> Signal uses a form of Diffie-Hellman to generate a common key. That exchange has to be signed. Without this signature you would have no way to know where a message came from, just like in the unsigned PGP case. Without this signature you would also be subject to MITM attacks (unlike the PGP case).
In what sense do you believe X3DH is "signed" ? Is this going to be more handwaving where you treat authenticated as "signed" just because that's how you'd do it in PGP ?
The recipient of a first X3DH message does know for sure it is from the sender, because they're the only one that could have performed the steps needed without the recipient's private keys.
But if (like the recipient) you know their private keys, you can just forge messages that are entirely plausible.
Consider an initial message purportedly from Alice to Bob but let's suppose we're Bob and we're forging it. How do we go about this?
We need to do three DH calculations. First one, is Alice's long term Identity crossed with Bob's prekey. Alice would do this by knowing her Identity private key, but we know Bob's private prekey, so we use that with Alice's public long term Identity.
Second one, Alice's ephemeral crossed with Bob's long term Identity. Alice would do that using her ephemeral private key, but we know Bob's private Identity key, so we use that with any key X.
Third one, Alice's ephemeral crossed with Bob's prekey. Alice would use her ephemeral private key again, but we know Bob's private prekey, so we use that with key X again.
We now have a secret key we can use to encrypt our forged message "from Alice" and we write key X into the headers where Alice would have written her public ephemeral. Our message is indistinguishable from genuine. Only Bob (who forged it) and Alice (who knows she didn't send it) know this is a forgery and neither of them can prove it.
> In the Signal case you would have to claim that your trusted correspondent forged a message from you. That wouldn't really work.
If the Secret Police are content that you are a Blasphemer or you plotted against the Emperor or whatever without proof, nothing would "really work", regardless of the technology used or not used. That's not interesting here because we can't do anything about it. If the Star Chamber will have you executed based on hearsay, that's just as true for literal whispers as for a Signal message.
In contrast PGP provides them with unshakeable proof even though it didn't need to.