Need help to inconsistency problem

Question

I am using theThe Logic Book) I need help to find the inconsistencies. using SD

1.(E V F)=>(G & -I)

2.(G V F)=>I

3 -F=> E

I tried everything to show it I think showing -I and I might do it But deriving it has got me stumped Any help would be appreciated

I’m voting to close this question because forum is not a homework site — Swami Vishwananda, Mar 10 '21 at 11:00
This is not a homework problem. I do it for fun and learning. I am bored in lockdown. l am 58 years old. @SwamiVishwananda. Did you downvote me. If so upvote me — larry mintz, Mar 10 '21 at 14:48
According to a book review (albeit of the 1st ed.), your book uses some kind of Fitch style system for proofs... which from a google search inside the book it calls SD... which is not a term anyone else uses for that FYI. — the gods from engineering, Mar 10 '21 at 16:57
Another FYI: I suggest you read about resolution and analytic tableau (elsewhere) as these methods have a strategy for finding a contradiction (or more generally, a counter-model). — the gods from engineering, Mar 10 '21 at 17:28
Already went through that chapter. Pretty good at it. Still somewhat shaky on using SL: Sentential Logic Derivations (Chapter 5) — larry mintz, Mar 11 '21 at 02:32

score 1 · Answer 1 · answered Mar 10 '21 at 00:04

1

Well, consider ~F -> E is equivalent to (E V F). Together with (E V F)->(G & -I), you obtain G & ~I. Then G and G V F -> I gives you I. So now we have I and ~I. Formalize that as you wish.

answered Mar 10 '21 at 00:04

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Thanks for the hint – larry mintz Mar 10 '21 at 02:02

score 0 · Accepted Answer · answered Mar 11 '21 at 03:34

Examine the premises

   1.|  (E V F)=>(G & -I)  Premise
   2.|  (G V F)=>I         Premise
   3.|_ -F=> E             Premise

By inspection, if you could derive -F then you may derive a contradiction (I & -I) using these premises by a straightforward argument.

Well you can derive -F at root context by negation introduction: since under an assumption of F the same contradiction is entailed by the same argument.

Tip: Since we have the same argument, we should avoid duplication of effort by first deriving (E V F) => (I & -I).

   4.|  |_ E V F
     |  :
  10.|  |  I & -I
  11.|  E V F => (I & -I)

Once you have filled in that argument, you can complete the proof.

  12.|  |_ F
  13.|  |  E V F
  14.|  |  I & -I
  15.|  -F
  16.|  E
  17.|  E V F
  18.|  I & -I

Need help to inconsistency problem

2 Answers2