Does the doctrine of no soul is means that there is nothing permanent to continue? So is the teaching of reincarnation, rebirth misleading as there is a notion there is a continuity after death? Also death and birth is visible at the physical level, [but the rebirth process is not. What is the implication on this in the doctrine?]
I agree with you.
The Pali suttas do not refer to a continuous consciousness at the termination of life.
Instead, the suttas explain 'birth' ('jati') & 'death' ('marana') refer to the birth & death of 'self-views'.
Therefore, 're-birth' refers to the re-arisings of 'self-view'.
For example, you make $100,000. Re-birth is the happy view: "I made $100,000; I am rich". You lose $100. Death is the unhappiness of 'I lost $100; I am poor".
I think you're asking, partly, about rebirth -- I asked a question similar to that here, you might find the answers to that question helpful.
Also, various people posted elsewhere that the suttas don't tend to talk about "rebirth"; they just talk about "birth" -- for example, in this answer:
The Buddha never, afaik, used a term that could be translated as "rebirth". In fact, the idea of anything being reborn goes against orthodox early Buddhist teachings. Throughout the Buddha's teachings, it is made clear that at the breakup of the body there is birth, not rebirth - as in birth of new things, not the return of anything old.
But your current question is mostly asking about "the doctrine of no soul", rather than asking about rebirth. If you're asking "what is the doctrine of no soul?" I think that the most famous sutta on that subject is The Anatta-lakkhana Sutta. I find that sutta difficult to understand though (possibly because it's refuting a "doctrine of soul" or Atman that I wasn't really familiar with in the first place).
Instead I found the clearest explanation of the doctrine was in the Alagaddupama Sutta, which I talked about a bit in this answer.
In brief I think that the "doctrine of no soul" might be summarized as, "there is no satisfactory doctrine of soul".
You ask, as well, "what is the implication of this doctrine?"
One of the (mistaken) implications might be that if there's no rebirth then there's no heaven and no hell, and no reason to be ethical. Buddhism warns that this would be "Wrong view":
And what is wrong view?
'There is nothing given, nothing offered, nothing sacrificed. There is no fruit or result of good or bad actions. There is no this world, no next world, no mother, no father, no spontaneously reborn beings; no contemplatives or brahmans who, faring rightly & practicing rightly, proclaim this world & the next after having directly known & realized it for themselves.'
This is wrong view.
So the doctrine may appear to contradict, or be related somehow, to e.g. doctrine about karma.
Another implication of the doctrine is the concept of "self" (e.g. here the title of the The Anatta-lakkhana Sutta is translated as On the No-self Characteristic).
Also, some (later) schools of Buddhism try to explain what you might have been asking about (i.e. explaining the workings of karma), for example Wikipedia on store-house consciousness says,
The store-house consciousness accumulates all potential energy for the mental (mana) and physical (rupa) manifestation of one's existence (namarupa). It is the storehouse-consciousness which induces transmigration or rebirth, causing the origination of a new existence.
(See also What is storehouse consciousness?).
Another sutta that you may find be relevant to this topic is the Mahatanhasankhaya Sutta.
Incidentally the translator's (Thanissaro Bhikkhu's) introduction to that sutta says,
Although the Buddha never used any word corresponding to "rebirth" in his teachings, he did describe birth as a process following on death again and again as long as the appropriate conditions are present.
... which more-or-less agrees with Ven. Yuttadhammo's answer which I quoted above.
And it says,
However, a being — in the Buddha's sense of the term — not only takes birth after the death of the body, it can also take birth, die, and be reborn many times in the course of a day — as attachment develops for one desire, ends, and then develops for another desire. This is why the processes leading to rebirth after death can be observed and redirected in the present moment during life.
... which more-or-less agrees with Dhammadhatu's answer answer to the question.
Anyway, the Mahatanhasankhaya Sutta starts with one of the monks saying,
As I understand the Dhamma taught by the Blessed One, it is just this consciousness that runs and wanders on [from birth to birth], not another.
The other monks see that as a pernicious view, and bring him to the Buddha, who corrects him, saying,
Haven't I, in many ways, said of dependently co-arisen consciousness, 'Apart from a requisite condition, there is no coming-into-play of consciousness'?
I think that maybe the theory of "dependent origination" or "dependent co-arising" may be related to anatta too: there's no such thing as an independent self (nor an eternal self), instead things like consciousness arise and end together with feelings, contact, sense-objects, etc.; and there are different types of consciousness associated with different senses.
Dharmafarer's introduction to this suttas starts with,
The average ordinary person is very likely to view things as wholes and entities, even as being fixed and eternal. For example, he may think that there is a permanent “self,” “soul,” “person” or “being” when what he is really experiencing is only a series of mental events comprising a super-rapid series of discrete thought-moments.
Later in the introduction it says,
The bhavanga (lit “existence-factor”) or “life-continuum” is a concept that evolved primarily in the Abhidhamma commentarial tradition to explain the continuity of consciousness and personal identity in the absence of a permanent self (which is denied by the anatta doctrine). The life-continuum flows on like a stream (sota) from one existence to the next. It is sometimes called bhavanga,citta (existence-factor consciousness) or “consciousness continuum” and is the foundation of all experience, both conscious and unconscious.
A simple answer might be found in the Sabbasava Sutta, which warns that there are questions:
Was I in the past? Was I not in the past? What was I in the past? How was I in the past? Having been what, what was I in the past? Shall I be in the future? etc. etc.
... and views:
he view I have a self arises in him as true & established, or the view I have no self... or the view It is precisely by means of self that I perceive self... or the view It is precisely by means of self that I perceive not-self etc. etc.
... and warns that these kinds of thinking do not lead to freedom:
This is called a thicket of views, a wilderness of views, a contortion of views, a writhing of views, a fetter of views. Bound by a fetter of views, the uninstructed run-of-the-mill person is not freed from birth, aging, & death, from sorrow, lamentation, pain, distress, & despair. He is not freed, I tell you, from suffering & stress.
... which maybe leads back to my original summary of anatta: that "there is no satisfactory doctrine of soul".
There is rebirth in Buddhism, for sure, but rebirth of what?
When we say "rebirth", we mean that something that we can peg onto as YOU, is born again after physical death. But what is YOU? The soul or self or the SAME consciousness that wanders on throughout this lifetime?
The Buddha taught that all phenomena is not self. The Buddha also taught that all conditioned and/or compounded things (including consciousness) is impermanent. It is not the SAME consciousness that wanders from childhood to old age. It changes. (See MN 38)
So, in Buddhism, there is no rebirth of the self or consciousness.
In fact, if a person injures his brain or experiences dementia or amnesia or coma, then his personality, his memories and even his consciousness could be impaired or non-existent for the remainder of his physical life. In this case, even in his lifetime, we cannot guarantee that he is totally present at every moment. Then, what more for the next lifetime?
So, if a soul or self or consciousness is not reborn, then what is reborn?
There is definitely rebirth in Buddhism, but what is reborn?
- Suffering. The continuity of suffering, regardless of physical death, is rebirth.
- The continuity of conditioned processes when the condition for them to exist continues, is also rebirth. For e.g. karmic cause and effect, and its relationship to an individual's perceived continued existence in this lifetime. Also, the physical and mental processes of the five aggregates, for e.g. the "stream" of consciousness (MN 38) and the thought "I am the thinker" (Snp 4.14).
The Theravada perspective is that at every moment there is death and rebirth. In between, we have the continuity of conditioned processes, that bridges every moment. With the continuity of conditioned processes, we have the continuity of suffering, so long as ignorance is present.
When ignorance is permanently ended, craving is permanently ended and suffering is permanently ended. This permanent ending of ignorance is referred to as liberation in Buddhism.
Usually when people talk about rebirth, they imply that the SAME consciousness wanders on in this lifetime and onto the next lifetime. In the following quote from this answer, we read an excellent explanation by Damith, of why this is not true.
Stream of consciousness OR Continuity of consciousness means individual consciousness which arises (uppada), exists (titi), ceases (bhanga) rapidly as an effect of many causes. It's just a stream, not a long lasting single one.
Think of a stream of water. When you look at a single point in the stream, you see many water droplets appear, exist, and disappear. Take a single droplet out of it which passed by(*) that point. If you consider the past, present, and future of that droplet, you'll see a stream. In reality, it's not the same droplet nor another droplet you see in the past, present, and future. Stream of consciousness is analogous to this example. Please try to understand it in that way.
(*) Note: "passed by" - This phrasal verb can be only used when we talk in conventional truth. The reader may think that rūpa (form/matter) can move in space which is not a true in ultimate reality. Neither any nāma (name) nor rūpa can move. What happens is the arose rūpa existed and ceased then and there. After that new rūpa arise nearby. This is called "Deshāntarotpatti".
I'm not saying that there is no YOU (self) at all. There is a YOU, but it's not a permanent YOU. It's changing from moment to moment, depending on conditions, and compounded by different mental and physical processes.