1

I’m reading Introduction to Mathematical Logic by Elliott Mendelson. He gave these three axioms:

enter image description here

And then he tried to prove that ¬ ¬ R → R :

enter image description here

My question is: what happened in line number 1 (when applying axiom (A3))?

3

For simplicity, rewrite axiom (A3) as follows:

(¬A → ¬D) → ((¬A → D) → A),

where (see page 3) A and D are statement forms (i.e. place-holders that stand for formulas whatever; see Uses of schema in Logic).

Il Line 1, we perform the substitution of B in place of A and of ¬B in place of D, getting:

(¬B → ¬¬B) → ((¬B → ¬B) → B).

  • Thank you for your help I'm new to this subject, so I'm still a little confused or maybe a lot. You said “…we perform the substitution of B in place of A and of ¬B in place of D…” I understood the first substitution from B to A but I didn’t understand the second substitution, from ¬B to D? In other words, we have two signs (¬¬)? Before I asked this question I did this: if ¬¬B is D and B is A, then the proof, in line 1, will go like this (as I understood with my little brain before I asked this question): (¬B→¬¬¬B)→(( ¬B→¬¬B)→B) What am I missing here? – Mohammad Altamimi Nov 28 '17 at 13:54
  • 1
    @MohammadAltamimi - we have exactly to put ¬B in place of D in every occurence where D is present. This means that the D occurrence will become ¬B and the ¬D occurrence will become ¬¬B – Mauro ALLEGRANZA Nov 28 '17 at 15:20

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.