Annual Day Rule and Truth-Functional Entailment
The annual day committee of a college has fixed one rule: 'If the electricity board grants a load sanction and the grounds committee clears the drainage work, then the annual day will be held on the main lawn.' On the morning of the event, the principal announces that the annual day will not be held on the main lawn because of rain. Three students in the logic club draw different conclusions. Prakash says, 'So the electricity board did not grant the load sanction and the grounds committee did not clear the drainage work.' Meera says, 'So either the electricity board did not grant the load sanction or the grounds committee did not clear the drainage work.' Rina says, 'So the electricity board did not grant the load sanction.' The club has to advise the principal's office which conclusion is logically forced by the rule and the announcement. They have a truth table with variables L, D and M, and 10 minutes before the notice must be printed.
1. Symbolize the rule and the principal's announcement; symbolize each student's conclusion. Which conclusion is truth-functionally entailed?
Read the worked answer
Let:
- L = the electricity board grants the load sanction.
- D = the grounds committee clears the drainage work.
- M = the annual day is held on the main lawn.
The rule is: (L and D) implies M, written (L ∧ D) → M. The announcement is: not M, written ¬M.
By modus tollens, ¬M and (L ∧ D) → M give ¬(L ∧ D). By De Morgan's law, ¬(L ∧ D) is equivalent to ¬L ∨ ¬D.
- Prakash concludes ¬L ∧ ¬D - both failed.
- Meera concludes ¬L ∨ ¬D - at least one failed.
- Rina concludes ¬L - the electricity board alone failed.
Only Meera's conclusion is truth-functionally entailed. The rule and the announcement force that at least one of the two conditions was not met; they do not force which one, and they do not force both.
2. Use a truth-table row or short truth assignment to show that Prakash's and Rina's conclusions do not follow. State the row explicitly.
Read the worked answer
Choose the assignment L = true, D = false, M = false.
Check the premises:
- L ∧ D = true ∧ false = false.
- (L ∧ D) → M = false → false = true.
- ¬M = true.
So both premises are true in this row.
Now check the conclusions:
- Meera: ¬L ∨ ¬D = false ∨ true = true.
- Prakash: ¬L ∧ ¬D = false ∧ true = false.
- Rina: ¬L = false.
This single row makes the premises true while Prakash's and Rina's conclusions are false. Therefore neither conclusion is entailed. Meera's disjunction remains true in every row in which the premises are true.
3. The principal's office wants a single sentence for the notice that is logically forced and does not overstate. Draft it and explain why it is safe.
Read the worked answer
A safe sentence is:
'Either the electricity board did not grant the load sanction or the grounds committee did not clear the drainage work, or both.'
This is exactly ¬L ∨ ¬D, the De Morgan equivalent of the forced conclusion ¬(L ∧ D). It does not say that both failed, so it avoids Prakash's stronger claim. It does not name the electricity board alone, so it avoids Rina's claim. The inclusive 'or both' makes clear that the sentence allows either one failure or two.
The rain announcement explains why the event was not held on the main lawn, but the formal entailment comes from the committee rule and ¬M. If the rain statement were formalized as an extra premise, it would not change the conclusion already forced by the rule and the announcement.