The question assumes Muhammad (saw) was not a prophet of mercy as he murdered his critics. Secondly, it assumes Muhammad (saw) was a deceiver who ordered his followers to deceive and kill Ka'b in cold-blooded murder.
Both of these assumptions are incorrect as I explain in this answer.
Social and Historical Background
The poet Ka'b ibn al-Ashraf was a Jew who lived in Medinah, a member of the tribe of Nabhan. Ka'b was the leader of Banu al-Nadir.
When the Prophet (saw) arrived at Yathrib (later known as Medinah) the first Muslim state was established. Islam was not a personal faith, rather a way of life underpinned by a state; Muhammad (saw) was not just a Prophet but also a leader and ruler of the state. As the Prophet (saw) was a representative of a recognized state, anyone fighting him was initiating war with that state.
He, as the leader of the state, made a treaty with the tribes of Yathrib (as small neighbouring states), including the Jewish tribe of Banu al-Nadir. The treaty included all the different tribes and groups at Yathrib, including Muslims and Jews, to constitute one nation while ensuring the freedom of religious beliefs and practices for all citizens of the new nation, protection was required for both parties, alliances with enemies were prohibited for either party. The one who violated the treaty could suffer capital punishment for treason.
The main articles of this treaty comprised:
ولَا يُنْصَرُ كَافِرٌ عَلَى مُؤْمِنٍ ... وَإِنَّهُ مَنْتَبِعَنَا مَنْيَهُودَ، فَإِنَّلَهُ النَّصْرَوَالْأُسْوَةَ ... وَإِنَّهُ لَا يُجِيرُ مُشْرِكٌ مَالًا لقريش وَلَا نفسا، ... (ولا يحل) أَنْيَنْصُرَ مُحْدِثًا وَلَا يُؤْوِيهِ... وَإِنّ الْيَهُودَ يُنْفِقُونَ مَعَ الْمُؤْمِنِينَ مَا دَامُوا مُحَارَبِينَ ... وَإِنَّ عَلَى الْيَهُودِ نَفَقَتَهُمْ وَعَلَى الْمُسْلِمِينَ نَفَقَتَهُمْ، إِلَّا مَنْ ظَلَمَ وَأَثِمَ فَإِنَّهُ لَا يَوْتَغُإِلَّا نَفْسَهُ وَأَهْلَ بَيْتِهِ. ... وَإِنَّ بَيْنَهُمُ النَّصْرَ عَلَى مَنْ حَارَبَ أهل هَذِهِ الصَّحِيفَةِ، ... وَإِنَّهُ لَا تُجار قُرَيْشٌ وَلَا مَنْ نَصَرَهَا وَإِنَّ بَيْنَهُمْ النَّصْرَ عَلَى مَنْ دَهَمَ يَثْرِبَ
"Whoever follows us from the Jews, we will provide him with support and victory …
It is not allowed to financially or personally co-operate with Quraysh …
It is not allowed to give support or shelter to enemies …
Jews should spend with believers as long as they afford war …
Whoever oppresses (betrayed) or does wrong will be destroyed with his family …
Jews cover their financial expenses and Muslims afford their financial expenses ...
They are to give support to one another when anyone fights against the allies of this treaty...
No protection is to be given to Quraysh nor any of its allies. They (Jews and Muslims) are to co-fight against anyone that attacks Yathrib (Medina)." (Ibn Kathir, al-Bidayah wa an-Nihayah, Vol. 4, pp. 554-558, asunnan al-Kubra, Sirat Ibn Ishaq)
Whilst we see Muslims fulfilled the agreed articles, the Jewish tribes -including Ka'b- violated them and so deserve punishment.
For example, when Quraysh attacked Medinah at Badr in the year 2 AH, the Muslims confronted the invading army alone despite being less than a third of their enemy. The Muslim army crushed the invaders, increasing the spite of the onlooking Jewish elites. Ka'b ibn al-Ashraf, a party of the Medinah treaty, began using his poetry to lament Quraysh's defeat and to openly incite hatred and enmity against Islam and Muslims.
Poetry during that time in Arabia exerted the same influence over peoples akin to contemporary media. People could start a war lasting for decades over a stanza composed against them. It was the most influential political activity that existed.
Not only did Ka'b compose satirical poems against Muhammad (saw) and his religion, he also went to Mecca and openly joined the enemies of Islam. He incited Quraysh to attack Medinah and swore in the Ka'bah he would fight the Muslims when Medinah was invaded.
Not content with this, after he returned to Medinah, he planned to murder the Prophet by underhand means.
Did the Prophet (saw) order to kill Ka'b only for criticising Him?
The answer is 'No'.
Once we read the entire story from various early sources, we are acquainted Ka'b was not killed for just criticising the Prophet (saw) or Muslims, but rather he was executed because of a number of major capital crimes.
The historiographical sources give many reasons behind the death of Ka'b other than the reason of insulting the Prophet (saw) or Muslim women.
1. Ka'b ibn al-Ashraf: A Traitor and Warmonger:
Al-Baghawi reported on the authority of Al-Kalbi – Abu Salih – Ibn Abbas:
وكَانَ بَيْنَهُمْ وَبَيْنَ رسول اللَّه صلى اللَّه عليه وسلم، عَهْدٌ إِلَى مُدَّةٍ، فَنَقَضُوا ذَلِكَ الْعَهْدَ، وَانْطَلَقَ كَعْبُ بْنُ الْأَشْرَفِ فِي سِتِّينَ رَاكِبًا إِلَى أَهْلِ مَكَّةَ: أَبِي سُفْيَانَ وَأَصْحَابِهِ، فَوَافَقُوهُمْ، وَأَجْمَعُوا أَمْرَهُمْ، وَقَالُوا: لَتَكُونَنَّ كَلِمَتُنَا وَاحِدَةً. ثُمَّ رَجَعُوا إِلَى الْمَدِينَةِ.
"… As there had been a treaty been stipulated between the Messenger of Allah, the Jews (Banu Nadir) broke the treaty by Ka'b Ibn al-Ashraf going to Makkah, accompanied with sixty horsemen, in order to incite the Makkan leaders until they agreed to join their forces to fight the Messenger of Allah." (Tafsir al-Baghawi, commentary on Surah 3:12)
Ka'b composed verses showing enmity and hatred against Muslims and lamented the defeat of Quraysh at Badr. Among these verses were:
طَحَنَتْ رَحَى بَدْرٍ لِمَهْلِكِ أَهْلِهِ ... وَلِمِثْلِ بَدْرٍ تَسْتَهِلُّ وَتَدْمَعُ
قُتِلَتْ سَرَاةُ النَّاسِ حَوْلَ حِيَاضِهِمْ ... لَا تَبْعَدُوا إنَّ الْمُلُوكَ تُصَرَّعُ
وَيَقُولُ أَقْوَامٌ أُسَرُّ بِسَخَطِهِمْ ... إنَّ ابْنَ الْأَشْرَفِ ظَلَّ كَعْبًا يَجْزَعُ
نُبِّئْتُ أَنَّ بَنِي الْمُغِيرَةِ كُلَّهُمْ ... خَشَعُوا لِقَتْلِ أَبِي الْحَكِيمِ وَجُدِّعُوا
وَابْنَا رَبِيعَةَ عِنْدَهُ وَمُنَبِّهٌ ... مَا نَالَ مِثْلَ الْمُهْلِكِينَ وَتُبَّعُ
نُبِّئْتُ أَنَّ الْحَارِثَ بْنَ هِشَامِهِمْ ... فِي النَّاسِ يَبْنِي الصَّالِحَاتِ وَيَجْمَعُ
لِيَزُورَ يَثْرِبَ بِالْجُمُوعِ وَإِنَّمَا ... يَحْمَى عَلَى الْحَسَبِ الْكَرِيمُ الْأَرْوَعُ
Badr's mill ground out the blood of its people..At events like Badr you should weep and cry
The best of the people were slain round their cisterns... Don't think it strange that the princes were left lying.
Some people whose anger pleases me say "Ka'b bin al-Ashraf is utterly dejected".
I was told that all the Banu'l-Mughira were humiliated... And brought low by the death of Abu'l-Hakim
And the two sons of Rabi'a with him, and Munabbih and the others did not attain (such honour) as those who were slain
I was told that al-Harith ibn Hisham is doing well and gathering troops to visit Yathrib with armies,
For only the noble, handsome man protects the loftiest reputation.
(Ibn Ishaq, Sirat Rasul Allah)
Ka'b's poems incited hatred and violence against Muslims. He urged the Arab tribes to take revenge and fight Muhammad.
Also, al-Wahidi reported after the Battle of Uhud, Ka'b Ibn al-Ashraf went to Mecca, accompanied by seventy riders, to ally themselves with Quraysh against the Messenger of Allah and break the covenant which they had with Allah's Messenger.
Ka'b then went to the Ka'bah and clung to it and promised he would do his best to fight Muhammad. (Al-Wahidi, Asbab al-Nuzul, Commentary on Surah 4:51)
2. The assassination of the Prophet (saw):
Ibn Hajar while commenting on the killing of Ka'b in Sahih al-Bukhari, he reported on the authority of 'Ikrima that Ka'b invited the Prophet (saw) to feast and plotted with his men to murder him when he came. (Ibn Hajar, Fath al-Baal-Bari fi Sharh Sahih al-Bukhari)
So, Ka'b undertook different capital crimes that made him an outlaw; the violation of the treaty he signed as he joined the enemies and inciting Arabs against Muslims; weaponising poetry against the Muslim state, thereby becoming an outspoken enemy of an ongoing war against Islam. Not only that, he also plotted to treacherously murder the Prophet (saw).
All these activities were in clear violation of the treaty, necessitating capital punishment for treason against the state of Madinah i.e., "Whoever oppresses (betrayed) or does wrong will be destroyed with his family."
Did the Prophet deceive Ka'b?
Another assumption made about the execution of Ka'b is the Prophet (saw) approved Maslamah to lie and then get Ka'b killed.
At first glance, it seems a valid assumption, however, by examining the whole story and considering the context of the events, there was no injustice in killing Ka'b in this way and these means were not wrong.
Ka'b Ibn Ashraf plotted to murder the Prophet (saw) and made his effort to incite war against the Muslims state. Muhammad (saw) as the political leader of the new state ordered the death of K'ab. Although there is no clear evidence in the hadith to suggest the Prophet (saw) gave any directions as to the manner in which that sentence was to be carried out, there was no other method to which resort could be had under such circumstances.
In a battlefield and war it is all about intelligence. So, by no means can this assassination be deemed a negative deception.
Muhammad ibn Maslamah, the leader chosen by the Prophet (saw) to carry out the sentence, chose the best and most effective method under such circumstances to avoid bloodshed. If they had attacked Ka'b openly, there would have been more bloodshed and the whole Jewish tribe of Bani Nadir would probably have suffered along with Ka'b.
Abu Hanifah (ra) treated sabb al-rasul on part of a believer as a matter of apostasy and accepted the person's repentance and he held that non-Muslims who insult the Prophet (saw) "are not to be killed, because their disbelief is worse".
Besides, the Prophet (saw) did pardon some poets who insulted him such as Zuhair ibn Abi Salma, Abdullah ibn al-Zab'ari, Hubayrah ibn Wahb al-Maghzumi and others. Among them, some who later embraced Islam whilst others died in disbelief.
So the case is for the political leader to decide the suitable solution for a problem the state faces.
A critic may judge the case of Ka'b's execution by the phonology of the twenty-first century underpinned by liberal values as opposed to divine values, however they would be mistaken, as divine values are privileged over liberal values. Fourteen centuries ago, such methods of execution were normal and unobjectionable. Even in Europe and the US, treason till attract capital punishments and assassinations of proven enemies of the state are undertaken often without comment.
Ka'b ibn al-Ashraf was not killed for just criticising or insulting the Prophet (saw) or the Muslim women as claimed by some writers.
Rather he was executed for capital crimes including the violation the treaty and high treason against the state of Medinah by inciting the Qurayshite enemies against Muslims. The Prophet (saw), the political leader of Medina, gave him the appropriate sentence.
Concerning the way how Ka'b was killed, although the Prophet (saw) did not give any direction on how to carry out the sentence, this was the best and most effective method under such circumstances in order to avoid additional bloodshed.
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