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In English, we can distinguish between **not wanting** to do something, and
**wanting to not** do something. It's not a distinction that we usually make,
but it is possible:

```

    P = "do anything"
    
    A. "I want to **not** do anything."    　 WANT(NOT(P))
    B. "I **don't** want to do anything."　   NOT(WANT(P))    
```

The first sentence means `WANT(NOT(P))` _"I want to do nothing"_.

The second sentence literally means `NOT(WANT(P))` _"I lack the desire to do
anything"_ , but by implication it usually also means `WANT(NOT(P))` _"I want
to do nothing"_. Since this is just an implication, it's possible for it to be
cancelled in certain contexts.

* * *

I was wondering if something similar is true of `〜したくない`. Take a look at this
sentence:

> 何もしたくない。

I think this usually means `WANT(NOT(P))` _"I want to do nothing"_. But is it
possible for it to mean `NOT(WANT(P))` _"I lack the desire to do anything"_ as
well?

Can 何もしたくない have either meaning?
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meaning ambiguity scope
Scope of negation with 〜したくない
Yes, it can have either meaning. English allows a distinction due to the fact
that there are two clauses involved (removing the negative, 'I want' and 'to
do (some/any)thing'), so it allows negation in either clause. Japanese has a
single clause, and so negation has to end up in a single location. So the
distinction is determined purely contextually - 何もしたくない can be either 'I want
to do nothing' or 'I don't want to do anything'. (And if you think about it,
these are extremely similar, and it's more of a quirk of English grammar that
you can somehow distinguish the two in English.)

If you wanted to specify the second, you might reword it as したいことがない, 'there
isn't anything I want to do'.
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The negative form …したくない means “WANT(NOT(…)).” For example, 山に登りたくない means the
speaker wants to avoid climbing a mountain.

To express “NOT(WANT(…)),” we have to use other constructs such as
山に登りたいのではない.

Compare the following examples.

  1. 竜とは **戦いたくない** が、姫を助ける方法はほかにない。
  2. 竜と **戦いたいわけではない** が、姫を助ける方法はほかにない。

In the first example, the speaker wants to avoid fighting against the dragon.
In the second example, the speaker is just saying that it is not that he/she
wants to fight against the dragon.

Your notation related to P = “do anything” is ambiguous because it does not
specify how the negation interacts with the quantifier implicit in word
_anything_. To avoid this, we should make the quantifier explicit. Then
何もしたくない means ∀x. WANT(NOT(do x)), that is, the speaker wants to be free from
all actions.
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It’s true that 何もしたくない only has one interpretation (“I don’t want to do
anything”) as other answers pointed out. But you aren’t out of options.

The reason we can’t use the second negation pattern is because we can’t put a
negative marker on し followed by たい because たい cannot agglutinate onto ない.

However, we can get around this.

If we make the ない adverbial via ないで, we can put a verb of existence after it
and stick the たい onto that verb. This gets around the agglutination
restriction.

So:

何もしないでいたい

“I want to be doing nothing”

There is a progressive or stative aspect to this sentence so it isn’t a
perfect equivalent of the English “I want to not do anything” but it conveys
the idea. However, it does not work well for many verbs.

Instead of trying to express “I want not to” try expressing “I’d rather not”
“I’d prefer not to” “I want to avoid”. Stuff like “〜するのはちょっと…” “〜する気分じゃない”
“するのを避けたい” “〜しないでおこう” etc.
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