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Rules of inference: cheque book argument

Short answer

p→q and ¬q give ¬p (modus tollens). s→p with ¬p gives ¬s (modus tollens). r∨s and ¬s yield r (disjunctive syllogism). r→t and r give t (modus ponens). So the cheque book is on the breakfast table.

Use rules of inference to show that the hypothesis "If my cheque book is in office, then I have paid my telephone bill", "I was looking for phone bill at breakfast or I was looking for phone bill in my office", "If I was looking for phone bill at breakfast then my cheque book is on breakfast table", "If I was looking for phone bill in my office then my cheque book is in my office", "I have not paid my phone bill" imply the conclusion "My cheque book is on my breakfast table."

Propositions

  • p: my cheque book is in the office
  • q: I have paid my telephone bill
  • r: I was looking for the phone bill at breakfast
  • s: I was looking for the phone bill in my office
  • t: my cheque book is on the breakfast table

Hypotheses and conclusion

  1. p → q
  2. r ∨ s
  3. r → t
  4. s → p
  5. ¬q

Conclusion: t.

Proof

StepStatementReason
1p → qHypothesis
2¬qHypothesis
3¬pModus tollens from 1, 2
4s → pHypothesis
5¬sModus tollens from 4, 3
6r ∨ sHypothesis
7rDisjunctive syllogism from 6, 5
8r → tHypothesis
9tModus ponens from 8, 7

So the cheque book is on the breakfast table. ∎

I also checked all 32 truth assignments by computer: every assignment that makes the five hypotheses true makes t true, so the argument is valid.

✓ Answer checked by Claude Opus 5.5 (high effort)

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