Lesson · Ethics and the Arithmetic of Welfare · Metaethics in outline

Is and Ought

An argument whose premises only describe and whose conclusion prescribes has a counterexample row, and the lab finds it. Adding a bridging premise makes the argument valid — and the work moves to that premise: what it says, who accepts it, and what it costs to assume it.

Open the interactive lesson Practice

premises: facts conclusion: an ought the row p = T, q = T, o = F refutes it add (p ∧ q) → o and it is valid the bridge is the premise to examine

A gap, and what closes it

Core ideas

What this lesson establishes

Hume’s observation is about what an argument can deliver, not about what is true. The three ideas below keep those apart.

A gap in vocabulary, not a verdict on morality

If the conclusion says something the premises never say, validity needs a premise that connects them. An argument can lack that premise and still have a conclusion that is true. The gap is in the argument, not in the conclusion.

A bridge makes the argument valid and nothing more

Add the premise that links the facts to the duty and the argument is valid. Validity says that if the premises are true the conclusion is. It does not say the new premise is true, and that is where the dispute moves.

The lab sees letters, not meanings

To the lab o is a sentence letter. It does not know o is an ought; it knows that o appears in the conclusion and in no premise. That is the logical shape of Hume’s point, and it is all the lab can check.

The material

From what is to what ought to be

One argument, its counterexample row, and the premise that closes the gap.

Definition · The is–ought gap

An argument crosses the is–ought gap when every premise describes how things are and the conclusion says how they ought to be. Hume’s remark was that such arguments usually slip an “ought” in between the last description and the conclusion without announcing it.

Take Jones, who promised to repay Smith. Let p be “Jones promised to repay Smith”, q be “Smith has not been repaid” and o be “Jones ought to repay Smith”. The argument is p, q ∴ o, and it sounds entirely natural.

The lab builds all eight rows for p, q and o and finds exactly one on which both premises are true and the conclusion is false: p = T, q = T, o = F. Nothing is being claimed about Jones. The row only describes a case in which the facts hold and the duty does not follow from them.

p q o | premises both true? | conclusion T T T | yes | T T T F | yes | F (the counterexample row) any other row: a premise is false, so no refutation

It is easy to hear this as saying something stronger, so the limit of the point is worth stating. The row does not show that o is false, or that there are no duties, or that morality is opinion. It shows that these two premises do not force this conclusion.

Definition · Bridging premise

A bridging premise is an added premise that connects the descriptive premises to the normative conclusion. Here it is (p ∧ q) → o: if Jones promised to repay and has not repaid, then Jones ought to repay.

With the bridge in place the lab finds no counterexample row. Every row that makes p, q and the bridge true also makes o true, so the argument is valid. The bridge is not a description of anyone’s behaviour. It says what follows from the facts, and it is an ought-bearing claim.

Three replies are available, and each can be put at full strength. The moralist accepts the bridge as a moral principle, argued for on its own grounds. Searle replies that promising is an institution: to say the words under the right conditions is, by what the word means, to undertake an obligation, so the bridge records a fact about the institution rather than a moral commitment. The skeptic answers that this only relocates the bridge: why should anyone act inside the institution at all?

Example · Searle’s derivation

Let u be “Jones uttered the words of a promise in the right conditions”, m be “Jones made a promise” and o be “Jones ought to repay”. The premises are u, u → m and m → o, and the conclusion is o. The lab finds the argument valid. Whether m → o is a fact about what promising is or a moral claim in disguise is exactly what the lab cannot tell you, since both are sentence letters to it.

So the price of the bridge is threefold. It has to be argued for, because validity gives it no support. It has to be defended for every case it covers, not only for Jones. And it can be denied without inconsistency by someone who grants every fact: the counterexample row is a situation in which the facts hold and the bridge fails.

Interactive

Validity by cases

Every row, every counterexample, and the form

Choose the argument

The table lists every assignment of true and false to the sentence letters. The verdict is read off it: valid exactly when no row is a counterexample.

Argument

Premises, separated by semicolons

Conclusion

Rows shown

Rows—

Counterexample rows—

Verdict—

Form—

Sentence letters are words such as p or q; type ~ for not, & for and, | for or, -> for if-then and <-> for if and only if, or use the symbols.

Method

Testing an argument across the gap

Five questions, in this order. The third one is a table, not a judgment.

Sort the sentences

Mark each premise and the conclusion as descriptive (says how things are) or normative (says how they ought to be). If the conclusion is normative and no premise is, the gap is in play.

Give each sentence its own letter

Write the descriptive premises as p, q and the conclusion as o. The conclusion’s letter should appear in no premise; that is the formal shape of the gap.

Run the table

A counterexample row, with every premise true and the conclusion false, confirms that the premises do not force the conclusion.

Write the bridge

State the premise that would close the gap, usually a conditional from the facts to the duty, and check the argument is now valid.

Price the bridge

Ask who would accept it and why, whether it holds in every case it covers, and whether it is itself a description or an ought. Only then decide whether to accept the conclusion.

What the row shows

Score 0 / 0

Choose the answer you can justify.

Jones and the debt

Formalise the argument, find the row, then close the gap and ask what closing it cost.

p Jones promised to repay Smith q Smith has not been repaid o Jones ought to repay Smith p, q therefore o: invalid, row p=T q=T o=F add the bridge (p and q) imply o p, q, bridge therefore o: valid, no such row

The second verdict is not an endorsement. The lab has shown that the conclusion follows if the bridge is true. Whether it is true is the question the argument was supposed to settle and has instead been rewritten to contain.

Common mistakes

Reading Hume as saying that moral claims are falseThe counterexample row p = T, q = T, o = F says only that these premises do not force this conclusion. With the bridge, o follows, and is as true as the premises are. A conclusion can be unsupported by some premises and supported by others; the gap is a fact about one argument.

Treating the bridge as a harmless extraThe bridge does the work. Whoever accepts the argument has accepted it, and it is the premise a critic should look at first. An argument that quietly assumes its bridge has assumed the point at issue.

Taking “valid” as the end of the matterThe Searle preset is valid and the dispute goes on, because validity does not touch the truth of m → o. The lab checks the step from premises to conclusion; it never checks the premises.

Check your understanding

✓

Finish when you can name the premise that carries the ought.

Given an argument from stated facts to a stated duty, you should be able to classify its sentences, produce the counterexample row, write the bridging premise that closes the gap and say in one sentence what accepting that premise commits the arguer to.

Note: The lab uses one letter for each sentence, so it cannot see inside “ought”. A stricter treatment needs a logic of obligation; “Moral Dilemmas and Deontic Consistency” later in this course uses one only as a consistency check.