Construct L1 symbolisations for each of the following sentences.
S1. The mountain moon sets too soon.
S2. Lefty can't sing the blues.
S3. Fido is not hungry.
S4. The game is cancelled only if the field is wet.
S5. If Olivia is a logician, then Olivia doesn’t love contradictions.
S6. If Rudolf leaves the party, then Jen will be upset if she isn’t already dancing.
S7. If Frank passes only if Olivia doesn't study, then she graduates only if he doesn't.
Using ∃LOGIC construct annotated derivations showing that the following arguments are valid.
1:1 P ∴ ¬¬P
1:2 P. (P→Q) ∴ Q