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, rewriting if necessary to make all symbolic conditionals point from left to right. Be sure that the unanalyzed components of your answer are complete and independent sentences; also try to respect any grouping in the English.

They won’t get home early if they go through the city during rush hour.

Unless it was raining, the picnic was postponed only if it was unusually cold.

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. (Your truth-table for a counterexample should show the truth-value of each compound component of sentence under the main connective of that component, and it should indicate the final truth-value of each sentence.)

B → (A → C) ⊨ (A ∧ B) → C

A → ¬ C, C → B ⊨ ¬ B → A

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.)

If Al went through Phoenix, he stayed with Barb’s family.

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.

Ann spoke to Bill, and he introduced himself to Carol.

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.

Ga → a = c, Rb(fa) ⊨ b = c → (Ga → Ra(fb))

they won’t get home early if they go through the city during rush hour

they won’t get home early ← they will go through the city during rush hour

¬ they will get home early ← they will go through the city during rush hour

unless it was raining, the picnic was postponed only if it was unusually cold

¬ it was raining → the picnic was postponed only if it was unusually cold

¬ it was raining → (¬ the picnic was postponed ← ¬ it was unusually cold)

if Al went through Phoenix, he stayed with Barb’s family

Al went through Phoenix → Al stayed with Barb’s family

[ _ went through _ ] Al Phoenix → [ _  stayed with _ ] Al (Barb’s family)

Wap → Sa([ _’s family] Barb)

Ann spoke to Bill, and he introduced himself to Carol

Bill is such that (Ann spoke to him, and he introduced himself to Carol)

[Ann spoke to x, and x introduced x to Carol]_x Bill

[Ann spoke to x ∧ x introduced x to Carol]_xb

[[ _ spoke to _ ] Ann x ∧ [ _ introduced _ to _ ] x x Carol]_xb