Questions: Deductive Closure and Knowledge

Question 1

The brain-in-a-vat scenario creates a problem for deductive closure because:

• A. It shows that 'I am sitting reading' does not actually entail 'I am not a brain in a vat'
• B. It suggests we know ordinary things but cannot know skeptical hypotheses are false, even though ordinary knowledge entails this
• C. It demonstrates that knowledge requires certainty, which we lack for ordinary beliefs
• D. It shows that deductive inference is unreliable when applied to philosophical scenarios

Answer: (B)

The structure of the problem is: (1) you know ordinary proposition P ('I am sitting reading'); (2) P entails Q ('I am not a brain in a vat'); (3) by closure, you know Q. But there is strong intuition that you cannot know Q — your evidence doesn't distinguish the real situation from the vat scenario. Something has to give: closure, ordinary knowledge, or the claim that you can't know Q. Option A is wrong because the entailment holds — being a brain in a vat would mean you are not actually sitting reading. Option C mischaracterizes the problem, which is about closure, not about certainty.

Question 2

A philosopher accepts deductive closure and also accepts that you know ordinary things (your car is in the driveway, your hands are before you). What must they conclude about skeptical hypotheses?

• A. Skepticism is correct — we know nothing, because all knowledge depends on ruling out skeptical alternatives
• B. We do know that skeptical hypotheses are false — closure requires it, given that ordinary knowledge entails this
• C. Closure only applies to analytic entailments, not to contingent propositions about external reality
• D. Knowledge of ordinary things and knowledge of skeptical hypotheses are logically independent

Answer: (B)

This is the forced conclusion of accepting both closure and ordinary knowledge. If you know P (ordinary facts) and P entails Q (not-brain-in-a-vat), then by closure you know Q. You cannot consistently accept closure, accept ordinary knowledge, and deny that you know skeptical hypotheses are false. This seems counterintuitive — and that's the puzzle. Option A accepts closure but rejects ordinary knowledge (the skeptic's position). Option B is the position you must hold if you keep both closure and common sense.

Question 3

Nozick's tracking account of knowledge accepts deductive closure but adds extra conditions to restrict what follows from known propositions.

• T. True
• F. False

Answer: False

Nozick's tracking account is precisely one of the main ways to DENY closure. On the tracking view, knowledge requires that your belief be sensitive to the truth: if P were false, you would not believe P. You can track the truth of 'my car is in the driveway' (if it weren't, you'd see an empty driveway), but you cannot track the falsehood of 'I am a brain in a vat' (your experiences would be identical in both worlds). So tracking knowledge of ordinary P does not extend to tracking the negation of skeptical hypotheses — closure fails.

Question 4

Accepting deductive closure forces philosophers to adopt skepticism, since hardly anyone can know that skeptical scenarios are false.

• T. True
• F. False

Answer: False

This is only one of three responses available. Skeptics do accept closure and deny ordinary knowledge. But a second option is to accept both closure AND ordinary knowledge — and conclude that we DO know skeptical hypotheses are false (even if this seems surprising). A third option is contextualism: in ordinary conversational contexts the standards are low and we know everyday facts; in skeptical contexts the standards rise. Closure does not force skepticism; it creates a trilemma where each horn has costs.

Question 5

What is the trilemma generated by applying deductive closure to skeptical scenarios? What must any response to the problem sacrifice?

Model answer: The trilemma: (1) accept closure + accept ordinary knowledge → must accept we know skeptical hypotheses are false (counterintuitive); (2) accept closure + accept that we can't know skeptical hypotheses are false → must deny ordinary knowledge (skepticism); (3) deny closure → can hold ordinary knowledge + cannot know skeptical hypotheses are false, but must explain why a seemingly obvious logical principle fails. Every response sacrifices something: either the denial of skeptical hypotheses, ordinary knowledge, or closure itself.

This trilemma is what makes deductive closure philosophically important rather than merely technical. It shows that three prima facie plausible commitments — closure, common-sense knowledge, and the inexplicability of skeptical scenarios — are jointly inconsistent. Any adequate epistemology must resolve the tension, and the different schools (closure-deniers like Nozick, contextualists like Cohen, common-sense epistemologists like Moore) each pay a different price for their resolution.