Analyze each sentence below in as much detail as possible, presenting the result in both symbols and English notation (using both ... and, etc., as well as ∧, etc.). Be sure that the unanalyzed components of your answer are complete and independent sentences; also try to respect any grouping in the English.

Sam was cool, but he was not both calm and collected.

Tom spoke to either Al or Barb but to neither Carol nor Dave.

Synthesize an English sentence (the more idiomatic the better) that has the following analysis:

¬ E ∨ F (E: Ed worked last weekend; F: Fred worked last weekend)

Use derivations to check whether each of the entailments below holds. If one fails, present a counterexample by providing a table in which you calculate the truth values of the premises and conclusion on an extensional interpretation (i.e., an assignment of truth values) that divides an open gap.

Do not use attachment or detachment rules in 4-6. That is, do not use Adj or the rules MTP, MPT, and Wk of §4.3; instead use only the basic rules for exploiting resources, planning for goals, and closing gaps.

In 7 you may use attachment and detachment rules (and their use can simplify the derivation).

Sam was cool, but he was not both calm and collected

Sam was cool ∧ Sam was not both calm and collected

Sam was cool ∧ ¬ Sam was both calm and collected

Sam was cool ∧ ¬ (Sam was calm ∧ Sam was collected)

Tom spoke to either Al or Barb but to neither Carol nor Dave

Tom spoke to either Al or Barb ∧ Tom spoke to neither Carol nor Dave

(Tom spoke to Al ∨ Tom spoke to Barb) ∧ ¬ Tom spoke to either Carol or Dave

(Tom spoke to Al ∨ Tom spoke to Barb) ∧ ¬ (Tom spoke to Carol ∨ Tom spoke to Dave)

¬ E ∨ F (E: Ed worked last weekend; F: Fred worked last weekend)

¬ Ed worked last weekend ∨ Fred worked last weekend

Ed didn’t work last weekend ∨ Fred worked last weekend

Either Ed didn’t work last weekend or Fred did