Analyze each sentence below in as much detail as possible, presenting the result in both symbols and English notation (i.e., both using symbolic notion, such as ‘∧’, and using the notation, such as ‘both … and’, that employs English words). Provide a key to your abbreviations of unanalyzed components, and be sure that they are complete and independent sentences. Try to respect any grouping in the English.

Al presented his suggestion but didn’t get both the management and the union to agree to it.

Either Bob didn’t make the call or neither Carol nor Dave was home

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

(E ∧ ¬ F) ∨ (¬ E ∧ F)
(E: Dave contacted Ed; F: Dave contacted Fred)

Use derivations to check whether each of the claims of entailment 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. (Your table should indicate the value of any compound component by writing this value under the main connective of the component, and you should be sure to indicate the final value of each premise and the conclusion.)

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.

You may use attachment and detachment rules in 7. While those rules can be used to shorten the derivation somewhat, it can also be completed without using them.

Al presented his suggestion but didn’t get both the management and the union to agree to it

Al presented his suggestion ∧ Al didn’t get both the management and the union to agree to his suggestion

Al presented his suggestion ∧ ¬ Al got both the management and the union to agree to his suggestion

Al presented his suggestion ∧ ¬ (Al got the management to agree to his suggestion ∧ Al got the union to agree to his suggestion)

Either Bob didn’t make the call or neither Carol nor Dave was home

Bob didn’t make the call ∨ neither Carol nor Dave was home

¬ Bob made the call ∨ ¬ either Carol or Dave was home

¬ Bob made the call ∨ ¬ (Carol was home ∨ Dave was home)

[¬ C ∧ ¬ D is also correct as an analysis of the second disjunct; and ¬ (B ∧ (C ∨ D)) is technically a correct analysis of the whole sentence, though it is much farther from the English.]

(E ∧ ¬ F) ∨ (¬ E ∧ F) (E: Dave contacted Ed; F: Dave contacted Fred)

(Dave contacted Ed ∧ ¬ Dave contacted Fred) ∨ (¬ Dave contacted Ed ∧ Dave contacted Fred)

(Dave contacted Ed ∧ Dave didn’t contact Fred) ∨ (Dave didn’t contact Ed ∧ Dave contacted Fred)

(Dave contacted Ed but not Fred) ∨ (Dave didn’t contact Ed but did contact Fred)

Either Dave contacted Ed but not Fred, or he didn’t contact Ed but did contact Fred