Here's the formal argument from the first pages.

$(1)\,(x)(y) [(x=y) \supset (Fx \supset Fy] \\(2)\,(x) \Box (x=x) \\(3)\,(x)(y) (x=y) \supset [ \Box (x=x) \supset \Box [(x=y)] \\ (4)\,(x)(y) [(x=y) \supset \Box [(x=y)]$

What's posited here is that if two things are identical then they are necessarily identical.

So what? Well, if it is right - and it seems it is - then in no possible world is Clark Kent not Superman. For we know the Kent is Superman, and we can swapping x for Superman and y for Kent, we show that necessarily Kent is Superman.

But surely it may have turned out that Superman had taken on a different secret identity?

That's the pull of the argument attributed to David Wiggins.

(the maths I used, for those wanting to use Mathjax is
(1)\,(x)(y) [(x=y) \supset (Fx \supset Fy] \\(2)\,(x) \Box (x=x) \\(3)\,(x)(y) (x=y) \supset [ \Box (x=x) \supset \Box [(x=y)] \\ (4)\,(x)(y) [(x=y) \supset \Box [(x=y)]
)

I'm on page 15 out of 30. It's an easy read if you're aquatinted with Naming and Necessity, which you can find here:

https://thephilosophyforum.com/discussion/4545/naming-and-necessity-reading-group

https://thephilosophyforum.com/discussion/4857/naming-and-necessity-lecture-three/p1

Some thoughts about terminology. Kripke introduces the term de re modality, which I think to newcomers can be confusing. Care to address this @Banno?

Further, on around page 15, Kripke brings up the statement, that the the 37'th president of the United States is non-rigid and Nixon is rigid. This would sound confusing to someone at first. I can attempt answering this on the basis that one doesn't need to invoke their imagination too wildly and simply designate that there could be a possible world where Humphrey won the election, thus it is contingent. However within the reference frame of our world Nixon won the the 37 elections for the president of the United States, making him a rigid designator. It's important to note that the possibility of 37th election of the United States sets up the possibility that out (this is our criteria condition for identity) of two possible candidates one or the other won. So, following from this we can only reference who won based on the feature of the world obeying causality (Kripke relies on a causal theory of reference).

I'm not sure if this is a good place to start, so please let me know @Banno.

Edit:

I would like to reiterate where people go on a wild goose chase, that counterfactuals like Nixon could have lost the binary relation of either winning or losing the presidency or winning it with respect to the causal chain of events in our world, not some other, and hence the status of Nixon being a rigid designator is assigned by our world not any other. That's just one instance where a counterfactual could have arisen given the epistemic criteria that fulfills the condition of him winning or losing the election. Treat everything else as et. cetera.

And, if it needs to be said in the positive that, yes, the causal chain of events would be the framing relation that allows Nixon to obtain as a rigid designator, in our world.

The key topic of the paper is 'How are contingent identity statements possible?", or as Kant may have put it, "How are synthetic a priori judgements possible?"

Kant's synthetic a priori judgement is more in agreement with a Kripke necessary identity statement than a contingent identity statement, though Kant's synthetic a priori is more about knowledge by acquaintance than Kripke's knowledge by description.

Contingent identity statements versus Kant's a priori judgements
Kripke writes in the introduction: "“How are contingent identity statements possible?” This question is phrased by analogy with the way Kant phrased his question “How are synthetic a priori judgments possible?”. However, he later writes: "To state finally what I think, as opposed to what seems to be the case, or what others think, I think that in both cases, the case of names and the case of the theoretical identifications, the identity statements are necessary and not contingent"

As regards contingent identity statements, such as "Hesperus is Phosphorus", from Hume's principle of constant conjunction, we logically infer that Hesperus is Phosphorus, and therefore is logically contingent rather than logically necessary.

As regards Kant's synthetic a priori judgement, which is about a priori pure and empirical intuitions, it is not about the identity statement "the postbox is red", rather it is about the identity statement "this is red", and as a priori "knowledge", logically necessary.

Synthetic a priori judgements
I've always thought the phrase "synthetic a priori" was wrong, as it mixes two fundamentally different things. To my understanding, within language are synthetic and analytic propositions, some knowledge can be a priori and some a posteriori and within logic is the necessary and contingent. It is as if one said "anger is a heavy thing", not to be understood literally but metaphorically.

"Synthetic a priori" means no more than humans are born with certain innate abilities, such as the innate ability to be able to distinguish between a loud and quiet noise, something hot and something cold, etc. Children don't need to go to school to be able to distinguish between a sweet and sour taste, as this is instinctive.

The term a priori knowledge is not correct either, in that humans don't have a knowledge of the colour red before seeing it for the first time, but they do have the ability to see the colour red before ever seeing it for the first time. As an analogy, a wine glass passively shatters when the frequency of an opera singer's voice matches the natural resonant frequency of the wine glass, it is not the case that the wine glass is an active participant.

The term "synthetic a priori" should be understood as an idiomatic expression rather than as a literal guide to Kant's doctrine of "transcendental idealism".

Better copy of Identity and Necessity
There is a web site, but one needs to sign in through your library.
https://academic.oup.com/book/36436/chapter/320710138

Anyway, I have to go now to see what Santa Claus has left under the tree.

What's posited here is that if two things are identical then they are necessarily identical. — Banno

Right, and our spacetime is necessarily what it is and could not have been different. Bye bye free will?

As regards contingent identity statements, such as "Hesperus is Phosphorus", from Hume's principle of constant conjunction, we logically infer that Hesperus is Phosphorus, and therefore is logically contingent rather than logically necessary. — RussellA

Sorry, but if you read the paper, Kripke posits the logicality on the empirical finding that Hesperus is Phosphorus. It's only a contingent identity statement upon observation that would allow us to conclude that it is Venus, in fact. Hence it is de facto a synthetic a priori upon examination via observation.

Right, and our spacetime is necessarily what it is and could not have been different. Bye bye free will? — litewave

Could you elaborate?

The term "synthetic a priori" should be understood as an idiomatic expression rather than as a literal guide to Kant's doctrine of "transcendental idealism". — RussellA

Kantian transcendental idealism, not needing any inverted commas, is predicate on the possibility of synthetic a priori relations, and cannot stand without them, so must be understood as a literal guide to it, whether or not one regards the philosophy itself as legitimate.

In Kant, identity is the ground for truth in analytic judgements, which are never contingent. Synthetic judgements do not rely on identity, and the truth of them relies on the relation between its conceptions, hence is contingent. Kripke wants to unite the contingent with identity, which Kant deemed, if not impossible, then at least logically insufficient in regard to a brand new philosophy.

So maybe Kant’s term isn’t a mere idiom after all. Which is neither here nor there with respect to the thread.

Could you elaborate? — Shawn

Does Kripke believe that it is possible for us to act differently than we actually act? It seems he doesn't, because our actions are parts of spacetime and spacetime is necessarily identical to itself, which means it cannot be different than it is and so our actions cannot be different than they are.

That's the pull of the argument attributed to David Wiggins. — Banno

Could you provide that argument? I'm reading the paper again and see that Wiggins argument seems to point out the fact that sometimes we refer to the same object with two different proper names. But, he (Kripke) goes on to say that we discover these truths after observation, as in the case of Hesperus and Phosphorus being Venus.

I would argue that that is too strictly a transcendental argument. Every day we make choices where seemingly we could have done something otherwise. Even taking your argument to the extreme, there could be a possible world where causality would have allowed for a different event cone to allow a counterfactual to arise.

Every day we make choices where seemingly we could have done something otherwise. — Shawn

Yes, seemingly.

Even taking your argument to the extreme, there could be a possible world where causality would have allowed for a different event cone to allow a counterfactual to arise. — Shawn

Quantum-mechanical indeterminacy could do that. But a world with a different outcome of a quantum measurement would be a different world, with a different identity, than our world. And I cannot be in both worlds, if by "I" we understand someone who is conscious of being only in one world. The "I" in the different world would be my copy, a counterpart.

But a world with a different outcome of a quantum measurement would be a different world, with a different identity, than our world. — litewave

Yes, a world which is called a possible world ...

And I cannot be in both worlds, if by "I" we understand someone who is conscious of being only in one world. The "I" in the different world would be my copy, a counterpart. — litewave

The issue of personal identity is somewhat justified by the fact that my identity is consistent with the world that my personal I obeys causality in. The degree to which I have influence over the causal chain of events culminating in my identity is not a philosophical question, in my opinion.

Anyways, since the issue raised by @litewave often crops up in these discussions, here's a link. Fraught with controversy, I believe:

https://plato.stanford.edu/entries/identity-transworld/

...de re modality... — Shawn

Yeah, it's an important, but fraught, distinction. I suspect focusing on it will cause yet more confusion. It's a distinction that is important because of ambiguities in natural languages, that simply do not carry over to formal modal languages.

The de dicto interpretation has a wide scope for the modal operator:
$De\,dicto: \, \Box \exists (x) F(x) \\De\,re: \, \exists (x) \Box F(x)$
The discussion on p.164 is to the effect that if we can have necessary properties for individuals then (1) must hold; that (1) says the same as that F(x), even if F is (x)(x=x).

So that page is mostly a justification for the soundness of the argument (1-4).

For the rest, small steps. I would like to look a the paper with care, rather than moving ahed too quickly, or jumping to Kant or to the mind-body problem.

Yes, a world which is called a possible world ... — Shawn

But the actual world is still necessarily identical to itself and therefore cannot be something different than it is. My actions in a merely possible world may be different than my actions in the actual world but my actions in the actual world cannot be different than my actions in the actual world.

, The issue addressed int he article is, how best are we to talk about necessity and identity? The article seeks to sort out the many confusions by looking at how we might pars natural languages in in a possible world semantics.

This is germane to issues of free will and so on, but in a way logically precedes it. That is, we ought work out how best to talk about such issues prior to addressing them. Work out the grammar before we engage in the conversation.

All this by way of requesting that we stick to the text for now, rather than wander off on a tangent.

In the early part he's basically explaining the problems with looking at proper names as descriptions.

We learn empirically that Hesperus is Phosphorus. This couldn't be so if those two words were descriptions.

We learn empirically that Hesperus is Phosphorus. This couldn't be so if those two words were descriptions. — frank

But we did learn that the star we see in the morning is the same as the star we see in the evening - they are both Venus.

So we can learn that two descriptions are of the same individual.

Now, necessarily, Hesperus is Phosphorus. But is is necessary that the morning star is the evening star?

If you really need a physical grounding for entertaining the notion of possible worlds, then assume the many worlds interpretation...AND there's no issue with things being deterministic given the MWI. You can still have counterparts and alternate versions of you.

The discussion on p.164 is to the effect that if we can have necessary properties for individuals then (1) must hold; that (1) says the same as that F(x), even if F is (x)(x=x).

So that page is mostly a justification for the soundness of the argument (1-4). — Banno

The discussion on pg. 164 in N&N?

No, in Identity and Necessity

Kripke wants to unite the contingent with identity... — Mww

That's what I would call a category mistake.

No, in Identity and Necessity — Banno

Oh, okay.

The discussion on p.164 is to the effect that if we can have necessary properties for individuals then (1) must hold; that (1) says the same as that F(x), even if F is (x)(x=x). — Banno

I think it's important to make a distinction here about properties of objects and properties of individuals. Would it be accurate to label descriptions as also constituting properties also?

Hmm. Individuals as in individual constants... a,b,c...

and properties of individuals as in fa, fb, ga...

Descriptions are presumably formula that pick out at least one individual. "The man who invented bifocals" would be something like "The x such that x is a man and x invented bifocals".

Much of the next page is concerned with examples of two descriptions of the same individual.

Let's look at "the author of Hamlet", "H". This picks out one individual. "The author of Hamlet might not have written Hamlet" has ambiguous parsings.

It might be that Shakespeare did not write Hamlet; that the individual who did write Hamlet might possibly have not written Hamlet. Roughy
$\exists (x) (Hx . \diamond \sim Hx)$

It could not have been that the person who wrote Hamlet, whoever that was, did not write Hamlet; it is not possible that the person who wrote Hamlet did nto write Hamlet.

$\sim\diamond (\exists (x) (Hx . \sim Hx)$

The ambiguity dissipates because the scope of the existential and modal quantifiers is explicit.

And hence to Russell's solution to the apparent problem of substituting descriptions not argument (1-4). The argument seem to be something like that, if P is being the first post master and B being the inventor of bifocals, our conclusion is something like

$\exists (x)(Px ) \exists ( y)( By)(x=y)$

"...here, x and y are both Benjamin Franklin, and it can certainly be necessary that Benjamin Franklin is identical with himself" p.166

So descriptions will not result in the sort of paradox Wiggins seems to have had in mind.

That's my reading to bottom of p.166. Kripke then passes on to proper names.

The ambiguity dissipates because the scope of the existential and modal quantifiers is explicit. — Banno

Can you elaborate about the scope? There is a large footnote about this in the text.

Note 5? Seems to be about the differences in how the formula are to be written rather than anything of direct import.

Quine had issues with extensionally opaque contexts, including modal considerations. Seems he wouldn't allow them as knowledge claims. He wouldn't have accepted $\Box Ha$, it seems.

Not in the scope of my thinking.

Note 5? — Banno

Note 4.