# Proof by natural deduction advice

Any advice on shortening my right to left proof will be appreciated.

• You can use disjunction elimination to derive A v B. Get A -> (A v B) and (B & C) -> (A v B) for that. Then derive A v C in a similar way. The proof should be a lot shorter. – Eliran Oct 13 '19 at 3:49
• This is math problem. And shouldn't be in this forum. – Joseph Lutz Oct 14 '19 at 19:02
• I am in a symbolic logic philosophy class and this is homework from that class. – Sanjeev Oct 21 '19 at 14:41
• This topic is in an overlap of mathematics and philosophy, so it is quite reasonable that it is tagged here. On the other hand, asking at mathematics.stackexchange.com will have the advantage of the mathjax environment being available to typeset math questions and answers. – Graham Kemp Oct 22 '19 at 2:49
• Have you had any success in applying the advice? – Graham Kemp Oct 29 '19 at 23:17

Your proof is valid but rather difficult to follow. It is quite disorganised.

It would be clearer to collate the contexts together so that the reader does not have to track back and forth.

(I also prefer using just one disjunction elimination, although using two is not wrong.)

| _ 1 ( 1) A v (B & C)         P
||  2 ( 2) A                   H
||  2 ( 3) A v B              vI 2
||  2 ( 4) A v C              vI 2
||_ 2 ( 5) (A v B) & (A v C)  &I 3,4
||  3 ( 6) B & C               H
||  3 ( 7) B                  &E 6
||  3 ( 8) A v B              vI 7
||  3 ( 9) C                  &E 6
||  3 (10) A v C              vI 9
||_ 3 (11) (A v B) & (A v C)  &I 8,10
|   1 (12) (A v B) & (A v C)  vE 1,2..5,6..11


There is no need to use indirect proofs for the converse. Nesting proofs by cases will suffice, since the premise simplifies into two disjunctions wit a common disjunct.

Here is a skeleton for you to fill out. It should be clear what three hypothecia are needed, and how to derive the goal from them.

|   1   ( 1) (A v B) & (A v C)   P
|   1   ( 2) A v B              &E 1
| _ 1   ( 3) A v C              &E 1
||  2   ( 4)                     H
||_ 2   ( 5)
|| _3   ( 6)                     H
||| 4   ( 7)                     H
||| 3,4 ( 8)
|||_3,4 ( 9)
||_ 3   (10) A v (B & C)        vE 3,4..5,7..9
|   1   (11) A v (B & C)        vE 2,4..5,6..10