1. In each of the following, show that the schema is valid by carrying out a deduction using our ten-rule TFDS+:
- (-(p.q).(r ⊃ p)).(r ⊃ q) ⊃ -r
- -(-(p⊃q).-(q⊃p))
- -p.-q ⊃ (p⊃q).(q⊃p)
2.Show each of the following by carrying out a deduction using our ten-rule TFDS+:
- “(-p ⊃ q).(-p ⊃ -q)”implies “p”
- The three schemata “p.q ⊃ r”, “r⊃s”, and “q.-s” jointly imply “-p”
- The four schemata “(p⊃q) ⊃ r”, “s ⊃ ‒p”, “t”, and “-s.t ⊃ q” jointly imply “r”
- By carrying out two deductions using our ten-rule TFDS+,show that“p ⊃ q.r”is equivalent to “(p⊃q).(p⊃r)”.
- In each of the following, show that the schema is valid by carrying out a deduction using TFDS++:
(You may assume that if (i) X implies Y and (ii) Y implies X, then X is equivalent to Y.)
- (p⊃q)∨(p⊃-q)
- (p.(p⊃r) ∨ q.(q⊃r)) ⊃r
5. Show each of the following by carrying out a deduction using TFDS++:
- The two schemata“-p” and “p∨q” jointly imply “q”
- The three schemata “(p⊃q) ∨ (t⊃s)”, “t.-q”, and “r ⊃ (t ⊃ -s.p)” jointly imply “–r”
6.By carrying out two deductions using TFDS++, show that“p.(q∨r)”is equivalent to “p.q ∨ p.r”.(You may assume that if (i) X implies Y and (ii) Y implies X, then X is equivalent to Y.)
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