1
A definition makes a claim about every case
To say a thing is good exactly when it is pleasant is a biconditional. It is true or false of each case you can describe, so a single case where the two come apart is enough to refute it.
Lesson · Ethics and the Arithmetic of Welfare · Metaethics in outline
Defining “Good” and the Open Question
A definition of “good” predicts, for every case you can describe, whether it is good. Write the cases as a table, evaluate a candidate definition on every row, and classify each disagreement as too broad or too narrow. The lab also searches for the short formulas that fit the cases you accept.
a definition: good ⟺ some condition holds
says good, table says no: too broad
says no, table says good: too narrow
a fit to five cases is a candidate only
Testing a definition against cases
Core ideas
What this lesson establishes
The method is the one used for any definition: propose, then try to break it with a case. The three ideas below say what counts as breaking it.
2
There are two ways to fail
A definition is too broad when it counts a case good that the table says is not, and too narrow when it leaves out a case the table says is good. It can be both, and the two repairs pull in opposite directions.
3
The verdicts are premises
The case table is judgment, not measurement. You may reject a verdict, but you then owe a reason, and the lab shows what the other cases cost you when you do.
The material
Definitions against cases
Four conditions, five cases, and two candidate definitions.
G. E. Moore’s open question argument starts from a contrast. If “a bachelor is an unmarried man” is a real definition, then asking of an unmarried man whether he is a bachelor is a closed question: no competent speaker can sensibly doubt it. But asking of a pleasant thing whether it is good is open. Someone who asks it is not confused. So “good” does not simply mean “pleasant”.
The lab does not run that test. It runs the method the test motivates, which is the method of cases: describe a case, settle its verdict before consulting any definition, and see which definitions agree.
Definition · Definition tested by cases
A proposed definition is a formula in the conditions P (pleasant), D (desired), A (authentic: the person’s beliefs about the life are true) and H (harmless to others). It fits a case when the formula is true exactly when the case is good.
Five cases, with each condition marked 1 if it holds and 0 if not, and the verdict the case table records.
case P D A H | verdict
a good life 1 1 1 1 | good
a hard-won project 0 1 1 1 | good
the experience machine 1 1 0 1 | not good
the satisfied sadist 1 1 1 0 | not good
the unwanted cure 0 0 1 1 | good
Test hedonism, “good = pleasant”, which is the formula P. It is true of the good life, and of the experience machine and the satisfied sadist, which the table says are not good: too broad. It is false of the hard-won project and the unwanted cure, which the table says are good: too narrow. It fits only the first case. The lab calls it too broad and too narrow.
Desire fares differently. “Good = desired”, the formula D, fits the good life and the project but is too broad on the machine and the sadist and too narrow on the cure. A different shape of failure, from different cases.
The pairs search then asks which conjunctions or disjunctions of two conditions fit all five cases. It finds A ∧ H: good when the life is authentic and harms no one. That is a candidate, and a modest one. It fits these cases, and a sixth case could break it. A philosopher would also ask whether it still passes the open question.
What if you reject a verdict? The hedonist can say the machine life is good. In the lab, click that verdict and change it. Hedonism then fits two cases instead of one, and it still fails on the other three. Rejecting a verdict is allowed, but it is a specific move with a specific price, which is what makes it different from a shrug about taste.
The limit is stated once. The lab checks formulas against the table you give it. It does not decide which cases belong in the table, or whether four conditions are the right ones to describe a life.
Interactive
The method of cases
A definition tested against every case in the table
Method
Testing a definition
Settle the cases first. A definition judged before the cases is a definition judged on how it sounds.
1
Write the cases as rows
Choose the conditions the definition is built from and mark each case 1 or 0 on each condition. A case is one row.
2
Settle each verdict independently
Decide whether the case is good before looking at any definition. The verdict column is your data, and it is allowed to be disputed later.
3
Evaluate the definition on every row
Compute the formula’s value for each case and mark the rows where it differs from the verdict.
4
Classify each disagreement
Definition says good, table says not: too broad. Definition says not, table says good: too narrow. Report the first row of each kind.
5
Ask what the failure shows
Either revise the definition to fit the cases, or defend the verdict you are refusing. Both are legitimate; the second needs a reason beyond the definition.
Too broad or too narrow
Score 0 / 0Choose the answer you can justify.
Hedonism on five cases
Evaluate the formula P on each row and compare it with the verdict.
case P verdict result
a good life 1 good fits
a hard-won project 0 good too narrow
the experience machine 1 not too broad
the satisfied sadist 1 not too broad
the unwanted cure 0 good too narrow
One agreement in five, with failures in both directions. The machine is the classic case: pleasant, desired by the user, not authentic. The search finds A ∧ H as a fit to all five, which is why the lab reports a candidate rather than a definition.
Common mistakes
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Treating a counterexample to a definition as a disagreement about tasteThe verdict column is the data, and a disagreement is a specific dispute about one cell. Changing the machine’s verdict to good lifts hedonism from one agreement to two and leaves three failures. Rejecting a verdict is a move with a price, not an exit.
2
Reading a fit as the discovery of what good meansA formula that matches the table is a candidate. A ∧ H fits five cases and the next case may break it. The search lists what the cases allow, not what “good” means.
3
Mixing up the two directionsAsk which side the definition is on. If it says good where the table says not, it has reached too far, and the repair is a stricter condition. If it says not where the table says good, it has left something out, and the repair is a weaker one.
Check your understanding
✓
Finish when you can classify each failure and say which it is.
Given a definition and a table of cases, you should be able to evaluate the definition on every row, name each disagreement as too broad or too narrow, and say whether you would revise the definition or dispute the verdict.
Note: “Defining art” is a legitimate instance of the same method, and so is defining knowledge, as in Knowledge and Evidence. What changes is the cases, not the method.