Three Science-Magazine Submissions: Induction, Affirming the Consequent, and a Confounded Experiment
A school science magazine has received three 500-word submissions for its annual issue; the deadline is Friday and two reviewers must choose one. Submission 1 argues: 'Every swan we have observed is white; therefore all swans are white; therefore the statement all swans are white is scientifically proven.' Submission 2 argues: 'Astrology makes testable predictions about personality; all scientific theories make testable predictions; therefore astrology is scientific.' Submission 3 reports an experiment: four identical pots, same soil, water and light; pot A gets 10 ml fertilizer, pot B gets 20 ml, pot C gets 30 ml, pot D gets none; all plants grow taller with more fertilizer, so fertilizer is the cause. Reviewers notice pot D was kept in a darker corner. The editor asks you to rank the submissions for publication and explain the logical and methodological errors, using Mill's methods and Popper's falsification.
1. Assess Submission 1's use of induction and its claim that the hypothesis is scientifically proven.
Read the worked answer
Submission 1 uses enumerative induction: from a finite set of observed white swans it infers a universal statement. The inference is not deductively valid; the premise can be true while the conclusion is false. The historical discovery of black swans in Australia is the standard counterexample. Induction can give support or confirmation, but it cannot give proof of a universal empirical statement.
The phrase 'scientifically proven' conflates confirmation with proof. Under Popper's account, universal scientific hypotheses are not verifiable by accumulated instances; they are falsifiable by a genuine counterexample. The statement 'all swans are white' is scientific because it forbids something - the existence of a non-white swan. But the observation of many white swans does not prove it; one black swan refutes it.
A better submission would say: 'The hypothesis that all swans are white has been corroborated by all observed cases so far, but it remains open to falsification.' That preserves the inductive support while avoiding the false claim of proof. The reviewer should also note that the sample is biased if observations are drawn only from one region.
2. Formalise Submission 2's argument and identify its logical error.
Read the worked answer
Submission 2 has the form: All scientific theories are things that make testable predictions; astrology is a thing that makes testable predictions; therefore astrology is scientific. In categorical form, the middle term is the predicate of both premises, so the argument is invalid. It commits the fallacy of affirming the consequent: if S then P; P; therefore S. Making testable predictions is a necessary condition of scientificity, not a sufficient one.
There is also an ambiguity problem. 'Testable predictions' may mean precise, risky predictions that could fail, or vague statements that can be reinterpreted after the fact. Astrology's predictions are often vague and post hoc; if they are made precise, they have repeatedly failed. So the premise 'astrology makes testable predictions' is either false or equivocal.
The reviewer should reject the argument, but not merely by saying astrology is false. The logical point is that a shared feature does not establish membership in a class. A scientific theory must also be coherent, empirically adequate, and able to survive serious attempts at falsification. Submission 2 should be returned for revision because its central inference is invalid.
3. Evaluate Submission 3's causal claim and recommend a better experimental design.
Read the worked answer
Submission 3 appears to use Mill's method of difference: pot D lacks fertilizer and grows least, while pots A–C receive increasing amounts and grow more. The dose-response pattern strengthens the causal suggestion. But the design is confounded: pot D is in a darker corner, so light, not fertilizer, may explain the difference. The comparison between D and A–C mixes two variables. The method of difference requires that the two instances be alike in all relevant respects except the alleged cause.
A better design would keep light, temperature, soil, water, and pot size constant; randomly assign several pots to each fertilizer dose, including a control; measure growth blind; and repeat the trial. If the dose-response remains under identical light conditions, the causal claim is better supported.
Ranking for publication: Submission 3 is the most publishable because it is empirical and fixable; Submission 1 is second because it is clear but epistemically mistaken; Submission 2 should be rejected for formal invalidity. The editor should ask for a revised version of Submission 3 with a proper control and a caution about correlation versus causation.