Table of Contents

Lab 4 - Propositional Logic / Inference

Problem Description

(Based on “The Adventure of Silver Blaze,” an original Sherlock Holmes mystery by Arthur Conan Doyle)

A prize-winning racehorse named Silver Blaze has been stolen from a stable, and a bookmaker named Fitzroy Simpson has been arrested as the prime suspect by good old Inspector Gregory. Sherlock Holmes, however, after ample use of his magnifying glass and some of the strongest black tobacco this side of the Atlantic, finds the true thief by reasoning from the following premises:

Who stole Silver Blaze?

Tasks

  1. Encode all the given information as a knowledge base in propositional logic.
  2. Write down which propositional symbols you used and which facts in the environment they represent.
  3. Use the inference rules and equivalences below to infer who is the thief. For each inference step note which sentences and which inference rule / equivalence you used!

Hints

Inference Rules and Equivalences

  1. <latex>$\{ \alpha \Rightarrow \beta, \alpha \} \: \vdash \: \beta $</latex>
  2. <latex>$\{ \alpha \Rightarrow \beta, \neg \beta \} \: \vdash \: \neg \alpha $</latex>
  3. <latex>$\{ \alpha \land \beta \} \: \vdash \: \alpha $</latex>
  4. <latex>$\{ \alpha , \beta \} \: \vdash \: \alpha \land \beta $</latex>
  5. <latex>$\{ \alpha \} \: \vdash \: \alpha \lor \beta $</latex>
  6. <latex>$\{ \alpha \lor \beta, \neg \alpha \} \: \vdash \: \beta $</latex>
  7. <latex>$ \alpha \Leftrightarrow \beta \: \equiv \: \beta \Leftrightarrow \alpha $</latex>
  8. <latex>$ \alpha \Leftrightarrow \beta \: \equiv \: (\alpha \Rightarrow \beta) \land \beta \Rightarrow \alpha $</latex>
  9. <latex>$ \alpha \Rightarrow \beta \: \equiv \: \neg \alpha \lor \beta $</latex>
  10. <latex>$ \alpha \land \beta \: \equiv \: \beta \land \alpha $</latex>
  11. <latex>$ \alpha \lor \beta \: \equiv \: \beta \lor \alpha $</latex>
  12. <latex>$ \neg (\alpha \land \beta) \: \equiv \: \neg \alpha \lor \neg \beta $</latex>
  13. <latex>$ \neg (\alpha \lor \beta) \: \equiv \: \neg \alpha \land \neg \beta $</latex>
  14. <latex>$ \neg \neg \alpha \: \equiv \: \alpha $</latex>