Natural Deduction in Propositional Logic Help?

Question

Use the first four rules of inference to derive the conclusions of the following symbolized arguments. (please please help. And explanation would be nice too!)

(1) 1. ~C>(A>C)
2. ~C /~A



(2.) 1. F v (D>T)
2.~F
3. D /T

(3.) 1. (K & B) v (L>E)
2. ~(K&B)
3. ~E /~L


(4.) 1. J > (K>L)
2. L v J
3. ~L /~K

Answer 1

The symbols are a little difficult, so I'm putting English here

2.

a) F or (D implies T)  Given

b) not F  Given

c) D implies T (Disjunctive Syllogism)

d) D

e) therefore T (Modus Ponens)

 

Answer 2

This a good question. Natural deduction can be difficult and takes practice. In this way it is much like algebra, but where you already know the answer and you are trying to figure out the steps to get from the original problem to that answer.

 

The solution steps are in bold. The acronyms representing the rule of inference used and related steps are in the right column.

 

1. ~C>(A>C)
2. ~C            /~A

3. A ⊃ C        1,2 MP (If ~C then, from line 1, we know that (A>C) is true)

4. ~A            2,3 MT (If ~C then A cannot be true)



1. F v (D>T)
2. ~F
3. D                   /T

4. D ⊃ T          1,2 DS (Disjunctive syllogism)

5. T                3,4 MP (modus ponens)

1. (K & B) v (L>E)
2. ~(K&B)
3. ~E                       /~L

4. L ⊃ E                   1,2 DS (same as previous problem)

5. ~L                       3,4 MT (modus tolens)

1. J > (K>L)
2. L v J
3. ~L                  /~K

4. J                    2,3 DS

5. K ⊃ L             1,4 MP

6. ~K                 3,4 MT