Which of these phrases are equivalent, if any, and why

Some say the following two phrases are equivalent because of Raising (linguistics)!

Example 1

He doesn't believe that bigfoot exists
He believes that bigfoot doesn't exist

Are those two phrases really equivalent? In case they are, are the phrases from the examples below also equivalent? If so, why?

Example 2

He doesn't love that bigfoot exists
He loves that bigfoot doesn't exist

Example 3

He doesn't hate that bigfoot exists
He hates that bigfoot doesn't exist

Example 4

He doesn't know that bigfoot exists
He knows that bigfoot doesn't exist

To me none of the examples contains two equivalent phrases.
The example that seems more likely to have equivalent phrases, to me, is example 1, but, if the guy i'm talking about in example 1 doesn't know what bigfoot is, how can he hold a belief of any sort about the existence or nonexistence of bigfoot (phrase 2)? I mean, phrase 1 doesn't state that he believes something about bigfoot, so phrase 1, in this case, seems to be right.

IMPORTANT SIDENOTE: I don't mean to state or imply that all possible sentences that can be written in the stated format will not be equivalent and I also don't mean to state or imply that the linguistic principle of raising is somehow incorrect.

Solution 1:

There is a good discussion of this issue in The Cambridge Grammar Of The English Language (page 839) where Geoffrey Pullum analyses the following sentences:

ia. Mary doesn't like you.

ib. Mary dislikes you.

As Pullum states:

Consider [ia]. This would presumably be true if Mary had never even heard of you, and in this case [ib] would be false. But it would normally be quite pointless to say [ia] if this were the case, so this scenario can be ignored in interpreting it.

Extrapolating to your example:

He doesn't believe that bigfoot exists.

is true if he has never heard of the bigfoot, but 'would normally be quite pointless to say'.

Pullum moves on to a careful analysis of verbs and adjectives which have three degrees of subordinate negative implicature (weak, medium, strong). As examples he lists:

i a. I'm not willing to be included. / b. I'm willing to not be included. [weak]

ii a. I don't want to be included. / b. I want to not be included. [medium]

iii a. I don't insist on being included. / b. I insist on not being included. [strong]

Pullum states:

It is clear that [b] is an implicature of [a] only in case [ii].

In other words, want is a verb of medium implicature, so that I don't want to be included has the same meaning as I want not to be included.

Pullum lists several other verbs of medium implicature under the catgories wanting, advice, probability, opinion, and perception. The opinion group includes the verb to believe. The verbs love, hate and know do not appear in the list of medium-implicature verbs.

So, in summary, we can say that the sentences in your Example 1 are equivalent, but not those in 2, 3, and 4 because believe is a verb of medium subordinate negative implicature, but love, hate, know are not.

Solution 2:

The position of the negation is flexible only with certain verbs. 'Believe', in your first example, allows this sort of flexibility and both sentences of this example are essentially synonymous. The same would have been the case with 'appear' (It does not appear that bigfeet exists/It appears that bigfeet does not exist) or 'seem' (It seems that bigfeet does not exist/It does not seem that bigfeet exist).

The raising-to-subject verbs 'love', 'hate' and 'know' semantically do not allow flexibility in the position of the negation, and therefore the pairs of sentences in your examples 2, 3 and 4 have essentially different meanings.