(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:

• The horse was stolen either by Fitzroy or by its trainer John Straker.
• The thief had to have entered the stable the night of the theft.
• If a stranger enters the stable, the dog barks.
• Fitzroy was a stranger.
• The dog did not bark.

Who stole Silver Blaze?

Tasks:

1. Encode all the given information as a knowledge base <latex>$KB$</latex> 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!

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>