Scholarship Notice and the Square of Opposition
The scholarship office of a degree college in Bhubaneswar has issued a notice: 'All students receiving the merit scholarship are residents of the college hostel.' An audit later finds that 7 of the 40 merit scholars live with relatives outside the hostel, so the notice's universal claim is false. In a logic tutorial, three students respond. Anita says, 'Then no merit scholar is a hostel resident.' Bikash says, 'Then some hostel residents are not merit scholars.' Chandan says, 'Then some merit scholars are hostel residents.' The teacher asks the class to use the square of opposition and immediate inference rules, not general impressions, to decide which response follows. The tutorial group must also explain what can be inferred from the original universal statement if it had been true, and whether converting it or converting its contradictory is legitimate.
1. Symbolize the notice and the audit finding using S for merit scholars and H for hostel residents. State their relation on the square of opposition, and determine what can be known about the four standard categorical statements A, E, I and O.
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Let S = merit scholars and H = hostel residents.
The notice is an A statement: All S are H. The audit finding is an O statement: Some S are not H - specifically, 7 of the 40 merit scholars.
A and O are contradictories. If the O statement is true, the A statement must be false, and vice versa. Since the audit establishes O, we know:
- A is false.
- O is true.
- E ('No S are H') is undetermined. The 33 merit scholars who may be hostel residents could all be residents, or none might be; O alone does not settle this.
- I ('Some S are H') is undetermined. It may be true, but O is compatible with I being false if every merit scholar lives outside the hostel.
So the only definite results are A false and O true. E and I remain open.
2. Evaluate Anita's, Bikash's and Chandan's inferences. For each, name any fallacy of immediate inference involved and state whether it is valid.
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Anita: From the audit's O statement, she concludes E ('No merit scholar is a hostel resident'). O and E are not equivalent. 'Some S are not H' does not rule out 'Some S are H'. Her move treats a particular negative as a universal negative. Invalid.
Bikash: He converts O ('Some S are not H') into 'Some H are not S'. Simple conversion of an O proposition is not valid. Counterexample: let S = {a, b} and H = {a}. 'Some S are not H' is true because b is outside H, but 'Some H are not S' is false because the only H member, a, is in S. Invalid - illicit conversion of O.
Chandan: He infers I ('Some S are H') from O. O and I are subcontraries: they can both be true, but O alone does not force I. Counterexample: let S = {a, b} and H = {c}. 'Some S are not H' is true, but 'Some S are H' is false. Invalid.
All three responses go beyond what the audit licenses.
3. Suppose the original notice had been true. Which of these follows: (i) All hostel residents are merit scholars; (ii) Some hostel residents are merit scholars; (iii) No merit scholars are non-hostel residents; (iv) All non-hostel residents are non-merit scholars? Justify each.
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Assume All S are H is true and that S is not empty - there are 40 merit scholars.
(i) All H are S - does not follow. This is simple conversion of an A proposition, which is invalid. The set of merit scholars could be a proper subset of hostel residents.
(ii) Some H are S - follows by conversion by limitation. Since there are merit scholars and all of them are hostel residents, at least one hostel resident is a merit scholar. The existential assumption is met by the 40 scholars.
(iii) No S are non-H - follows by obversion. Obverting 'All S are H' yields 'No S are non-H'. This is a valid immediate inference.
(iv) All non-H are non-S - follows by contraposition. The contrapositive of 'All S are H' is 'All non-H are non-S'. This is valid for A propositions.
Thus (ii), (iii) and (iv) follow; (i) does not.