Fact 1. If $\Gamma \vdash_{HK} A$, then $\Gamma \cup \Delta \vdash_{HK} B \supset A$.
2. Prove Fact 1 analytically, i.e., using the strategy that we used to prove the facts on p. 2 of the handout "week7a".
Important. Write your proofs clearly; state what your assumptions are and justify the various steps in your proofs.
p.2 of handout
â–· Fact 2. $\{\neg A, A\} \vdash_{HK} B$.
Proof.
1. $\vdash_{HK} A \supset (\neg B \supset \neg A)$
2. $\{\neg A\} \vdash_{HK} \neg B \supset \neg A$
3. $\{\neg A\} \vdash_{HK} (\neg B \supset \neg A) \supset (A \supset B)$
4. $\{\neg A\} \vdash_{HK} A \supset B$
5. $\{\neg A, A\} \vdash_{HK} B$
A1 is a theorem in HK
Deduction Theorem
A3 is a theorem in HK
Metatheoretic Modus Ponens 2,3
Deduction Theorem
â–· Fact 3. If $\Gamma \cup \{\neg A\} \vdash_{HK} \neg B$, then $\Gamma \cup \{B\} \vdash_{HK} A$.
Proof.
1. $\Gamma \cup \{\neg A\} \vdash_{HK} \neg B$
2. $\Gamma \vdash \neg A \supset \neg B$
3. $\Gamma \vdash_{HK} (\neg A \supset \neg B) \supset (B \supset A)$
4. $\Gamma \vdash B \supset A$
5. $\Gamma \cup \{B\} \vdash_{HK} A$
hyp
Deduction Theorem
A3 is a theorem in HK
Metatheoretic Modus Ponens 2,3
Deduction Theorem
