Represent the propositional structure of each of the following sentences. First identify the atomic component sentences (i.e. sentences that do not contain connectives) and abbreviate each with a distinct capital letter. We have suggested letters after the sentences. Then represent the form of the original sentence using the symbols ∨,∧,¬,→ for the connectives “or”, “and”, “not”, “if…then…”. Make sure to include parentheses, if necessary to disambiguate.

It’s not true that if Ron doesn’t do his homework then Hermione will finish it for him. (R,H)

Harry will be singed unless he evades the dragon’s fiery breath. (S,E)

Aristotle was neither a great philosopher nor a great scientist. (P,S)

Mark will get an A in logic only if he does the homework or bribes the professor. (A,H,B)

Dumbledore will be killed, and either McGonagall will become head of school and Hogwarts will flourish, or else it won’t flourish. (D,M,H)

Harry and Dumbledore are not both right about the moral status of Professor Snape. (H,D)

Prove that the following argument forms are valid. The premises are to the left of the ⊢ symbol, the conclusion is to the right. You should number the lines of your proof, and each line must either be a premise (i.e. an assumption) or be justified by one of the following rules of inference: ∧I, ∧E, ∨I, MP, MT, or DN.

*P*→ (*Q*→*R*),*P*→*Q*,*P*⊢*R**P*⊢ (*P*∨*R*) ∧ (*P*∨*Q*)*P*⊢*Q*∨ (¬¬*P*∨*R*)¬¬

*Q*→*P*, ¬*P*⊢ ¬*Q**Q*→ (*P*→*R*), ¬*R*∧*Q*⊢ ¬*P*

Explain what’s wrong with the following “proof”.

`(1) P∨(Q∧R) A (2) P∨Q 1 ∧E`

If you were allowed to make up logic rules, do you think the following would be a good rule?

Any time a conjunction

*A*∧*B*occurs on a line, even as a part of a larger sentence, you may rewrite that line with only*A*in place of the conjunction*A*∧*B*, or with only*B*in place of*A*∧*B*.Explain your answer in a paragraph (no more than half a page).