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

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

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

1 Answer 1


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? Nov 28, 2017 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 Nov 28, 2017 at 15:20

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