Three Arguments on the Notice Board
The student council posts three arguments on the notice board and asks the debating society to label them valid or invalid before a scholarship interview. Argument A: 'All scholarship holders are meritorious. Some meritorious students are athletes. Therefore some athletes are scholarship holders.' Argument B: 'All scholarship holders are meritorious. No athlete is a scholarship holder. Therefore no athlete is meritorious.' Argument C: 'All scholarship holders are meritorious. Some athletes are not meritorious. Therefore some athletes are not scholarship holders.' A first-year student says all three are invalid because the premises do not mention the same people. A second-year student says A and B are valid because the words 'all' and 'no' are strong. The debating society must produce a verdict with a method, not just answers, and must explain why a single counterexample is enough to prove invalidity but not enough to prove validity.
1. Put A, B and C into standard syllogistic form, identify the middle term, and decide validity using distribution or countermodels.
Read the worked answer
Standard form: A: All S are M. Some A are M. Therefore some A are S. (S = scholarship holders, M = meritorious, A = athletes.) This is invalid: the middle term M is undistributed in both premises. Countermodel: S = {1}, M = {1,2}, A = {2}. Premises true, conclusion 'some athletes are scholarship holders' false. B: All S are M. No A are S. Therefore no A are M. Invalid: the major term M is distributed in the conclusion but undistributed in the premise 'All S are M'. Countermodel: S = {1}, M = {1,2}, A = {2}. Premises true, conclusion false. C: All S are M. Some A are not M. Therefore some A are not S. Valid: if the A that is not M were S, then by 'All S are M' it would be M, contradiction. No existential fallacy because the second premise asserts that such an A exists.
2. Explain the distribution rules that decide A and B. Then evaluate the two students' reasoning.
Read the worked answer
A term is distributed when the proposition says something about every member of its class. 'All S are M' distributes S only; 'No A are S' distributes both A and S; 'Some A are M' distributes neither; 'Some A are not M' distributes M only. A is invalid because the middle term must be distributed at least once. B is invalid because the major term is illicitly distributed. The first-year student's objection is wrong: the arguments do share terms, but sharing terms is not enough; the middle term must connect the premises. The second-year student's appeal to strong words is also wrong: validity depends on form, not on the emotional force of 'all' or 'no'. A valid syllogism can have premises that are factually false, and an invalid one can have premises that are factually true.
3. Why does one counterexample prove invalidity, while no number of examples can prove validity? Distinguish deduction from induction here.
Read the worked answer
A single counterexample proves invalidity because validity means 'no possible case with true premises and false conclusion'. One consistent case is enough to refute that universal claim. Validity cannot be proved by examples; it needs a rule-based derivation or a semantic argument covering all interpretations. A valid argument can have false premises: validity concerns the link between premises and conclusion, not their actual truth. For example, C remains valid even if no athlete is in fact a scholarship holder, provided the premises are true. Deduction aims at conclusive form; induction aims at probable support and is assessed by strength, not validity.