge without verifying defeaters and competences.

Consequence

Supports the transmission of justification through logical relations and thereby grounds many ordinary knowledge ascriptions and practical inferences. It also sharpens skeptical challenges by showing how ordinary claims can imply skeptical hypotheses when premises are accepted without scrutiny.

Reversal

Closure is qualified when the subject lacks the epistemic competence to grasp or retain the entailment, when relevant defeaters exist, or when the entailment invokes highly hypothetical or epistemically remote conditions; in such cases the inferred proposition need not be known despite the antecedent being known.

Boundary

Clearly within: inferring from knowing 'the heater is off' and knowing 'if the heater is off then the room will be cold' to knowing the room will be cold, provided no defeaters. Boundary case: inferring external-world conclusions from perceptual beliefs—depends on assumptions about defeaters and epistemic access. Clearly outside: treating closure like formal deductive closure in a logical system without regard to psychological or justificatory conditions.

Semantic Tension

Tension with fallibilist or non‑closure views that emphasize that justification may not transfer in ordinary epistemic contexts, and with pragmatic accounts tying knowledge ascription to action rather than strict logical transmission.

Synthesis

Closure is a rule about the potential transmission of epistemic entitlement via recognized entailments; its force depends on the subject's ability to recognize the entailment and the absence of defeating factors, so it constrains but does not mechanically determine knowledge attributions.