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(1) Enlightenment is inherently desirable
(2) Self-Interested desires prevent us from attaining enlightenment
(3) In order to attain enlightenment, one must train oneself to give up self-interested desires
(4) One does not engage in deliberate action unless one desires the foreseen result of actions
I'll try to relate that to what the suttas say.
Enlightenment is inherently desirable
I'm not sure what "enlightenment" is -- i.e. what, in Buddhist doctrine, is translated as "enlightenment"?
Enlightenment in Buddhism [Wikipedia] suggests it's any of several terms:
If I try to summarise Buddhism in one word, it's not "enlightenment" but rather "dukkha-nirodha" i.e. "cessation of suffering".
Self-Interested desires prevent us from attaining enlightenment
I'm not sure it's useful to use "self" in language.
In particular it's "wrong view" -- a thicket -- to say for example "self exists" and/or "self doesn't exist".
If you need to use that language then the ideal is "practice which benefits both self and others (AN 4.95)".
I find it clearer to remove "self" from the sentence, so instead of seeing something as "my suffering" just see it as "suffering".
So the sentence becomes, "desires prevent us from attaining enlightenment".
And in reply to that, Buddhism distinguishes between wholesome and unwholesome desires -- i.e. taṇhā (which is referenced in the Second Noble Truth), and chanda which is defined for example here or in a bit more detail here, the point being that chanda (one type of desire) isn't necessarily a bad thing.
You may be right though, that if you don't know what enlightenment is, and don't know how to go about it, and want it anyway, then that "wanting" and so on may be the unwholesome form of desire, i.e. an example of the kind of thing that actually causes suffering and delusion.
I'm not sure how items 3 and 4 in the OP were meant to add to the alleged paradox, so I won't address them directly -- they seem to be more of 2, i.e. talking about "self-interested" and "desire".
You might be interested in the Brahmana Sutta (SN 51.15) which answers someone's saying that "it's impossible that one could abandon desire by means of desire" (using the word chanda for desire). The answer is that you desire e.g. to go to the park (I suspect that "park" there is a reference to "forest monastery" i.e. where the monks are), and so you go (i.e. do that), and when you get there then the corresponding desire is allayed (i.e. paṭippassaddha which I think of as a nirodha again).
I guess that, in summary, nirodha isn't impossible. Maybe you have to recognise what it is that sustains (perpetuates) dukkha, and not attach to that.
"There’s a passage where Ven. Ananda is being questioned by a brahman. The brahman has come to see Ananda in the park where he’s staying and asks him, “What is the purpose of this path you follow?” And Ananda says, “One of the purposes is to put an end to desire.” The brahman asks, “How do you do that?” Then Ananda explains a set of teachings that are called the bases for success: concentration based on desire, concentration based on persistence, concentration based on intent, and concentration based on your powers of analysis.
The brahman says, “That’s impossible. How can you use desire to get rid of desire?” And Ananda asks him, “Before you came to the park, did you have desire to come to the park?” “Well, yes.” “Now that you’re here, where is the desire?” “It’s gone.” “In the same way, you use desire to put an end to desire — if you do it right.”
~ Thanissaro Bhikkhu "The Desire to End All Desires" https://www.dhammatalks.org/Archive/Writings/Transcripts/190314_The_Desire_to_End_All_Desires.pdf
There is no basis, other than bad practice, to assume that 4 creates a logical problem. I say “bad practice” to indicate when a practitioner engages in seeking enlightenment for themselves. The correct practice is engaged in when the goal is to benefit all sentient beings. This is both a desirable goal for oneself along the way, as well as a benefit along the way for others. Its desirability gives motivation to practice to help others. And helping others by overcoming ones ego-centric focus enables success.
you make a fine and subtle point. this is also known as the paradox of becoming and there is actually a book titled the same.
the thing about enlightenment that is strange - at least as far as I understand it - is that it's part theory and part practice. The theory part is more or less taken care of - in the form of teachings, discourse, 8-fold path, Buddhism SE, etc. The realization of that is what is missing. Nobody else can do that for you, but fortunately the groundwork is already there. That doesn't make it easy, but it's better than being clueless.
You're conflating the Pali tanha with the English word desire. Tanha is better translated as thirst. It's an unthinking and karmically originated compulsion to do something or an adverse reaction prompting us to avoid something else. It has no relation to the noble intention to seek enlightenment. You can have tanha for enlightenment, but that path doesn't lead to enlightenment. Tanha has to be replaced by adhitthana which means resolution and determination. These are vastly different things.
