1

So, I have this proof:

Let a be an open sentential variable. Let K and C be prepositional variables asserted of sentences.

Given

  1. K(a) <=> C(a) & a
  2. C(a) <=> C( C(a))
  3. C(a & b) <=> C(a) & C(b)

If

  1. K(a)

Then

  1. C(a) & a //by 1,4
  2. C( C(a)) //by 2,5
  3. C( C(a)) & C(a) //by 5,6
  4. C( C(a) & a) //by 3,7
  5. C( K(a)) //by 1,8
  6. C( K(a)) & K(a) //by 4,9
  7. K( K(a)) //by 1,10

Thus

  1. K(a) -> K( K(a))
  2. K( K(a)) -> K(a) //by 1

And so

  1. K(a) <=> K( K(a)) //by 12,13

QED

Can anyone see any issues with this? The thing that I'm least sure about is the use of recursive.

Thanks

  • PS: You let K and N be propositional variables, then used K and C in the proof. – Graham Kemp Sep 23 '18 at 13:09
1

It is correct, but you've skipped over some steps, which obsured your reasoning.

For instance step 13. You just say K(K(a)) → K(a) by 1. Why?

Well, indeed, applying 1 gives you K(K(a)) ↔ C(K(a)) & K(a) and you can obtain the desired result from that, but you should spell this out. Show your working more.

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