The Syllogism Audit at the Dharwad Logic Fest
The logic association of a PU college in Dharwad holds a 'syllogism audit' for 24 inter-collegiate teams. Each team submits four arguments, and a first-year volunteer must mark them using mood, figure and the rules of distribution. The four arguments on the key are: (i) All scholarship holders are library members; some library members are day scholars; therefore some day scholars are scholarship holders. (ii) No mobile phone is a learning aid; all tablets are mobile phones; therefore no tablet is a learning aid. (iii) All poets are imaginative; some published authors are not imaginative; therefore some published authors are not poets. (iv) All journalists are graduates; all graduates are degree holders; therefore all degree holders are journalists. The volunteer has marked (ii) and (iv) as valid 'because the statements are true in real life', and (i) and (iii) as invalid 'because they sound odd'. Results are to be announced in 20 minutes, and the coordinator needs each verdict corrected, with the fallacy named for every invalid argument.
1. State the mood and figure of arguments (ii) and (iii), and verify each with the rules of distribution. Are they valid?
Read the worked answer
Argument (ii) - S = tablets, P = learning aids, M = mobile phones.
- Major premise (the one containing P): No mobile phone is a learning aid = No M is P - E.
- Minor premise: All tablets are mobile phones = All S is M - A.
- Conclusion: No S is P - E. Mood EAE; M is the subject of the major and the predicate of the minor, so figure 1 (Celarent).
Distribution check: E distributes both terms, so M and P are distributed in the major; the E conclusion distributes S and P. Nothing is distributed in the conclusion that was not already distributed in a premise. There is one negative premise and the conclusion is negative, so the quality rule is satisfied. Valid.
Argument (iii) - S = published authors, P = poets, M = imaginative persons.
- Major: All poets are imaginative = All P is M - A.
- Minor: Some published authors are not imaginative = Some S is not M - O.
- Conclusion: Some S is not P - O. Mood AOO; M is the predicate of both premises, so figure 2 (Baroco).
Distribution: A distributes its subject P; O distributes its predicate M; the O conclusion distributes P. P is distributed in the major as well, M is distributed in the minor, S never; no illicit process. One negative premise with a negative conclusion. Valid. Flag the traditional assumption that 'some' carries existence - the minor premise asserts that at least one published author exists. Under a Boolean reading without existential import, the step from the minor premise to the conclusion would need that assumption spelled out.
2. Arguments (i) and (iv) were rejected only because they 'sound odd'. Give the mood and figure of each and name the exact fallacy involved.
Read the worked answer
(i) S = scholarship holders, P = day scholars, M = library members.
- Major (contains P): Some library members are day scholars = Some M is P - I.
- Minor: All scholarship holders are library members = All S is M - A.
- Conclusion: Some S is P - I. Mood IAI, figure 1 (M subject in the major, predicate in the minor).
An I proposition distributes neither term, so M is undistributed in both premises → fallacy of the undistributed middle. IAI-1 is not among the valid moods of figure 1 (AAA, EAE, AII, EIO). Counterexample with true premises and false conclusion: let M = animals, P = cats, S = dogs - some animals are cats, all dogs are animals, but 'some dogs are cats' is false. The form is at fault, not the wording.
(iv) S = degree holders, P = journalists, M = graduates.
- Major: All journalists are graduates = All P is M - A.
- Minor: All graduates are degree holders = All M is S - A.
- Conclusion: All S is P - A. Mood AAA, figure 4 (M predicate in the major, subject in the minor).
A distributes only its subject, so P is undistributed in the major but distributed in the A conclusion → illicit major (illicit process of the major term). AAA-4 is not a valid mood of figure 4 (AAI, AEE, IAI, EAO, EIO). Both verdicts of the volunteer are therefore wrong, and both failures are failures of distribution, not of natural-sounding content.
3. The volunteer insists (iv) must be valid because the premises are true in real life and the conclusion uses the same three terms. Diagnose the confusion and state the principle that settles the case.
Read the worked answer
Three ideas are being run together: truth of a proposition, validity of a form, and soundness of an argument.
- Truth belongs to individual propositions; validity belongs to the link between premises and conclusion. A form is valid when it is impossible for all the premises to be true while the conclusion is false - a formal test, not a factual one.
- The volunteer's own example refutes him. 'All journalists are graduates' and 'All graduates are degree holders' are close to true in the real world, while 'All degree holders are journalists' is plainly false. An argument with true premises and a false conclusion cannot be valid; the situation is a textbook counterexample, not a proof of validity.
- Sharing the same three terms is irrelevant. What matters is their distribution: the major term P is distributed in the conclusion but not in the major premise, so the conclusion claims more than the premises license.
The controlling principle is: a deductive argument is valid if and only if no substitution instance of its form has true premises and a false conclusion. One actual counterexample is enough to expose the form as invalid, however plausible the sentences sound. Soundness would require validity plus true premises - and (iv), being invalid, is not made sound by true premises.