The behaviorist might grant that if there are mental states but we do not know anything about them. — Fooloso4

I believe this would mean they were no longer a behaviorist. Maybe we have different definitions. You said earlier that a behaviourist is an empiricist who deals (only?) with what can be observed.

When I say "I remember" that does not mean that I have a particular mental process going on. — Fooloso4

I believe it does. Otherwise, why would you say it?

That is a grammatical fiction. — Fooloso4

The grammatical fiction is the assumption that the word “remember” gets its meaning from a description of the particular mental process rather than from its expression; that the grammar of sensation language is based on a description of private mental/sensation objects instead of being based on the public expression of those sensations.

The grammar of "I remember" is not about some mental process. But that does not mean that the mental process of remembering is a fiction. — Fooloso4

Agreed.

The Moyal-Sharrock essay, Wittgenstein's Razor: The Cutting Edge of Enactivism (thanks Luke for the link) has a good quote about behaviorism:

Meaning, believing, thinking, understanding, reasoning, calculating, learning, following rules, remembering, intending, expecting, longing – there is hardly anything, traditionally thought to be emergent from, underwritten by, or reducible to, a mental process or state, that Wittgenstein has not subjected to the razor of enactivism; that is: shown to be primitively embodied or enacted rather than originating in propositions, theories of mind, or ghostly
processes.

This may sound like behaviourism, but it isn't. As Peter Hacker aptly sums up: — Wittgenstein's Razor: The Cutting Edge of Enactivism

... behaviourism was right about some matters. Logical behaviourism (e.g., Carnap and Feigl in the 1930s) was right to insist that there is an internal relation between mental attributes and behaviour. For the criteriafor ascribing mental attributes to others consist in their behaviour in the circumstances of life. Where it was wrong was to suppose that the mental is reducible to behaviour and dispositions to behave. Ontologicalbehaviourism (Watson and Skinner) was right to emphasise that language learning is based on training, and that it presupposes common behavioural reactions and responses. It was right to conceive of language learning as learning new forms of behaviour – learning how to do things with words. It was correct to conceive of understanding in terms of abilities and dispositions, rather than as a hidden mental state or process.
But the behaviourists were sorely mistaken to suppose that the mental is a fiction. One can think and feel without showing it, and one can exhibit thoughts and feelings without having them. Avowals of experience are indeed a form of behaviour, but what they avow is not behaviour — ibid. page 5

Chomsky made similar points. The reduction makes the subject disappear.

I believe this would mean they were no longer a behaviorist. — Luke

As a behaviorist they would deal only with the science of behavior. They reject talk about mental states and mental processes because these are not observable or knowable. What we say about them, including whether or not they exist is a fiction. Some behaviorists might take a hard line and deny mental processes others might dismiss the question of whether they exist.

In the context of PI 307:

“Are you not really a behaviourist in disguise? Aren't you at bottom really saying that everything except human behaviour is a fiction?”

Wittgenstein takes your “hard line” wrt behaviourists.

I believe it does. Otherwise, why would you say it? — Luke

Because I remember.

Let me phrase this more carefully:

When I say "I remember" I do not mean that I have a particular mental process going on.

The grammatical fiction is the assumption that the word “remember” gets its meaning from a description of the particular mental process rather than from its expression — Luke

Agreed.

When I say "I remember" I do not mean that I have a particular mental process going on. — Fooloso4

It sounds like you are not remembering. It’s like saying: ‘when I say “ouch” it does not mean I have a particular pain sensation.’ Then why do you say “ouch”?

Edit: maybe the word “particular” is causing confusion between us. I only mean that there is some sensation occurring (a particular one each time), not that the word describes the same particular sensation every time (e.g the same memory or the same pain)

You are right, the behaviorist would not grant that there are mental states. But it could be argued that:

To say anything about something we can know nothing about would be a fiction. — Fooloso4

306. ...“There has just taken place in me the mental process of remembering . . .” means nothing more than “I have just remembered . . .”

If you were to ask me, "Do I mean there has just taken place in me the mental process of remembering? I would say no, I mean I just remembered.

We can drop the talk of the mental processes. That is not the way we speak and talk of the mental process does not add to the meaning. I do not mean the mental process but rather that I remember.

When I say "I remember" I do not mean that I have a particular mental process going on. — Fooloso4

“To deny the mental process would mean to deny the remembering” - PI 306

But §308.

Since Wittgenstein does not make any distinctions between behaviorists, I thought it best to stick to what is in the text.

When I say "I remember" I do not mean that I have a particular mental process going on.
— Fooloso4

“To deny the mental process would mean to deny the remembering” - PI 306 — Luke

But I don't deny it. If I say I ate dinner I don't mean the process of mastication and digestion.

The grammatical fiction is the assumption that the word “remember” gets its meaning from a description of the particular mental process rather than from its expression
— Luke

Agreed. — Fooloso4

If you agree to this then it’s no longer clear to me where you think our disagreement lies.

307. “Aren’t you nevertheless a behaviourist in disguise? Aren’t you nevertheless basically saying that everything except human behaviour is a fiction?” If I speak of a fiction, then it is of a grammatical fiction.

My reading: Wittgenstein rejects the behaviourist idea that everything except human behaviour is a fiction (i.e. he does not deny that we have private experiences). He accepts the behaviourist idea only with regard to the determination of grammar and meaning. That is, he considers any account of the determination of grammar other than human behaviour to be fictional.

This seems to me the most straightforward reading of 307 and what he means by “grammatical fiction”.

Your account that the grammatical fiction refers to saying something about which we know nothing seems less straightforward and possibly wrong.