Question

One can prove that a=b → □(a=b) by using the substitution axiom for equality. However, since ¬(a=b) is not of the form α=γ, I don’t see how to prove ◊(a=b) → a=b, even though I know it’s valid in most modal logics. Is the substitution [a=b/¬(a=b)] actually the correct way to prove it, or am I missing something?

Answer 1

To prove ◊(a=b) → a=b, we cannot use the substitution axiom for equality directly. Instead, we can use the rule of transitivity and the equivalence between ◊¬(a=b) and ¬□(a=b) to derive the desired result.

Step-by-step explanation:

The substitution axiom for equality allows us to substitute equals for equals in logical proofs.

However, when we have a statement of the form ¬(a=b), we cannot directly apply the substitution axiom to prove ◊(a=b) → a=b. This is because ¬(a=b) is not in the form α=γ.

Instead, we can prove ◊(a=b) → a=b by using a different approach. One way is to use the rule of transitivity to show that ◊(a=b) → a=b. We start with ◊(a=b) and use the transitivity of equality to deduce that ◊(a=b) → □(a=b).

Then, using the fact that ◊¬(a=b) is equivalent to ¬□(a=b), we can conclude that ◊(a=b) → a=b. So the substitution [a=b/¬(a=b)] is not the correct way to prove it, and this alternative approach is more appropriate.