- 2070 Magh · 8 marks
Rules of inference: Randy and the job
Short answer
p: Randy works hard, q: dull boy, r: gets job. From p and p→q, q by modus ponens. q→¬r and q give ¬r by modus ponens. Therefore Randy will not get the job.
Construct an argument using rules of inference to show that the hypotheses "Randy works hard," "If Randy works hard, then he is a dull boy," and "If Randy is a dull boy, then he will not get the job" imply the conclusion "Randy will not get the job."
Propositions
- p: Randy works hard
- q: Randy is a dull boy
- r: Randy will get the job
Hypotheses and conclusion
- p
- p → q
- q → ¬r
Conclusion: ¬r.
Proof
| Step | Statement | Reason |
|---|---|---|
| 1 | p | Hypothesis |
| 2 | p → q | Hypothesis |
| 3 | q | Modus ponens from 1, 2 |
| 4 | q → ¬r | Hypothesis |
| 5 | ¬r | Modus ponens from 3, 4 |
So Randy will not get the job. ∎
(Alternatively: p → q and q → ¬r give p → ¬r by hypothetical syllogism, and then modus ponens with p gives ¬r.)
✓ Answer checked by Claude Opus 5.5 (high effort)
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