If the statement ‘most of the students are obedient’ is taken to be true,…
2012
If the statement ‘most of the students are obedient’ is taken to be true, which one of the following pair of statements can be claimed to be true ?
I. All obedient persons are students.
II. All students are obedient.
III. Some students are obedient.
IV. Some students are not disobedient.
Codes :
Answer: C. III & IV — Concept. A quantified categorical statement licenses only those inferences that its own quantifier and its own term order support. Three rules settle an item…
- A.
I & II
- B.
II & III
- C.
III & IV
- D.
II & IV
Show answer & explanation
Correct answer: C
Concept. A quantified categorical statement licenses only those inferences that its own quantifier and its own term order support. Three rules settle an item of this kind.
Weakening is allowed, strengthening is not: a claim about most members of a class entails the corresponding particular claim ("some"), because a majority is never empty, but it never entails the universal claim ("all"), because "most" expressly leaves room for a minority.
The subject class is not the predicate class: a claim of the form "Most S are P" describes the S, and says nothing about the members of P that lie outside S, so it cannot be turned round into a universal claim about P.
Obversion: "Some S are not non-P" is logically equivalent to "Some S are P", because the double negation on the predicate term cancels.
Application.
The given statement is "Most students are obedient". The subject class is students, the predicate class is obedient persons, and the quantifier "most" asserts that a majority of the students — more than half — lie in the obedient class.
Statement I, "All obedient persons are students", is a claim about the whole class of obedient persons. The given statement describes only the students and leaves obedient non-students entirely unconstrained, so by rule (ii) it cannot be derived.
Statement II, "All students are obedient", upgrades "most" to "all". Since "most" explicitly allows a minority of students who are not obedient, by rule (i) the universal cannot be derived.
Statement III, "Some students are obedient", weakens "most" to "at least one". A majority always contains at least one member, so by rule (i) this does follow from the given statement.
Statement IV, "Some students are not disobedient", denies disobedience of at least one student. "Not disobedient" is the double negation of "obedient", so by rule (iii) statement IV asserts exactly what statement III asserts, and it follows for the same reason.
The statements that can be claimed true are therefore III and IV, so the required pair is III & IV.
Cross-check. Build one model in which the given statement is true — of 100 students, 60 are obedient and 40 are not — and test each statement against it.
Statement | Result in the model |
|---|---|
I. All obedient persons are students | Breaks down as soon as one obedient person is not a student, say an obedient teacher, which the given statement never rules out |
II. All students are obedient | Breaks down because 40 of the 100 students are not obedient |
III. Some students are obedient | Survives, since the 60 obedient students witness the claim |
IV. Some students are not disobedient | Survives, since those same 60 students are not disobedient |
A single model of this kind is enough to refute I and II, while III and IV survive in every model of the given statement — indeed III and IV are two wordings of one and the same particular claim. Hence the pair III & IV.