Challenging Truth Tree problem

What is the basis for reason? And mathematics?

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airrudra@gmail.com
Posts: 1
Joined: Thu Oct 08, 2015 11:20 pm

Challenging Truth Tree problem

Post by airrudra@gmail.com » Thu Oct 08, 2015 11:23 pm

(F&~H)→G
~A→~H
~HvC
~(B→E)
C→D
----------
{~G→[F→(D&A)]}&B

Can someone solve this, I believe the answer is that it is valid. But I don't know how to negate a ~A on a separate branch. I have to work through it some more but would love some help thanks in advance =)

Averroes
Posts: 61
Joined: Thu Jul 20, 2017 8:48 pm

Re: Challenging Truth Tree problem

Post by Averroes » Wed Nov 15, 2017 4:55 pm

I am late on this, and I hope that the poster found a solution to his/her problem. Here is what I came up with.

Code: Select all

1. (1) (F & ~H)-> G                         Premise
2. (2) ~A -> ~H                               Premise 
3. (3) ~H v C                                  Premise
4. (4) ~(B ->E)                               Premise
5. (5) C-> D                                   Premise
6 (6) ~G                                        Assumption
7 (7) F                                          Assumption
1,6 (8) ~(F&~H)                              1, 6 Paradox of material implication
1,6 (9) ~F v ~~H                              8 DeMorgan
1,6 (10) ~F v H                               9 Subformula double negation
1,6,7 (11) H                                   7,10 disjunctive syllogism (DS)
1,3,6,7(12) C                                  3,11 DS
1,3,5,6,7 (13) D                               5,12 MP
1,2,6,7(14) A                                   2,11 PMI
1,2,3,5,6,7 (15) D&A                         13,14 &I
1,2,3,5,6(16) F -> (D&A)                    7,15 ->I
1,2,3,5 (17) ~G-> F -> (D&A)                6,13 ->I
4 (18)B & ~E                                    4, Negation implication
4(19) B                                            18, &E
1,2,3,4,5 (20){~G->[F->(D &A)]} & B      17,19 &I

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