Questions: Closure Principles Formalized

Question 1

You know that having two hands entails you are not a handless brain in a vat. You admit you cannot know you are not a brain in a vat. What does the closure principle force you to conclude?

• A. You do know you are not a brain in a vat, since closure transmits knowledge forward from your knowledge of having hands
• B. You do not know you have two hands, since closure by contraposition requires knowledge of the entailed proposition
• C. Closure does not apply here because skeptical scenarios are not genuine epistemic possibilities
• D. You can know you have two hands without the closure principle applying to skeptical entailments

Answer: (B)

By contrapositive, if Kₐ(p → ¬SK) and ¬Kₐ(¬SK), then ¬Kₐp. Closure doesn't just run forward (from premises to conclusions) — by contraposition it runs backward: failure to know the conclusion forces failure to know the premise. This is exactly the skeptical argument: since you can't rule out being a brain in a vat, closure forces you to admit you don't know you have hands. This is why Dretske and Nozick rejected closure — it collapses ordinary knowledge into skepticism.

Question 2

Which of the following correctly states the basic closure principle?

• A. If Kₐp is true, then p is true in all possible worlds
• B. If Kₐp and Kₐ(p → q), then Kₐq
• C. If p entails q and Kₐp, then Kₐq — regardless of whether the agent knows the entailment
• D. If Kₐp and p is necessarily true, then Kₐq for any q

Answer: (B)

The closure principle requires both that the agent knows the premise AND that the agent knows the entailment from premise to conclusion. Option C states a stronger principle that drops the second knowledge condition — this would make knowledge automatically close under all logical consequences, even those the agent doesn't recognize. The standard principle requires the agent to know the implication, making it a condition on the agent's epistemic state rather than just on logical relations.

Question 3

Philosophers who reject the closure principle typically do so in order to preserve ordinary knowledge claims while admitting ignorance of far-fetched skeptical scenarios.

• T. True
• F. False

Answer: True

This is precisely the motivation articulated by Dretske and Nozick. By denying closure, they can say: 'I know I have two hands' and 'I don't know I'm not a brain in a vat' without contradiction, because the second doesn't follow from the first. Rejecting closure is not rejecting logic — it is denying that the logical transmission of truth is matched by a transmission of knowledge in all cases.

Question 4

In possible-worlds semantics, the closure principle holds without exception: if p is true in most epistemically accessible worlds and p → q is true in most accessible worlds, then q is true in most accessible worlds.

• T. True
• F. False

Answer: False

The logical argument in the possible-worlds framework is valid, but closure can still fail epistemically if the agent fails to know a valid implication. The formal analysis shows the logic is sound, but whether the agent's epistemic state satisfies all the conditions of closure is a separate question. More importantly, the closure debate often focuses on whether the possible-worlds account of knowledge itself must be revised — contextualist and relevant alternatives theories modify the accessibility relation in ways that make closure fail even within the framework.

Question 5

State the closure principle formally using the Kₐ notation, then explain how its contrapositive form generates a skeptical argument against ordinary knowledge.

Model answer: Formal statement: If Kₐp and Kₐ(p → q), then Kₐq. Contrapositive: If ¬Kₐq and Kₐ(p → q), then ¬Kₐp. The skeptical argument: Let p = 'I have two hands' and q = 'I am not a brain in a vat.' Since p entails q, and I know this entailment, closure gives Kₐ(p → q). But I cannot know I am not a brain in a vat (¬Kₐq). By contrapositive closure, I therefore do not know I have two hands (¬Kₐp). The argument generalizes: any ordinary knowledge claim p entails 'I am not in a skeptical scenario,' so failure to know the skeptical denial propagates back to undermine all ordinary knowledge.

The power of the formal statement is that it makes the stakes inescapable: either accept closure and face skepticism, reject closure and explain why knowledge doesn't transmit in these cases, or find some other response (like arguing you can know the skeptical denial after all). Each horn of the dilemma has costs, and the formal clarity shows there is no cheap escape.