The Reference Section Notice Board
The librarian of a PU college in Udupi posts four statements on the reference section notice board and runs a one-question contest: A: All reference books are borrowable books. E: No reference books are borrowable books. I: Some reference books are borrowable books. O: Some reference books are not borrowable books. Two facts are released. First, the overnight-issue register shows that at least one reference title was lent to a student last term. Second, a shelf check finds 14 reference titles, of which 3 have never been issued to anyone since 2019. A student argues: 'Since 3 titles were never issued, statement O is true; therefore A is false, and since I is true from the register, E must be false too. So A and E are both false and O is true.' The librarian, who taught the traditional square of opposition with existential import, must decide which of the student's truth-value claims are correct, which inferences are invalid, and what would change under the modern Boolean reading.
1. Assess each step of the student's argument and give the final truth value of A, E, I and O on the traditional (Aristotelian) reading.
Read the worked answer
Step 1 - 'Since 3 titles were never issued, O is true.' Not established. O says that some reference books are not borrowable, a fact about the lending rule. 'Never issued since 2019' is a fact about past borrowing. A book may be fully borrowable yet never requested; the inference confuses a record of use with a property of the book. O stays undetermined on the evidence given.
Step 2 - 'I is true, so E is false.' Correct. I and E are contradictories: they cannot share a truth value. I is true from the overnight-issue register, so E is false.
Step 3 - 'So A is false.' Only conditionally. A is the contradictory of O, so A is false if O is true; since O has not been shown, A is not settled by that route. A is also the contrary of E, but from E being false nothing follows about A - contraries may both be false. Nor can we argue upward from I to A: subalternation carries truth downward only (A → I), never upward.
Final on the evidence given: I = true, E = false, A = undetermined, O = undetermined. If the shelf check had instead found a volume stamped 'not for issue', O would be true, A false, and the square would be fully determined: A false, E false, I true, O true - a legitimate case in which both contraries fail and both subcontraries hold.
2. Starting from statement A ('All reference books are borrowable books'), give the converse, the obverse and the contraposition, marking each as valid or invalid and stating any condition.
Read the worked answer
Let S = reference books, P = borrowable books; A is All S is P.
- Converse (swap the terms): All borrowable books are reference books - invalid as simple conversion. A converts only per accidens, to Some borrowable books are reference books (I), and that too only if S is not empty.
- Obverse (change the quality and negate the predicate): No reference book is a non-borrowable book - valid. Obversion is an equivalence for all four proposition types.
- Contraposition (obvert the converse of the obverse): All non-borrowable books are non-reference books - valid. Contraposition holds for A and for O propositions.
For comparison: E converts simply, so No borrowable book is a reference book follows validly from No reference book is borrowable; O does not convert at all, which is why the student cannot turn O into a statement about borrowable books. The pattern to hold on to: E and I convert simply, A converts only by limitation, O does not convert; obversion is always safe; contraposition works for A and O.
3. What changes if the librarian adopts the modern Boolean reading? Would your verdict about I and E change in this library?
Read the worked answer
Under the Boolean reading, universal propositions carry no existential import: All S is P is read as there is nothing that is S and not P, which is true when no S exists. The consequences are:
- Subalternation fails: A no longer implies I, and E no longer implies O, because the lower proposition asserts existence while the upper one does not.
- Contraries may both be true: with an empty S, both 'All S is P' and 'No S is P' hold.
- Subcontraries may both be false, and conversion per accidens is withdrawn.
In this library, however, the subject class is not empty: the shelf check counts 14 reference titles. Because existence is independently guaranteed, the traditional inferences used here remain safe. The Boolean difference bites only where the subject class may be empty - for example, a notice about 'books on order' before any of them arrive. So the practical verdict (I true, E false) stands unchanged, but the justification must now cite the 14 titles on the shelf rather than relying on the square of opposition alone.