Question

Which, if any, of the following proofs are correct demonstrations of the validity of this argument? A ⊃ (B ⊃ C) B ⊃ (~C ⊃ ~A) Proof 1 (1) A ⊃ (B ⊃ C) /B ⊃ (~C ⊃ ~A) Premise/Conclusion (2) (A • B) ⊃ C 1 Exp (3) (B • A) ⊃ C 2 Com (4) B ⊃ (A ⊃ C) 3 Exp (5) B ⊃ (~C ⊃ ~A) 4 Contra Proof 2 (1) A ⊃ (B ⊃ C) /B ⊃ (~C ⊃ ~A) Premise/Conclusion (2) B Assumption (3) A Assumption (4) B ⊃ C 1, 3 MP (5) C 2, 4 MP (6) A ⊃ C 3–5 CP (7) B ⊃ (A ⊃ C) 2–6 CP (8) B ⊃ (~C ⊃ ~A) 7 Contra

Answer 1

Explanation:

Answer:

See the argument below

Explanation:

I will give the argument in symbolic form, using rules of inference.

First, let's conclude c.

(1)⇒a by simplification of conjunction

a⇒¬(¬a) by double negation

¬(¬a)∧(2)⇒¬(¬c) by Modus tollens

¬(¬c)⇒c by double negation

Now, the premise (5) is equivalent to ¬d∧¬h which is one of De Morgan's laws. From simplification, we conclude ¬h. We also concluded c before, then by adjunction, we conclude c∧¬h.

An alternative approach to De Morgan's law is the following:

By contradiction proof, assume h is true.

h⇒d∨h by addition

(5)∧(d∨h)⇒¬(d∨h)∧(d∨h), a contradiction. Hence we conclude ¬h.