direct proofs in natural deduction

Choose two of the arguments below and write a direct proof using the eight rules of inference introduced in section 8.1 of the textbook. You can do argument 1 or argument 2, but not both, then any of arguments 3â€“6. Note that commas are used to separate the premises from each other.

1. ~M, (~M â€¢ ~N) â†’ (Q â†’ P), P â†’ R, ~N, therefore, Q â†’ R

2. ~F â†’ ~G, P â†’ ~Q, ~F v P, (~G v ~Q) â†’ (L â€¢ M), therefore, L

3. ~(Z v Y) â†’ ~W, ~U â†’ ~(Z v Y), (~U â†’ ~W) â†’ (T â†’ S), S â†’ (R v P), [T â†’ (RvP)] â†’ [(~R v K) â€¢ ~K], therefore, ~K

4. (S v U) â€¢ ~U, S â†’ [T â€¢ (F v G)], [T v (J â€¢ P)] â†’ (~B â€¢ E), therefore, S â€¢ ~B

5. ~X â†’ (~Y â†’ ~Z), X v (W â†’ U), ~Y v W, ~X â€¢ T, (~Z v U) â†’ ~S, therefore, (R v ~S) â€¢ T

6. (C â†’ Q) â€¢ (~L â†’ ~R), (S â†’ C) â€¢ (~N â†’ ~L), ~Q â€¢ J, ~Q â†’ (S v ~N), therefore, ~R

Natural deduction is so called because it is a model for how we naturally reason. This often comes as a surprise to students because all of the symbols seem anything but natural. The symbols, however, allow us to focus on the form of the argument without getting bogged down by content. Recall that each sentence letter represents a simple sentence in English.

After writing your direct proofs, construct a translation key for your argument by assigning each letter a simple sentence, and use that key to fill in the content of the argument.