How can I derive (G&H)->J using SD rules?

[Weezer1223]

Ok, so I'm doing some logic review on my own. It's been awhile since I've done derivations, so I'm a little rusty. I'm just trying to use SD rules and not SD+. I'd appreciate any help you all can offer. Thanks in advance.

Derive: (G&H)->J

1. E->(F->G) assum.
2. H->(G->I) assump
3. (F->I)->(H->J) assump
______________________________________________
4. |E assumption
5. | |G&H assumption
6. | |H 5 &E
7. | |G->I 6,2 ->E
8. | |F->G 4,1 ->E
9. | | |F asummp
10. | | |G 8,9 ->E
11. | | |I 7,10 ->E
12 | | F->I 9-11 ->I
13. | | H->J 3, 12 ->E
14. | | J 6, 13 ->E
15. | (G&H)->J 5-14 ->I



This is where I get stuck. I do get #15 out of the subderivation of E? Or am I going about this all wrong? Can you even derive this thing?

Thanks

 

[honestrosewater]

Haha, that is a good question to ask. By my check, the tableaux closes without even needing (1). Your proof looks fine, but all you can get out of (4) is ~E or a formula of the form E -> p, e.g., E -> ((G & H) -> J). The key seems to lay with I. You can get I from (2) by assuming (G & H). Can you see how to get a formula from (3) that, effectively, makes you choose between ~I or J?

On second thought, let me put it this way: The only way that F -> I can be false is for I to be ... ??

 

[NickJ]

Rule of thumb: whenever you're trying to prove a conditional claim, the first assumption you make should always be the antecedent of the conditional. You didn't follow this rule; that's why you're having trouble discharging your assumptions at the end.

And the argument is valid: I did a truth-table that shows no row where all premises true and conclusion false.

Derive: (G&H)->J

1. E->(F->G) assum.
2. H->(G->I) assump
3. (F->I)->(H->J) assump
______________________________________________
4. |G&H (assumption)
5. | H 4, &E
6. | G 4, &E
7. | G -> I 2, 5, ->E
8. || F assumption
9. || I 6, 7, ->E
10.| F -> I 8-9, ->I
11.| H -> J 3, 10 ->E
12.| J 11, 5 ->E
13.(G&H) -> J 4-12, ->I QED

Note that the first premise is irrelevant -- I never use it.

 

[honestrosewater]

Hello again, NickJ. FYI: We aren't allowed to give full solutions here unless the person is having serious problems.

 

[Weezer1223]

Thanks everyone. I appreciate it. You're hint hones helped out greatly. Thanks for taking the time NickJ to write out a sollution, even though you weren't supposed to.

This is a great site, and I'm looking forward to reading and posting here.

 

[NickJ]

FYI: We aren't allowed to give full solutions here unless the person is having serious problems.

Oops! Now I know...and knowing is half the battle.