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This looks like such an easy problem but still can't figure it out. How do I prove ¬(¬a = a)?

No given premises.

I got this so far (in Fitch):

proof so far

This is a subproof where I assume the negation of my goal and then try to reach the absurd/contradiction so I can state the negation of my assumption, which would be my goal.

Thanks in advance!

Yaeger
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  • Should I interpret that as ¬(¬(*a* = *a*)) ? Because it seems like "=" denotes equality (and not "if and only if"), and if "=" denotes equality, then *a* must be an element which means that ¬*a* is not a valid formula. – aioobe Sep 18 '14 at 13:30
  • No, you should interpret it as ¬(¬a = a) which is a perfectly valid sentence. ¬a = a is just a false sentence and the negation would make it true. – Yaeger Sep 18 '14 at 16:43
  • So *a* is a predicate? (Note that [the screenshot you posted](http://gyazo.com/feab1b469ffe2a9b9a90267db620f9f0) seems to agree with me as it says *"This formula is not well formed"*) – aioobe Sep 18 '14 at 16:48
  • I've explained my view of this in my answer. – aioobe Sep 19 '14 at 09:14

1 Answers1

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Looking at your screenshot I'd say your =Intro introduces a variable a (that is, a is an object of the domain, rather than a predicate).

I say this because

  1. in all books I've read, the =Intro rule is used for objects rather than predicates, and

  2. for predicates, equals is expressed as "if and only if" which is typically written as ↔ and not =.

So, in other words, the only sensible interpretation of ¬(¬a = a) is that = binds harder than ¬, and the whole formula should be interpreted as ¬(¬(a = a)).

Now you should be able to

  1. introduce a = a
  2. assume the contrary: ¬(a = a)
  3. arrive at a contradiction, ⊥, based on 1. and 2.
  4. Use ¬Intro on 2 and 3 to get ¬(¬(a = a)).
aioobe
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  • Thanks, this works. http://gyazo.com/81218fb304d7848a42349cc459428010. But, it's still very confusing. Why does ¬ have higher precedence than =? Intuitively this makes no sense to me. Sorry, it's just my second week of studying Logic. – Yaeger Sep 19 '14 at 09:55
  • You nailed it :-) No problem. I know it can be confusing. (I've been a teaching assistant in a CS logics course) – aioobe Sep 19 '14 at 10:35
  • You can ping me on email if you post more logics related questions. – aioobe Sep 19 '14 at 10:57
  • Thanks aioobe, I posted another Logic question. Would be great if you could take a look at it! (can't find your email) – Yaeger Sep 19 '14 at 18:56