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recognize when more than one explanation fits the same evidence
Paper packet. Every task here also exists on screen, where it is checked automatically; answers written on paper are not assessed by Nydus. When you are back at a device, enter your answers there.
You will recognize when more than one explanation fits the same evidence, explaining the inference and its limits in a supplied case.
An explanation can fit an observation without being the only explanation that fits. Compare predictions before selecting a new test.
| Term | What it means |
|---|---|
| Underdetermination | Available evidence leaves more than one account open. |
| Discriminating test | A test for which the rival accounts predict different outcomes. |
| Empirical equivalence | Agreement in observable consequences over a specified domain. |
| Auxiliary assumption | A background condition needed to derive a prediction from a hypothesis. |
A wet path may be explained by rain or by a sprinkler. The observation is compatible with both explanations, so it does not by itself select between them. This is a simple case of underdetermination: the available evidence leaves more than one account open. It does not mean that every imaginable account fits, that the explanations are equally plausible in every respect, or that further inquiry is impossible. It means that the stated evidence does less discriminating work than a unique conclusion would require.
Write each explanation as a hypothesis and list what it predicts under the supplied conditions. Let H1 say that rain wet the path and H2 say that the sprinkler did. Both may predict a wet path. They can differ on other observations: rain might wet an uncovered surface beyond the sprinkler's range, while a localized sprinkler need not. A useful next test seeks a difference in predictions instead of simply confirming the observation both hypotheses already explain.
The predictions must be genuinely supplied or justified. Do not invent an assumption that makes your favored hypothesis fit and its rival fail without acknowledging it. Rain patterns can be uneven, and a sprinkler can reach farther than expected. The classroom cases stipulate simple conditions so that the logic of comparison is visible. In real inquiry, the auxiliary assumptions supporting each prediction also need evidence. Underdetermination is often about an entire package of hypothesis and background conditions.
The first habit to learn is restraint: replace 'H explains E, so H is true' with 'H is compatible with E, but these alternatives also predict E'. This does not devalue explanation. It identifies the remaining inferential task. A good explanation earns support partly through how well it compares with alternatives, not merely because it can be connected to one observed result after the fact.
Another way: Keep observational and theoretical claims separate
An observation describes what the inquiry records under an agreed procedure. A hypothesis goes beyond that record by proposing a cause, structure, or general account. In a classroom detector case, 'the light turned on' can be the observation, while 'the sample contains a metal that activates this detector' is an explanatory hypothesis. The distinction is not always perfectly sharp, since observations involve instruments and interpretation, but it remains useful for tracing an argument.
Suppose two models both predict a value of ten at the measured input. Their agreement on that point does not establish that they are the same model. They may diverge at another input. A straight line and a curve can pass through the same observed point while predicting different values elsewhere. The common fit establishes compatibility with the observed value; it does not identify which structure generated it. More varied observations can make the comparison more discriminating.
This is different from simple inconsistency. If a model predicts a value of five under exactly the stipulated conditions and the verified observation is ten, it conflicts with the evidence in the simplified case. If two models both predict ten, the observation underdetermines the choice between them. Keeping these categories separate prevents a common mistake: treating failure to select one account as though it refuted all accounts, or treating one account's failure as proof that every remaining imaginable account is correct.
The term empirical equivalence is often used when rival accounts agree in their observable consequences over a specified domain. Specify that domain. Agreement on the three tests performed so far is weaker than agreement on every possible observation. A finite classroom table usually establishes only the former. Do not inflate 'no difference in this table' into a philosophical proof that no conceivable experiment could distinguish the accounts.
Another way: Design a discriminating test
A discriminating test is one for which the rival hypotheses predict different outcomes under the same stated conditions. If H1 predicts a red signal and H2 predicts a blue signal, observing the signal can distinguish them in the simplified case. If both predict red, that test adds no direct discrimination between those two accounts, even if it confirms something else. The right question is not merely whether a test produces data, but whether its possible results divide the live alternatives.
Construct a prediction table with hypotheses as rows and tests as columns. Fill the entries before inspecting the new result. A column whose entries are all the same cannot distinguish those rows. A column whose entries differ can potentially do so, subject to measurement accuracy and the validity of the background assumptions. This visual organization makes a common mistake obvious: repeating a test in which every rival predicts success may increase confidence in the shared prediction while leaving the rivalry unresolved.
After a result appears, eliminate only accounts that conflict with it under the stipulated conditions. If H1 and H2 both predict the observed value and H3 predicts another value, the test excludes H3 but leaves two possibilities. This is progress even though it is not a unique answer. The surviving set should be reported explicitly. Saying 'the test found nothing' would overlook the eliminated alternative, while saying 'H1 is proved' would overlook H2.
Real measurements often involve ranges and probabilistic predictions. Then a result need not strictly eliminate a hypothesis merely because it is unusual under that hypothesis. Instead it can change relative support. This lesson uses deterministic tables with exact observations to teach the comparison structure. The simplification is an assumption of the exercise, not a claim that every scientific test delivers a clean yes-or-no verdict. State it when transferring the method to less tidy evidence.
Another way: Other grounds for comparison
When available observations fit several explanations, investigators may compare simplicity, coherence with established evidence, explanatory scope, and independently supported mechanisms. These considerations can guide inquiry, but they require careful interpretation. 'Simpler' might mean fewer adjustable parameters, fewer unsupported assumptions, or a more unified mechanism. It does not automatically mean the shortest sentence or the account easiest for one reader to imagine. Explain the criterion before using it to favor a hypothesis.
Inference to the best explanation proposes that explanatory virtues can provide grounds for accepting an account. Critics ask whether being best among considered rivals ensures being true, since an unconsidered alternative might be better. This concern does not make comparison pointless. It limits the conclusion: best supported among the evaluated accounts under the stated criteria is more cautious than uniquely established truth. The hypothesis set itself is part of the inquiry and can be revised.
Auxiliary assumptions complicate testing. A prediction can depend on a central hypothesis, instrument functioning, initial conditions, and background rules. If the prediction fails, at least part of that conjunction needs attention, but the failure may not identify which part. For example, an unexpected signal could indicate an incorrect material hypothesis or a broken detector. Good inquiry uses independent checks to locate the failure rather than automatically protecting a favored hypothesis through arbitrary adjustments.
There is also a difference between a temporary practical limitation and an in-principle philosophical claim. Lacking the equipment to perform a discriminating test today does not establish that the rival accounts are forever indistinguishable. Conversely, if a case stipulates that two accounts have exactly the same observable consequences, proposing another observation without showing a predicted difference does not answer the stipulation. Match the scope of your response to the scope of the problem.
End with three statements: which hypotheses fit the evidence, which available test would separate them, and what assumption connects that test to the hypotheses. If no listed test separates them, say so without generalizing beyond the list. This method converts a vague feeling that 'we need more evidence' into a concrete account of which evidence would help and why. It also protects against announcing certainty whenever one appealing story fits the facts.
Another way: Do not repair every failure after the fact
An auxiliary change can be legitimate when independently motivated. A verified battery fault provides a reason to revisit a detector reading. By contrast, repeatedly inventing an uncheckable exception whenever a favored account fails makes comparison less informative. Record what motivated the revision and what new prediction follows from it. A revised account should remain open to tests that could count against it, rather than merely absorb every possible outcome through another unsupported adjustment.
A school caretaker finds the courtyard path wet at dawn. The two initial hypotheses are overnight rain and an automatic sprinkler running. Both explain the wet path. The caretaker first considers photographing the path again. Under the simple case assumptions, both hypotheses predict that it remains wet for the next few minutes, so that photograph would not distinguish them. It might document the condition, but its contribution should not be mistaken for selecting a cause.
The caretaker instead checks an uncovered bench outside the sprinkler's stipulated range. Under the case assumptions, overnight rain would wet both path and bench, while the sprinkler would wet only the path. A dry bench therefore favors the sprinkler explanation against the rain explanation in this simplified comparison. The conclusion depends on assumptions about uniform rain exposure, drying rates, and sprinkler reach. Those assumptions should be checked rather than left invisible.
Now introduce a third hypothesis: a cleaning crew washed the path. It also predicts a wet path and dry bench. The bench observation no longer uniquely selects the sprinkler. The caretaker can inspect the sprinkler's independently maintained activation log and the crew's work record. The added alternative does not make the bench check useless; it changes the surviving set. Evidence that distinguished two initial accounts may fail to distinguish a larger set.
A responsible report would say which observations were made, which hypotheses they exclude under the stated assumptions, and which remain open. It might conclude that a localized water source is better supported than uniform rain while reserving judgment between sprinkler and cleaning. This narrower conclusion still has practical value. It directs the next investigation toward the relevant records without claiming that the first plausible explanation has been proved simply because the path is wet.
H predicts E and E occurs does not deductively establish H. Check which rivals also predict E. A repeated shared prediction can remain useful evidence without discriminating between those rivals. If a test eliminates one account, retain all the others that fit; elimination is not automatically unique confirmation.
List the rival predictions for test A.
H1 predicts 4; H2 predicts 4; H3 predicts 9
Predictions must concern the same test conditions.
Record the verified result.
A gives 4
The classroom case treats this exact observation as settled.
Compare H1 with the result.
H1 fits
Its prediction matches the stipulated value.
Compare the remaining rivals.
H2 fits; H3 conflicts
Matching supports compatibility, while the mismatching prediction excludes H3 here.
Report the surviving set.
H1 and H2
The observation narrows the possibilities without uniquely selecting H1.
Identify the live alternatives.
H1 and H2
Only their differing predictions matter for this comparison.
Inspect predictions for test B.
Both predict 6
This column would leave the two accounts observationally tied.
Inspect predictions for test C.
H1 predicts 2; H2 predicts 8
The predictions differ under the same stated conditions.
Select the discriminating test.
C
Its possible outcomes separate the rival rows in the supplied table.
State the assumption behind elimination.
The measurements and background conditions are accurate
A mismatching result identifies a conflict only relative to these stipulations.
Record the initial observation.
The path is wet
All three proposed water-source accounts predict this result.
Record the second observation.
The distant uncovered bench is dry
The case stipulates that uniform rain would wet it.
Test the rain hypothesis.
Rain conflicts with the combined observations
Its bench prediction fails under the simplified assumptions.
Test the sprinkler hypothesis.
Sprinkler fits both observations
Its stipulated range excludes the bench.
Test the cleaning hypothesis.
Cleaning also fits both observations
Localized washing can wet the path without wetting the bench.
Give the bounded conclusion.
Two localized-source hypotheses remain
Eliminating rain does not select between sprinkler and cleaning.
H1 and H2 both predict the observed A value of five.
Two hypotheses remain
The current observation does not distinguish their rows.
At B both predict three; at C they predict seven and nine.
C is discriminating
Only C has different entries for the surviving hypotheses.
State the result's limitation.
At observed test A=4, H1 predicts 4, H2 predicts 4, H3 predicts 9. For H1/H2, next B predicts 6/6 and next C predicts 2/8. Count survivors and choose the discriminating next test.
| Response | |
|---|---|
| Count | |
| Test |
At A=1, H1 and H2 both predict 1. At B both predict 5; at C they predict 3 and 8. Complete the comparison.
Retain accounts consistent with A.
count accounts remain
Both initial predictions match the observed value.
Select the next test with differing predictions.
Test test
Shared predictions cannot distinguish the two rows.
Bound the expected result.
A discrimination between these supplied accounts
The comparison remains conditional on the table's measurement assumptions.
At A=7, H1 predicts 3, H2 predicts 7, H3 predicts 7. For H2/H3, B predicts 1/5 and C predicts 9/9. Count survivors and choose the discriminating next test.
Number of surviving hypotheses: count. Label of the listed discriminating next test: test.
At A=2, H1/H2/H3 predict 2/2/2. At B they predict 4/4/4; at D they predict 1/3/5. Count survivors and choose the listed next test that separates all three.
Number of surviving hypotheses: count. Label of the listed discriminating next test: test.
Observed test A gives 5. H1,H2,H3 predict 5,5,9. At test B the surviving pair predicts 2,2; at C it predicts 4,7. Give surviving count, discriminating test label and number of survivors that predict 7 at that test.
| Your reconstruction | |
|---|---|
| Survivors | |
| Next test | |
| Predict 7 |
Rain, sprinkler, and cleaning all predict the observed wet path. Under stipulated conditions, test B (path again) predicts wet/wet/wet; test C (source records) predicts rain-record/sprinkler-record/cleaning-record. Treat those records as exact and independent. Count current survivors and choose the next test separating them.
Number of surviving hypotheses: count. Label of the listed discriminating next test: test.
Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.
At observed A=6, H1/H2/H3/H4 predict 6/2/6/6. For surviving H1/H3/H4, test E predicts 0/0/0 and test F predicts 2/4/8. Count survivors and select the test separating all survivors.
Number of surviving hypotheses: count. Label of the listed discriminating next test: test.
Without rereading, explain how to recognize when more than one explanation fits the same evidence. Give a fresh case and identify what would change your analysis.
10. Finish the next-test choice, step 3
This concerns the supplied alternatives and exact tests
An unlisted hypothesis or a faulty measurement would require additional analysis.