Even More Logic Practice!

In today's brief lab, we'll continue practicing the skill of applying the rules of natural deduction to write rigorous, low-level proofs.

Note: this "lab" is not due, but will be good practice as you prep for the exam on Friday.

Problem: Logic Practice

For each claim:

In a sentence, describe what the claim is saying. Your description should be in the rough form "under assumptions ... we must show that ... ." You can reference the atomic propositions in the claims directly in your description, i.e., you don't need to instantiate the proof state into a real-world context.
Now instantiate the claim using real-world statements like "the sun is shining" or "it is cold outside" or "pigs can fly". Try to pick statements that make sense in the context of the claim itself, what is being assumed, and what you are trying to prove.
Finally, give a rigorous, natural deduction-style proof of the claim.

Claim 1

$\neg P \lor Q, \neg Q \lor R ⊢ P \to R$ .

Claim 2 (Related to Proof by Contradiction - we'll see this next week!)

$\neg P \to \neg Q, Q ⊢ P$ .

Claim 3

$P \land Q, \neg P \lor R, \neg Q \lor S ⊢ R \land S$ .