Changes
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1. The schemata schema is not valid if we can find just one assignment of truth-values for which it is false.(a) p = T, q = F, r = T makes the schemata schema false. Not valid.(b) The only way to make this false is for p ⊃ r to be false and p ⊃ q.r to be true. This fixes p to T and r to F. However fixing r to F makes p ⊃ q.r false as well (regardless of the value of q), so it is impossible to make the antecedent true and consequent false here. Thus schemata schema is valid.(c) Making the schemata schema false requires each all of the disjuncts being to be false. To make first one false, fix p to T and q to F (or the opposite, makes no difference). To make second one false, r must be made F since p is already fixed. Thus r and q have the same truth-value and the last disjunct evaluates to true. Again, impossible to make an assignment of truth-values giving us a false statement, so schemata schema is valid.
2. (i) An assignment of p = q = r = F gives a value of false to the entire schemata, so it is satisfiable but not valid. (ii) An assignment of p = T, q = r = F gives a value of false to the entire schemata so it is also not valid. Thus we cannot easily say if one implies the other based on validity.
(i) does not imply (ii) as p = T, q = r = F is false for (ii) but true for (i). (ii) does not imply (i) as p = q = r = F is false for (i) but true for (ii).
3. Neither the President (a) No. p ⊃ q is false if p = T and q = F, but (p⊃r)⊃(q⊃r) is happy nor true with those assignments.(b) No. p ⊃ q is Congress placated; The President false if p = T and q = F, but (r⊃p)⊃(r⊃q) is not happy or Congress true with those assignments (for example with r = F).(c) Yes. q = T, r = F, p = T makes the second schema false, and is placatedthe only assignment that does. It also makes the first schema false, so the first implies the second.
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