This is all somewhat interesting, and there are many ways we could go with it, but you may first have to convince me that it is on topic for this thread. — Leontiskos
One way it seems relevant is that understanding the sentence as weird and contradictory on a gut level...
I think this reflects the confusion in modern thought where it is presupposed that there can be statements without implicit speakers. — Leontiskos
pumps the intuition that it must function as an assertion. It would never be uttered in normal circumstances since part of its mechanism asserts something and then undermines the act of assertion.
If it is indeed contradictory in some sense, it is contradictory by virtue of a property it has as an asserted statement, and not its propositional content (what makes it true or false). Since extensionally it's fine. It can be raining while its speaker believes it is not raining, so I can assert "It's raining and I believe it is not raining" and it could true.
The reason that would be relevant is that it highlights a kind of contradiction which can occur by virtue of the acts and attitudes contained in the statement relating to the statement's propositional content, which suggests that there is a type of contradiction which is not governed by the phrase's propositional content. In the sense that a stairway implies distinct floors.
Maybe analysing it in terms of illocutionary forces is good. And it would probably be better to look at it in the form "Sally said "It is raining but I believe it is not raining"", since that dodges all the weird crap involving "I", since we know who is saying it.
"It is raining but I believe it is not raining" is asserted by Sally. It has assertoric force.
"It is raining", the first clause, is something which could be true or false. Sentences that begin with "It is" parse as assertions. EG:
Sally said "It is an egg".
Sally said "It is a nice day today".
Sally said "It is going to be 3 degrees Celsius this evening".
So reading the sentence, the first clause invites us to interpret that Sally has asserted that it is raining.
Then there's "but", which registers an opposition or contrast between what came before and what came after. The sentence is still odd with "and" instead, so I shan't make too much of the "but" in it.
The second clause is "I believe it is not raining", which invites us to interpret it as an assertion. Sentences which begin with "I believe" parse as
assertions of belief on the part of their speaker. EG:
Sally said "I believe it is an egg".
Sally said "I believe it is a nice day today".
Sally said "I believe it is going to be 3 degrees Celsius this evening".
When someone asserts something, they are often taken to believe it. I think that's a good default assumption when someone makes a simple claim, and you've no reason to otherwise doubt them. Though it isn't necessary that when someone asserts something, they believe it - they could be lying, they could misspeak, they could be confused, they could be deluded, they could have very unstable beliefs in the moment etc. The expectation is that when someone says "I believe (blah)", they count as asserting (blah) truthfully.
So when Sally says the second clause, "I believe it is not raining", a reading of the phrase in which Sally's assumed to be truthful and sincere associates the "I believe" in the sentence with asserting the claim "It is not raining". So the first clause appears to assert "It is raining", the second clause appears to assert "It is not raining", and those things clash together in our heads.
Nevertheless, Sally is not in a state of contradiction. For Sally only appears to assert that it is raining, and only appears to assert that it is not raining. With logical heads on, that feels like she has just asserted that it is raining and that it is not raining, which is a contradiction. But I think a better explanation of the weirdness in the sentence is that appearing to assert X is both logically and behaviourally consistent with appearing to assert not-X in some contexts. I'd bet this conflict of appearances and a scramble for context is something you sever if you remove the attempt to contextualise the statement (radical interpretation eh?). And moreover, I'd bet that this conflict of appearances is commonplace and essential out in the wild.
Eg, I've said "I don't believe it's raining!" while wincing up at a sky thick with summer rain. And I wasn't insane at the time, I was just complaining. Making sense of "belief" in that statement didn't require too much work on the part of my friend who was with me, since they'd known that was the only day with a good forecast that week.
A similar logic lets you provide a model for Sally's odd phrase. I'm sitting here now, I believe it's not raining since it wasn't forecast to rain this evening last time I checked. But my curtains are closed. I just said "It is raining and I believe it's not raining" aloud... and it turned out it wasn't raining after all, when I opened the curtain.
What I think makes Moore's paradox a good gateway in this discussion is that there's a whole context of cooperative use and interpretation, which contains a myriad of exploitable oppositions and contradictions, that just don't show up when you analyse the phrase as an instance of asserts(p & believes not-p) & asserts(p)=>believes(p). Particularly how you can make sense of it, and the kind of doubt you might have regarding Sally's faculties and situation. Maybe those are the kind of things
@J was looking to incorporate into a logic.
Though there remains the question of whether this can be incorporated into normal flavours of logic, whether it's something that can be formalised, whether it should be formalised... and so on.
As an aside, when I said the phrase aloud I felt a powerful compulsion to immediately open the curtain to check... Surely something we expect Sally to have done in my shoes!