A theory yields a prediction only when it is joined to auxiliary hypotheses and a bridge to the instruments. When the prediction fails, the theory, the auxiliaries, the bridge and the observation cannot all be true, and the smallest inconsistent subset is the whole set. The observation refutes the conjunction and nothing smaller.
Core ideas
The previous lessons let a theory forbid an outcome and treated one forbidden observation as the end of it. This lesson asks what exactly was forbidden.
Newton's laws say nothing about where Uranus will stand on a given night. The calculation also uses the list of planets that pull on it, their masses, and a rule that links the computed position to what a telescope reports. Each of these is an assumption added to the theory.
Write the theory, the added assumptions, the link to the instruments and the claim that the predicted outcome did not occur. These cannot all be true together, and that is all the observation establishes.
Dropping any single member of the set restores consistency. The logic is silent on which one to drop, and the choice rests on grounds the set does not contain.
The material
The prediction first, written as a derivation. Then the set it makes inconsistent, and the repairs that restore consistency.
A theory by itself forbids nothing observable. “Falsification and What a Theory Forbids” gave a hypothesis a likelihood of zero for some outcome, and that is the right way to model a theory together with the conditions of one particular test. The conditions are the subject here.
An auxiliary hypothesis is an assumption, other than the theory under test, that is used to derive a prediction from it. A bridge is the assumption that links a computed quantity to what an instrument will report.
Write h for the theory, a for the auxiliary that nothing unseen disturbs the system, b for the bridge, and e for the predicted observation. The prediction is a conditional: if the theory, the auxiliary and the bridge are all true, then e. The observation is that e did not occur.
Sentences 1 to 4 entail e, and sentence 5 denies it. So the five cannot all be true, and this is the whole logical content of a failed test. The lab runs the test of “Consistency and Belief Sets”: it finds no assignment of truth values to the four letters that makes all five true, and names the smallest subset that cannot be true as {1, 2, 3, 4, 5}: the whole set.
If every member of an inconsistent set is needed for the inconsistency, the set has no smaller inconsistent subset, and dropping any one member makes the rest consistent. A failed prediction therefore refutes the conjunction of everything used to derive it, and no single member by itself.
Duhem made this point for physics: an experiment can condemn only a whole theoretical group, never an isolated hypothesis, because the physicist reads the instrument through the theory being tested. Quine extended it to everything we believe, so that any statement can be held true come what may if enough is adjusted elsewhere, and the unit of empirical significance becomes the whole of science. The set above is Duhem's group written out in five lines. Where the two differ is whether its boundary can be drawn anywhere short of everything, and the lab, which takes the sentences it is given, cannot say.
The lab's drop control shows this. Drop sentence 1 and the other four have models; drop sentence 2 and they have models; the same holds for 3, 4 and 5. Five different repairs are equally consistent. Dropping sentence 5 means disbelieving the observation, and that is a repair too, though a costly one when the measurement was careful.
By the 1840s the orbit of Uranus disagreed with the position computed from Newton's laws and the seven known planets. Astronomers who dropped the auxiliary that no further planet exists, and kept Newton, computed where an eighth would have to be, and Neptune was found close to that place in 1846. The same repair was tried for Mercury's orbit: a planet inside Mercury's orbit was proposed, none was found, and the repair that worked dropped the theory, in 1915.
The second and third presets are two consistent sets made from this one failure. In the second, the theory is kept and the auxiliary is replaced by its denial, so an eighth planet exists. In the third, the auxiliary is kept and the theory is denied. Each has exactly one model, and the count is the same. Nothing in the arithmetic prefers either.
What does choose is outside the set. A repair that predicts something new and checkable, such as a planet at a stated place, can be tested on its own, and one that only restores the old prediction cannot. That is a reason to try the first kind of repair first, and it is a reason of method and not of logic.
The positions can be put at full strength. The falsificationist can say that science proceeds by a decision to hold the auxiliaries fixed for the purpose of a test, so that the theory is exposed. The holist can answer that no such decision is forced by the logic, and that a decision to hold something fixed can be revised by the next test. The lab agrees with both: the set is refuted, and the choice of what to blame is made outside it.
The lab's limit is that it tests only whether sentences can all be true together. It does not weigh them, so it cannot say that the bridge is better supported than the theory. The earlier lessons supply the means for that: the posterior of each member given the observation.
Interactive
Every model, and every minimal inconsistent subset
Method
Five steps, in this order, for any test that did not come out as predicted.
Write the theory, every auxiliary, and the bridge to the instrument as separate sentences. If you cannot say what the auxiliaries were, you have not yet said what was tested.
Join the theory, auxiliaries and bridge as the antecedent and the predicted outcome as the consequent. Then add the observation that the outcome failed.
Run the set through the consistency test. If each sentence is needed, the smallest inconsistent subset is the whole set.
For every sentence, ask whether the rest is consistent. Each yes is a repair that the logic permits.
For each repair, say what else it commits you to and whether that can be tested separately. Prefer the repair that makes a prediction of its own.
Take h as Newton's laws, a as “no eighth planet”, b as the bridge from computed to observed position, and e as the predicted orbit.
Both repairs are consistent, with the same number of models, and the arithmetic cannot say which was right. History chose the first for Uranus. For Mercury it tried the same repair first, a planet Vulcan, and the second repair, a new theory, was the one that worked.
Given a theory, its auxiliaries, a bridge and a failed prediction, you should be able to write them as a set of sentences, find the smallest inconsistent subset, list the repairs that restore consistency, and say what separates a repair that can be tested from one that cannot.
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