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 the package was sent, then it was lost.

Al finished the project only if he had help; but he started it unless there was a rush order.

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.

A → C, B → ¬ C ⊨ A → ¬ B

B → (A → C) ⊨ (B ∧ C) → 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.)

Al sold his car to the first caller, and he bought Dave’s truck.

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.

If Bill went to Chicago, then Ann didn’t reach him.

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.

a = fb, fc = d ⊨ (b = c ∧ ¬ Fc) → (a = d ∧ ¬ Fb)

If the package was sent, then it was lost.

the package was sent → the package was lost

Al finished the project only if he had help; but he started it unless there was a rush order

Al finished the project only if he had help 
∧ Al started the project unless there was a rush order

(¬ Al finished the project ← ¬ Al had help) 
∧ (Al started the project ← ¬ there was a rush order)

The first interpretation divides the first dead end gap, and both divide the second. It is enough to reach one of the two dead ends and to present one counterexample that divides that gap.

Al sold his car to the first caller, and he bought Dave’s truck

Al sold his car to the first caller ∧ Al bought Dave’s truck

[ _ sold _ to _ ] Al Al’s car the first caller ∧ [ _ bought _ ] Al Dave’s truck

S a (Al’s car) f ∧ B a (Dave’s truck)

S a ([ _’s car] Al) f ∧ B a ([ _’s truck] Dave)

If Bill went to Chicago, then Ann didn’t reach him

Bill is such that (if he went to Chicago, then Ann didn’t reach him)

[if x went to Chicago, then Ann didn’t reach x]_x Bill

[x went to Chicago → Ann didn’t reach x]_x b

[x went to Chicago → ¬ Ann reached x]_x b

[ [ _ went to _ ] x Chicago → ¬ [ _ reached _ ] Ann x]_x b

It is also possible to close the second gap at stage 5 using QED=.