The traditional analysis says that to know is to believe something true with justification. Each of the three conditions is claimed to be necessary and the three together sufficient, and a table of cases can test both claims. The skill taught here is finding the row where a definition and a verdict disagree.
Core ideas
An analysis of knowledge is a definition, and a definition can be wrong in exactly two ways. Cases are how you find out which.
A condition is necessary when every case that lacks it is a case of not knowing. One lucky guess, a belief held without any justification that happens to be true, is enough to show why justification is on the list.
The conditions are jointly sufficient when every case that has all of them is a case of knowing. This is the claim that later lessons break.
A case table records what you judge about each case before the definition is applied. The lab compares the definition with those judgements; it cannot say whether the judgements are right.
The material
A definition, four cases, and the one question the table asks of each row.
To believe something is not yet to know it, and to be right is not yet to know it either. A person who believes it will rain tomorrow, and is right, but only because they flipped a coin to decide what to believe, does not know it will rain. The oldest serious answer to what is missing is that the belief must also be justified.
A person knows that p exactly when three conditions hold: p is true (T), the person believes p (B), and the belief is justified (J). In symbols, knows ⟺ J ∧ T ∧ B.
Two claims are packed into that biconditional, and they fail differently. Each condition is necessary: nobody who lacks one of the three knows. And the three are jointly sufficient: anybody who has all three knows. A definition that fails in the first way, because some case lacks one of its conditions and is knowledge all the same, shuts out a case of knowledge and is called too narrow. One that fails in the second way, because some case has every condition and is not knowledge, lets in a case that is not knowledge and is called too broad. This lesson tests the first claim; the next lesson breaks the second.
Read the table row by row. Ordinary perception is a plain case: you see the cup, you believe it is there, it is there. The lucky guess lacks justification, the confident error lacks truth, and the unbelieved truth lacks belief, as when a student has good evidence that she passed and cannot bring herself to accept it. Each of the three failing rows is a reason one condition is on the list.
Written as a formula, the definition is J ∧ T ∧ B, and the lab writes the and-sign as an ampersand. It agrees with all four verdicts. Now remove a condition and test again. The definition T ∧ B says the lucky guess is knowledge, and the verdict says no. The definition J ∧ B says the confident error is knowledge, and the verdict says no. The definition J ∧ T says the unbelieved truth is knowledge, and the verdict says no. Each shortened definition is too broad, and the lab names the row.
State a definition. Collect cases whose verdict you are confident of. Apply the definition to every case. If the definition and a verdict disagree on some row, one of them has to give, and the rest of the lesson is about which. Here nothing disagrees, so the definition survives these four cases and no more than these four.
That last clause is the lab’s limit. Agreement on four rows is agreement on four rows. The next lesson is a case the table above does not contain.
Interactive
A definition tested against every case in the table
Method
Five moves, in this order. The order matters: the verdict comes before the definition.
Name each condition the analysis uses and give it a letter. A case is then a row of ones and zeros: has the condition, or lacks it.
Write what you judge about each case before looking at the definition. Otherwise the definition decides your verdict and the test tests nothing.
Apply the formula to each row. The definition is true on a row when it says the person knows.
A row where the definition is true and the verdict is no is too broad. A row where the definition is false and the verdict is yes is too narrow.
“It fails” is not a finding. “It is too broad on the lucky guess” tells the next author which way to repair it.
Take the four cases of the table and try the full definition, then each pair.
Every pair is too broad on exactly the row that the dropped condition was there to exclude. That is what it means for the three to be individually necessary on this table, and it is the whole of the argument for each: a case in which the condition is missing and knowledge plainly is too.
Given a proposed definition and a table of cases you should be able to evaluate the definition on every row, name the first row where it disagrees with the verdict, and say whether it is too broad or too narrow there. “It does not work” without the row and the direction does not carry over to the cases in “Gettier Cases and the Fourth Condition”.
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