Does the necessity of identity also implies $(a \neq b) \to \Box(a \neq b)$?

Question

I was going to ask this on the philosophy site but it doesn't support Latex which suggests me it's not a good match for this question. The question is about modal logic (where boxes mean "necessarily" and diamonds mean "possibly") and Kripke models.

I understand the necessity of identity (NI) to be the claim that

$a = b \to \Box(a = b)$

Assuming that is true, can it be proven that $a \neq b \to \Box(a \neq b)$?

I kind of have a draft of a proof but it starts by assuming that NI is itself necessary, which I'm not sure if it's a given since that NI alone is contentious AFAIK. I'm not sure if just assuming it is the same as assuming it as tautological in modal logic. My proof also assumes some axioms other than just K which I wonder if could be avoided. It goes like:

1. $\Box(a = b \to \Box(a = b))$
2. Contrapositive of 1: $\Box(\neg \Box(a = b) \to \neg(a = b))$
3. Rewrite 2 as: $\Box(\Diamond(a \neq b) \to a \neq b)$
4. K distribution axiom on 3: $\Box\Diamond(a \neq b) \to \Box(a \neq b)$
5. Assume the antecedent $a \neq b$
6. Apply axiom B of S5 to 5: $\Box\Diamond a \neq b$
7. Modus Ponens on 4 and 6: $\Box(a \neq b)$

If that's correct, can it be improved by not having to assume the necessary version of NI, or by not needing S5?

Answer 1

Your derivation is correct. Notice that the derivation does not demand identity calculus. The theorem of system K

$$(\Box(\psi\wedge(\neg\phi\leftrightarrow\psi))\wedge(\phi\rightarrow\Box\phi))\rightarrow(\psi\rightarrow\Box\psi)$$

suffices to derive the conclusion, because the crux of the argument is the derivation of $\phi\rightarrow\Box\phi$. Let us provide the background for it. It should be remarked that the considerations mentioned below are intensely dependent on one's metaphysical views, some shared almost universally, some not.

We have two principles that govern our identity calculus:

(A) The principle of indiscernibility of identicals:$\quad\forall x\forall y((x = y)\rightarrow\forall F(Fx\leftrightarrow Fy))$

(B) The principle of identity of indiscernibles:$\qquad\forall x\forall y(\forall F(Fx\leftrightarrow Fy)\rightarrow(x = y))$

where $F$ is any property attributable to $x$ and $y$. There is no scepticism about (A), while there is some about (B). Hence, we can unanimously rely on (A). Making use of it, as first explicitly argued by Kripke, we can deduce the thesis called the necessity of identity.

A nice paper (freely available on the Web) that examines the roots of the derivations of this thesis is "On a Derivation of the Necessity of Identity" by John P. Burgess; I shall follow his discussion. Burgess outlines Kripke's argument as

$\forall x\Box(x = x)\tag{1}$

$\forall x\forall y(x = y\rightarrow (\Box(x = x)\rightarrow\Box(x = y))\tag{2}$

$\forall x\forall y(x = y\rightarrow\Box(x = y))\tag{3}$

The steps can be annotated as follows:

(1) states the necessity of self-identity.

(2) is an instance of the principle of indiscernibility of identicals, using it in the form

$$\forall x\forall y(x = y\rightarrow(\phi(x/z)\rightarrow\phi(y/z)))$$

where $\phi$ denotes any formula in which a variable $z$ occurs free and the variables $x$ and $y$ are free for $z$. As usual, $x/z$ and $y/z$ indicate uniform replacement of $z$ by $x$ and $y$, respectively. We take $\phi$ as $\Box x = z$.

(3) is a substitution instance of the theorem

$$(\forall xFx\wedge\forall x\forall y(Gx y\rightarrow(Fx\rightarrow Hxy)))\rightarrow\forall x\forall y(Gxy\rightarrow Hxy)$$

by (1) and (2).