Analyze the sentences below in as much detail as possible using only connectives; that is, the unanalyzed components should all be sentences (rather than individual terms, predicates, or functors). Present the result in both symbolic and English notation. Be sure that the unanalyzed components of your answer are complete and independent sentences; also try to respect any grouping in the English.

If John was invited, then he attended if he was free.

Unless we find the key, we’ll get in only if we break the lock.

Use derivations to check whether each of the entailments below holds. You may use detachment and attachment rules. If an entailment fails, present a counterexample that divides an open gap.

B → C ⇒ (A ∧ B) → C

¬ (C → D) → (A → B) ⇒ A → D

Analyze the sentence below in as much detail as possible, giving a key to your abbreviations of unanalyzed expressions. In this case you should identify components that are individual terms, predicates, or functors; however, you do not need to present the result in English notation (i.e., symbolic notation is enough). Your analysis should be in reduced form (i.e., you should not use abstracts and variables), so be sure that the unanalyzed components of your answer are independent—in particular, that none contains a pronoun whose antecedent is in another. (Also be sure also that the individual terms you identify really are individual terms and are not quantifier phrases or general terms, like simple common nouns.)

Sam wrote to Linda, and she sent his book to him.

Analyze the sentence below using abstracts and variables to represent pronominal cross reference (instead of replacing pronouns by their antecedents). That is, use expanded form to the extent necessary so that each individual term in your analysis appears only as often as it appears in the original sentence. In other respects, your analysis should be as described for 5.

The rock hit the road, but it didn’t hit Oscar.

Use a derivation to show that the entailment below holds. You may use detachment and attachment rules. Be sure to indicate the alias sets whenever an equation is added to the resources.

Ra(fb), fa = gb ⇒ a = b → (Rb(ga) ∧ fb = gb)

If John was invited, then he attended if he was free

John was invited → John attended if he was free

John was invited → (John attended ← John was free)

Unless we find the key, we’ll get in only if we break the lock

¬ we will find the key → we’ll get in only if we break the lock

¬ we will find the key → (¬ we’ll get in ← ¬ we’ll break the lock)

The first two interpretations divide the first dead end gap, and the last two divide the second. It is enough to reach one of the two dead ends and to present one of the two counterexamples that divide that gap.

Sam wrote to Linda, and she sent his book to him

Sam wrote to Linda ∧ Linda sent Sam’s book to him

Sam wrote to Linda ∧ Linda sent Sam’s book to Sam

[ _ wrote to _ ] Sam Linda ∧ [ _ sent _ to _ ] Linda Sam’s book Sam

Wsl ∧ Sl([ _’s book] Sam)s

The rock hit the road, but it didn’t hit Oscar

The rock is such that (it hit the road, but it didn’t hit Oscar)

[x hit the road, but x didn’t hit Oscar]_x the rock

[x hit the road ∧ x didn’t hit Oscar]_x the rock

[x hit the road ∧ ¬ x hit Oscar]_x the rock