Given this set of premises:

  1. Something in the forest I hadn't observed was not the dark ruler
  2. Something which had been noted means worth to be remembered
  3. Something I had seen in the forest not worth to be remembered
  4. Something I had observed in the forest is something I had noted in notebook

and given that

$p$ : Something in the forest I had observed
$q$ : The dark ruler
$r$ : Something in the forest which had been noted in notebook
$s$ : Something in the forest that is worth to be remembered
$t$ : Something I had seen in the forest

Show that with only using rules of inference, conclusion $u$: "Something in the forest I had seen was not the dark ruler" can be derived from the given set of premises.

My first problem is to translate these premises to appropriate logic proposition. These premises are simply confuse me a lot. After hours of thinking, my thought drive to these translations:

  1. $\neg p \implies \neg q$
  2. $r \implies s$
  3. $t \implies \neg s$
  4. $p \implies r$

    u : $ t \implies \neg q $ (conclusion)

but I feel these translations are pretty bad, given that their meaning aren't close to the original meaning, according to my understanding. Anyway, I couldn't find another logic proposition that is appropriate to these premises, and I decided to go forward:

  1. $\neg s \implies \neg r$ (Transposition 2)
  2. $t \implies r$ (Hypothetical Syllogism 3, 5)
  3. $\neg r \implies \neg p$ (Transposition 4)
  4. $\neg r \implies \neg q$ (Hypothetical Syllogism 7, 1)
  5. $\neg t \vee r $ (Material Implication 6)
  6. $r \vee \neg m$ (Material Implication 8)

Boom. Not much I could do from here to arrive at the conclusion. All I have done is only turn things into much more complicated mess. I decided to do validity checking test using statements 9 & 10 (not using the original statements 1 - 4). Didn't turn out well. The result is the conclusion is not valid. I'm pretty sure I've made mistakes, either in the translation or in the deriving steps, but I couldn't figure out where.

Assumption used is "Something" is not the same as "Some" and "observe" is not the same as "see". I think predicate logic is not allowed here.

Any helpful answer will be very appreciated. Thanks in advance.

  • $\begingroup$ @MauroALLEGRANZA thanks for your answer, but could you explain how predicate logic is used in this case? I haven't studied it yet and I'm worried I'm not allowed to use it in this case... $\endgroup$ – bksbunnyhead Sep 12 '15 at 17:22
  • $\begingroup$ It works : from 3) and 5) by Syll derive $t \to \lnot r$; then contrapose 4) and derive (again by Syll) : $t \to \lnot p$. Finally with 1) derive $t \to \lnot q$. $\endgroup$ – Mauro ALLEGRANZA Sep 12 '15 at 17:23
  • $\begingroup$ You have a typo in line 6 : it must be $t \to \lnot r$. $\endgroup$ – Mauro ALLEGRANZA Sep 12 '15 at 17:25
  • $\begingroup$ @MauroALLEGRANZA oh, I haven't noticed it before. Deriving conclusion is indeed tricky task. Thanks again, and yep, i'm wrong at line 6. $\endgroup$ – bksbunnyhead Sep 12 '15 at 17:33

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