user23662: 
In the same way, my friend, purity in terms of virtue is simply for the sake of purity in terms of mind. Purity in terms of mind is simply for the sake of purity in terms of view. Purity in terms of view is simply for the sake of purity in terms of the overcoming of perplexity. Purity in terms of the overcoming of perplexity is simply for the sake of purity in terms of knowledge & vision of what is & is not the path. Purity in terms of knowledge & vision of what is & is not the path is simply for the sake of purity in terms of knowledge & vision of the way. Purity in terms of knowledge & vision of the way is simply for the sake of purity in terms of knowledge & vision. Purity in terms of knowledge & vision is simply for the sake of total Unbinding through lack of clinging.
MN 24

Why is the purification of by overcoming doubt i.e purification leading to Stream Entry not already the doing the next two of the purifications subsequent to it? (excluding the purification leading to Nibāna without clinging). Why are those other two purifications needed before stream entry? Surely if one has purified any one of these three it would imply the the other two?

user8157: AN 10.13 says:

What are the five lower fetters? Identity view, doubt (vicikicchā),
misapprehension of precepts and observances, sensual desire and ill
will.

MN 24 uses the word "kaṅkhā" (translated in the question as "perplexity"), which, according to the dictionary, can mean either:

doubt; uncertainty; or

wish, desire, longing (per SN 7.18).


Regardless of the meaning of "kaṅkhā", SN 24.1 says a person that has given up "kaṅkhā" is called a 'stream-enterer', as follows:

When a noble disciple has given up doubt in these six cases, and has
given up doubt in suffering, its origin, its cessation, and the
practice that leads to its cessation,
Yato kho, bhikkhave, ariyasāvakassa imesu ca ṭhānesu kaṅkhā pahīnā
hoti, dukkhepissa kaṅkhā pahīnā hoti, dukkhasamudayepissa kaṅkhā
pahīnā hoti, dukkhanirodhepissa kaṅkhā pahīnā hoti,
dukkhanirodhagāminiyā paṭipadāyapissa kaṅkhā pahīnā hoti—
they’re called a noble disciple who is a stream-enterer, not liable to
be reborn in the underworld, bound for awakening.
ayaṁ vuccati, bhikkhave, ariyasāvako sotāpanno avinipātadhammo niyato
sambodhiparāyano”ti

Therefore, based on SN 24.1, there appears no distinction between the fetter of vicikicchā & kaṅkhā.
Bhikkhu Bodhi offers footnote 288 in his Majjhima Nikaya explaining these 7 stages from the Visuddhimagga (which I personally do not agree with). Bhikkhu Bodhi does however appear to caste some skepticism upon MN 24 by saying this teaching is found nowhere else in the suttas except for in DN iii 28 (which adds two more stages).
Therefore, I can only offer my personal explanation/interpretation of MN 24, which is:

Purification of view (diṭṭhivisuddhi) is predominately intellectual, such  as intellectually comprehending the teachings.

The above intellectual comprehension gives rise to faith & purifies the mind from doubt (kaṅkhāvitaraṇavisuddhatthā). Due to the absence of doubt, there is the faith or confidence to attempt to practise the path.

The above practise of the above learned teachings gives rise to knowledge & vision (ñāṇadassana) of the path, which is direct insight rather than intellectual acceptance. In other words, there is the knowledge & vision that whenever the mind is free from craving, attachment & self-view, the mind is free from suffering & tastes The Deathless.

The above purity of knowledge & vision of the path (maggāmaggañāṇadassanavisuddhatthā) gives rise to knowledge & vision of what the path is and what the path is not. Thus, even though this practitioner may develop  samatha & bliss, they know samatha & bliss is not the path. Instead, what is the path is the continuous prevention & dropping of any craving, attachment & self-view and ultimately dispassion (viraga).

Therefore, by knowing with direct insight what is the true path and what are the potential pitfalls of a wrong path is , the practitioner purifies their knowledge & vision of the true path.



23662: <blockquote>
<p>In the same way, my friend, purity in terms of virtue is simply for the sake of purity in terms of mind. Purity in terms of mind is simply for the sake of purity in terms of view. Purity in terms of view is simply for the sake of purity in terms of the overcoming of perplexity. Purity in terms of the overcoming of perplexity is simply for the sake of purity in terms of knowledge &amp; vision of what is &amp; is not the path. Purity in terms of knowledge &amp; vision of what is &amp; is not the path is simply for the sake of purity in terms of knowledge &amp; vision of the way. Purity in terms of knowledge &amp; vision of the way is simply for the sake of purity in terms of knowledge &amp; vision. Purity in terms of knowledge &amp; vision is simply for the sake of total Unbinding through lack of clinging.<br />
<a href="https://www.accesstoinsight.org/tipitaka/mn/mn.024.than.html" rel="nofollow noreferrer">MN 24</a></p>
</blockquote>
<p>Why is the purification of by overcoming doubt i.e purification leading to Stream Entry not already the doing the next two of the purifications subsequent to it? (excluding the purification leading to Nibāna without clinging). Why are those other two purifications needed before stream entry? Surely if one has purified any one of these three it would imply the the other two?</p>
 8157: <p>AN 10.13 says:</p>
<blockquote>
<p>What are the five lower fetters? Identity view, <strong>doubt</strong> (<strong>vicikicchā</strong>),
misapprehension of precepts and observances, sensual desire and ill
will.</p>
</blockquote>
<p>MN 24 uses the word &quot;<a href="https://suttacentral.net/define/ka%E1%B9%85kh%C4%81" rel="nofollow noreferrer">kaṅkhā</a>&quot; (translated in the question as &quot;perplexity&quot;), which, according to the dictionary, can mean either:</p>
<ol>
<li><p>doubt; uncertainty; or</p>
</li>
<li><p>wish, desire, longing (per <a href="https://suttacentral.net/sn7.18/en/sujato?layout=linebyline&reference=pts&notes=asterisk&highlight=true&script=latin#2.1" rel="nofollow noreferrer">SN 7.18</a>).</p>
</li>
</ol>
<p>Regardless of the meaning of &quot;<a href="https://suttacentral.net/define/ka%E1%B9%85kh%C4%81" rel="nofollow noreferrer">kaṅkhā</a>&quot;, SN 24.1 says a person that has given up &quot;<a href="https://suttacentral.net/define/ka%E1%B9%85kh%C4%81" rel="nofollow noreferrer">kaṅkhā</a>&quot; is called a 'stream-enterer', as follows:</p>
<blockquote>
<p>When a noble disciple has given up <strong>doubt</strong> in these six cases, and has
given up doubt in suffering, its origin, its cessation, and the
practice that leads to its cessation,</p>
<p>Yato kho, bhikkhave, ariyasāvakassa imesu ca ṭhānesu <strong>kaṅkhā</strong> pahīnā
hoti, dukkhepissa kaṅkhā pahīnā hoti, dukkhasamudayepissa kaṅkhā
pahīnā hoti, dukkhanirodhepissa kaṅkhā pahīnā hoti,
dukkhanirodhagāminiyā paṭipadāyapissa kaṅkhā pahīnā hoti—</p>
<p>they’re called a noble disciple who is a stream-enterer, not liable to
be reborn in the underworld, bound for awakening.</p>
<p>ayaṁ vuccati, bhikkhave, ariyasāvako sotāpanno avinipātadhammo niyato
sambodhiparāyano”ti</p>
</blockquote>
<p>Therefore, based on SN 24.1, there appears no distinction between the <em>fetter of vicikicchā</em> &amp; <em>kaṅkhā</em>.</p>
<p>Bhikkhu Bodhi offers footnote 288 in his Majjhima Nikaya explaining these 7 stages from the Visuddhimagga (which I personally do not agree with). Bhikkhu Bodhi does however appear to caste some skepticism upon MN 24 by saying this teaching is found nowhere else in the suttas except for in DN iii 28 (which adds two more stages).</p>
<p>Therefore, I can only offer my personal explanation/interpretation of MN 24, which is:</p>
<ol>
<li><p>Purification of view (diṭṭhivisuddhi) is predominately intellectual, such  as intellectually comprehending the teachings.</p>
</li>
<li><p>The above intellectual comprehension gives rise to faith &amp; purifies the mind from doubt (<strong>kaṅkhā</strong>vitaraṇa<strong>visuddha</strong>tthā). Due to the absence of doubt, there is the faith or confidence to attempt to practise the path.</p>
</li>
<li><p>The above practise of the above learned teachings gives rise to knowledge &amp; vision (ñāṇadassana) of the path, which is direct insight rather than intellectual acceptance. In other words, there is the knowledge &amp; vision that whenever the mind is free from craving, attachment &amp; self-view, the mind is free from suffering &amp; tastes The Deathless.</p>
</li>
<li><p>The above purity of knowledge &amp; vision of the path (maggāmaggañāṇadassanavisuddhatthā) gives rise to knowledge &amp; vision of what the path is and what the path is not. Thus, even though this practitioner may develop  samatha &amp; bliss, they know samatha &amp; bliss is not the path. Instead, what is the path is the continuous prevention &amp; dropping of any craving, attachment &amp; self-view and ultimately <em>dispassion</em> (<em>viraga</em>).</p>
</li>
<li><p>Therefore, by knowing with direct insight what is the true path and what are the potential pitfalls of a wrong path is , the practitioner purifies their knowledge &amp; vision of the true path.</p>
</li>
</ol>
