Use derivations to check each of the claims below; if a derivation indicates that a claim fails, present a counterexample that divides an open gap. Since d is a claim of tautologousness, it is established by a derivation that begins with only a goal and no initial premises.

Construct derivations for each of the following. These exercises are designed to make attachment rules often useful. The derivations can be constructed for the English sentences in e-g without first analyzing them since you generally need to recognize only the main connective and the immediate connectives in order to know what rules apply; however, the abbreviated notation provided by an analysis may be more convenient.

Tom will go through Chicago and visit Sue

Tom won’t go through both Chicago and Indianapolis

Tom won’t visit Ursula without going through Indianapolis

Tom will visit Sue but not Ursula

Either we spend a bundle on television or we won’t have wide public exposure

If we spend a bundle on television, we’ll go into debt

Either we have wide public exposure or our contributions will dry up

We’ll go into debt if our contributions dry up and we don’t have large reserves

We won’t have large reserves

We’ll go into debt

If Adams supports the plan, it will go though provided Brown doesn’t oppose it

Brown won’t oppose the plan if either Collins or Davis supports it

The plan will go through if both Adams and Davis support it