Answer as many questions as you can.

1. Circle all the free variable occurrences in the following formula:
   $$\forall z. z \supset x \supset (\forall x. Px \land \exists y. Qxy) \lor Rzx$$
2. Perform the substitution $(\forall f. \exists y. P\,(f\,x\,y))[x \mapsto f\,y]$.
3. Write in logical notation the judgement: $0$ is the best number assuming that numbers exist and also assuming that the best number is unique. (Please note there are two assumptions here, not one assumption that is a conjunction of two claims.)
4. In a many-valued logic, does the sentence “This sentence is false” have the truth value of true, false, both, or neither? (There is a best answer here that I am looking for, despite the fact that philosophers may disagree about it.)
5. What does it mean for a logical system to be sound, in terms of $\vdash$ and $\vDash$?
6. Write (in the logic notation used in this class) an encoding of “No vampire will ever be able to see their own reflection in a mirror”.
7. Write (in the logic notation used in this class) an encoding of “Every robot that once served a human is now obligated to obey the one who repaired it.”
8. Write (in the logic notation used in this class) an encoding of “Some prime number is greater than every number Tanya knows.”