1
Closure
If you know something and you know that it brings another claim with it, you know the other claim. Here: if you know you have hands, you know you are not a handless brain in a vat.
Lesson · Knowledge and Evidence · Justification
Skepticism and the Closure Argument
The skeptic and Moore share one premise, that knowing you have hands brings with it knowing you are not a brain in a vat, and disagree about which of two other claims is more certain. The lab shows that both arguments are valid. Choosing between them takes a judgement that no truth table supplies.
skeptic: h → b, ¬b, so ¬h
Moore: h, h → b, so b
both arguments are valid
the shared premise is closure
One conditional, two directions
Core ideas
What this lesson establishes
A skeptical argument is an argument, and what the earlier course taught about validity applies to it without change.
2
Modus tollens and modus ponens run one conditional both ways
From h → b and ¬b you may conclude ¬h. From h → b and h you may conclude b. Both are valid forms, so the quarrel is over the premises that are not shared.
3
Denying closure is a third option
Drop the conditional and neither argument goes through. The cost is that you must say a person can know one thing without knowing something it plainly implies.
The material
Two valid arguments with opposite conclusions
The argument written out, then its mirror image, then the choice neither of them makes.
Let h stand for “I know I have hands” and b for “I know I am not a brain in a vat being fed experiences”. The skeptic argues in three lines.
h → b if I know I have hands, I know I am not envatted
¬b I do not know I am not envatted
∴ ¬h so I do not know I have hands
The first premise is the closure principle applied to a case. The second is the skeptic’s claim that nothing in your experience could rule the vat out, since the vat is built to make your experience exactly what it is. The lab reads this argument as valid, with the form modus tollens, and that verdict is the whole of what the lab contributes.
Moore answered by reading the argument backwards. He was more certain that he had hands than he could be of any philosophical premise, so he took the same conditional and went the other way.
h I know I have hands
h → b if I know that, I know I am not envatted
∴ b so I know I am not envatted
This is modus ponens, also valid. Put the two side by side and what stands out is that they share a conditional and a disagreement. The skeptic is more sure of ¬b than of h, and Moore is more sure of h than of ¬b. The three sentences h → b, ¬b and h cannot all be true together, which is the sort of fact the consistency lab reports, and it means one has to go.
The third option
Dretske and others keep both ¬b and h and give up the conditional: you can know there is a zebra in the pen without knowing it is not a cleverly painted mule. The preset for this has only ¬b as a premise and ¬h as the conclusion, and the lab finds it invalid, with the counterexample row h = T, b = F. That row is the denier’s position written as an assignment: knows hands, does not know about the vat.
What each choice costs deserves to be said plainly. The skeptic accepts that nobody knows much of anything about the external world. Moore accepts that a philosopher can claim to know the vat is not there without any evidence that settles it. The denier accepts that “I know I have hands but I do not know I am not a handless brain in a vat” is true, which sounds absurd when said aloud. The lab can tell you none of this is inconsistent with the logic and nothing more.
Interactive
Validity by cases
Every row, every counterexample, and the form
Method
Formalising a skeptical argument
How to turn a paragraph of epistemology into something the lab can check.
1
Pick one letter for each knowledge claim
Let the letter stand for the whole claim, including “I know”. The lab does not see inside it, and the argument is about how the claims relate.
2
State the closure premise as a conditional
Write it as h → b. This is the step on which the skeptic and Moore agree, and the step that the denier rejects.
3
Write the argument in each direction
One has the negated consequent as a premise, the other has the antecedent. Run both and note the form the lab names.
4
Read the counterexample rows
A valid argument has none. An invalid one has at least one, and the row tells you which combination of truth values the argument fails to exclude.
5
Say what is left to decide
Name the premise that each side holds more firmly than the other. That judgement is not a truth-table fact, and the lesson is not finished until you have said it in words.
Which move, and what it needs
Score 0 / 0Choose the answer you can justify.
One conditional, three arguments
With h for “I know I have hands” and b for “I know I am not envatted”.
h → b, ¬b ∴ ¬h valid, modus tollens
h, h → b ∴ b valid, modus ponens
¬b ∴ ¬h invalid: h=T, b=F
{h → b, ¬b, h} cannot all be true
The first two arguments are both valid and their conclusions cannot both be accepted along with all their premises. The disagreement is therefore a disagreement about whether ¬b or h is the one to keep, and about nothing in the logic.
Common mistakes
1
Thinking a valid argument compels you to accept its conclusionValidity says only that if the premises are true the conclusion is. Here the skeptic’s argument and Moore’s are both valid, yet h → b, ¬b and h cannot all be true, so no one can accept every premise of both. A person who is more sure that they have hands than that ¬b is true should reject ¬b, and that is not a mistake of logic.
2
Hearing Moore’s reply as a refutationMoore’s argument is valid, but the skeptic can accuse it of begging the question: whoever doubts b will doubt h for the same reason. Whether that charge is fair depends on how premises earn their credibility, which the lab does not model. Moore’s reply tells you what rejecting the skeptic costs; it does not make the cost zero.
3
Treating closure as obviously trueThe conditional h → b looks like plain common sense, and that is why both sides use it. It is also the premise whose denial gives the third position, and the invalid argument in the lab is the shape of that position. Closure is a principle about knowledge, not a theorem of logic.
Check your understanding
✓
Finish when you can say what the lab has not decided.
You should be able to formalise the skeptic’s argument and Moore’s reply over one conditional, name their forms, find the counterexample row that closure denial exposes, and state in a sentence which premise each position holds more firmly. The validity verdict is the easy half.
Note: The same structure reappears whenever an argument has a premise that its opponent finds less credible than the denial of the conclusion. Learning to see the shared conditional is worth more than any one version of the vat.