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Islamic researcher, graduated from Al-Azhar University, Islamic Studies in the English Language. I also studied at Temple University in the US.
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In a Nutshell:
Fard Ayn is an obligation prescribed on each and every individual (such as the daily prayer, fasting,  … etc.) that must be performed by oneself. Fard Kifayah is a collective obligation upon all the Ummah (according to each one's capacity) to perform it (such as jihad, janaza prayer and burial, building mosques, building hospitals, undertaking ijtihad … etc.) and if this obligation was dismissed, all the Ummah will be considered as sinful (each one according to his capacity).
Sometimes this Fard Kifayah becomes a Fard Ayn upon a certain person or a group if he is the only one capable of executing the obligation, if he is appointed by the ruler or if he initiated it.
Background

All of the ahkam taklifiyyah (rules) in the Shari'ah fall under one of five categories: Fard (obligatory), Mandoob (recommended), Mubah (permissible), Makruh (disliked) and Haram (forbidden).

The ahkam (rules) relating to duties can be classified into two categories: Ayn (individual obligation) and Kifayah (collective obligation).

Fard Kifayah and Fard Ayn

Fard Ayn is the obligation prescribed upon each and every individual, such as the prayer, fast, hajj, zakat …etc. So, every Muslim has to pray and no one could legally, as a general rule, relieve him of this obligation by performing it on his behalf (so long he is alive and able). (Asnawi, al-Tamheed, Vol. 1, p. 74, Kudari, Usul al-Fiqh, p. 41)

A more comprehensive juristic definition of this duty is defined as:

فَالْوَاجِبُ الْعَيْنِيِ : هُوَ مَا طَلَبَ الشَّارِعُ فِعْلُهُ مِنْ فَرُدٍ مِنْ كُلِ أَفْرَادِ الْمُكَلَّفِينَ ، وَلَا يُجْزِئُ قِيَامُ مُكَلَّفٌ بِهِ عَنْ آخِرِ كَالْصَّلاَةِ وَالزَّكاةِ وَالْحَجِّ وَالْوَفَاءِ بِالْعُقُودِ

"Fard Ayn: is what the legislator (Allah) orders every (mukalaf) individual to do and it is not legally sufficient another individual does it in his place, such as the prayer, zakat, hajj, fulfilment of promises (or contracts)." (Khalaf, Ilm Usul al-Fiqh, p. 104)
On the contrary, Fard Kifayah is an obligation prescribed on all those legally responsible collectively and if some performed it, the rest would be relieved from it; when these obligations are left unestablished, all (according to the capacity of each) would be considered as sinful. Examples would include jihad al-talab, establishing the Islamic state and community, ghusl of the dead Muslim, prayer and the burying of the dead, building mosques/hospitals/courts/bayt al-mal, ensuring ijtihad is done, preservation of Qur'an and the Shari'ah … etc. (Shafi'i, al-Umm, Vol. 1, p. 312, Ibn Qudamah, al-Mughni, Vol. 9, p. 162, Zarkashi, al-Bahr al-Muheet, Vol. 1, p. 242, Aal ibn Taymiyah, al-Muswadah, pp. 30-31)

The definition was later put in a comprehensive form:

وَالْوَاجِبُ الْكِفَائِيِ : هُوَ مَا طَلَبَ الشَّارِعُ فِعْلُهُ مِنْ مَجْمُوعِ الْمُكَلَّفِينَ ، لَا مِنْ كُلِّ فَرَدٍ مِنْهُمْ، بِحَيْثُ إِذَا قَامَ بِهِ بَعْضُ الْمُكَلَّفِينَ فَقَدْ أَدَّى الْوَاجِبَ وَسَقَطَ الْإِثْمَ وَالْحَرِجَ عَنِ الباقين ، وَإِذَا لَمْ يَقُمْ بِهِ أَيُّ فَرَدٍ مِنْ أَفْرَادِ الْمُكَلَّفِينَ أُثِمُوا جَمِيعًا بِإهْمَالِ هَذَا الْوَاجِبَ

"The Kifayah obligation is what the legislator ordered a group of mukallafeen (those legally competent), not each individual, in a way that if done by some mukallafeen, the order is fulfilled and the sin or burden is relieved from the rest of them.
If no one established it from the mukallafeen, they all would be sinful because of the neglect of this obligation." (Khalaf, Ilm Usul al-Fiqh, p. 105)
The eighth-century Usuli scholar ibn al-Lahaam illustrated how the Kifayah differs from Ayn:

الْمَقْصُودُ فِي فَرْضِ الْكِفَايَةِ إيقَاعُ الْفِعْلِ مَعَ قَطْعِ النَّظَرِ عَنِ الْفَاعِلِ
"What is meant in Fard Kifayah is the performance of the act regardless of the doer (or actor)." (Ibn al-Lahaam, al-Qawa'id wa al-Fawa'id al-Usuliyah, p. 186)
So, Allah does not identify the person who is required to do it, rather the action that has to be done is specified.

Imam Ahmed gave an applicable example:

الْغَزْوُ وَاجِبٌ عَلَى النَّاسِ كُلُّهُمْ، فَإِذَا غَزَا بَعْضُهُمْ أَجْزأ عَنْهُمْ
"The invasion (Jihad al-Talab) is an obligation (wajib) upon all the people, (but) if some of them invaded, it is legally sufficient." (Ibn al-Najaar al-Hanbali, al-Kawkab al-Muneer, Vol. 1, p. 173)
The daily prayers are Fard Ayn, prescribed on each individual, but building mosques is Fard Kifayah. So, the former is an act that will benefit the individual personally (even religiously) but the latter helps in organising the community and facilitating the collective performance of the prayer. (Ibn al-Lahaam, al-Qawa'id wa al-Fawa'id al-Usuliyah, Vol. 1, p. 186)

Fard al-Kifayah also includes crafts, professions, industry and everything Muslims need that must be collectively made by them. (Ibn al-Najar al-Hanbali, Sharh al-Kawkab al-Muneer, Vol. 1, pp. 374-375)

Who is entitled to the undertake Fard al-Kifayah?

Scholars of Usul al-Fiqh disagree on who is entitled to perform the Fard al-Kifayah. Some scholars (such as al-Gazali, Ahmed, Ibn Qudamah, Ibn al-Hajib, al-Amidi, al-Isnawi and others) argue all from the Ummah are entitled to perform it and if some established it the others are relieved from accounting or punishment (al-Ithm), but if all of them left it, they all will be punished. (Al-Isnawi, Nihayat al-Sool, Vol. 1, pp. 40-118, Ibn al-Laham, al-Qawa'id wa al-Fawa'id, pp. 253-255)

The sixth-century leading Maliki jurist ibn al-Hajib noted:

الْوَاجِبُ عَلَى الْكِفَايَةِ عَلَى الْجَمِيع، وَيَسْقُطُ بِالْبَعْض،  إِثْمُ الْجَمِيعِ بِالتَّرْكِ بِاِتِّفَاق.
"The Kifayah obligation is prescribed upon all and is relieved when some do it. It is agreed that all of them will be sinful if they left it (unestablished)." (Ibn al-Hajib, Mukhtasar Muntaha al-Sool, p. 31, Asfahani, Sharh Mukhtasar ibn al-Hajib, Vol. 1, p. 339)
The other group of Usuliyeen (such as al-Shafi'i, Shatibi, Qarafi, Razi, Subki, Baydawi and others) argued only a group of them is addressed by the order and will be either rewarded or punished. This group of scholars argued only a group of people who are capable of establishing the hukm are entitled to undertake it collectively.

Imam Shatibi noted:

طُلِبَ الْكِفَايَةِ وَارِدٌ عَلَى الْبَعْضِ ... عَلَى مِنْ فِيه أَهْلِيَّةُ الْقِيَامِ بِذَلِكَ الْفِعْلِ الْمَطْلُوب لَا عَلَى الْجَمِيعِ عُمُومًا
"The Kifayah obligation is required from some … it depends on his qualification to do this required task and it is not upon all the masses." (Shatibi, al-Muwafaqat, Vol. 1, p. 278)
A similar notion was mentioned by Imam Shafi'i, the founder of the Shafi'i madhab, regarding if Muslims did not establish it:

وَلَوْ ضِيعُوهُ مَعًا، خِفْتُ أَلَا يَخْرُجُ وَاحِدٌ مِنْهُمْ مطيقٌ فِيه مِنَ المأثَمِ .
"If they neglected it altogether, I fear no one of them capable will be released from the sin." (Shafi'i, al-Risalah, p. 366)
A similar notion was also adopted by the seventh-century Usuli al-Qarafi. (Qarafi, Tanqeeh al-Fusul Fi Ilm al-Usul, p. 42)

Duty on all of the Ummah:

This group of scholars used a number of arguments to support their argument. They argued jihad is Fard Kifayah but Allah referred to all of the Ummah:

كُتِبَ عَلَيْكُمُ الْقِتَالُ
"Fighting has been enjoined upon you (alaikum)." (Qur'an 2:216)
They deduced from it that Allah refers to all the Ummah by the pronoun (alaikum – upon you), but this is improbable because not all the Ummah are required to fight as the children, women, elderly, disabled and others are not required to. So, the khitab (order) does not include all of them, but only included the group capable of fighting.

They also argued if all of Ummah would be sinful for leaving it, they all are entitled to do the action as Allah would not punish them for something they are not entitled to do. (Gazali, al-Mustasfa min Ilm al-Usul, Vol. 2, p. 93, Shinqeeti, Usul al-Fiqh, p. 353, al-Ijji, Sharh Mukhtasar al-Muntaha, Vol. 2, pp. 152-155)

 But this argument is also unstable because it is built on the conclusion deduced from the premises i.e. if we proved it is for all the ummah, all of them will be sinful but if not, only the group entitled with the issue will be accounted. So, the argument they presented in circular because the premises are the conclusion as well.

Duty on some of the Ummah:

They present a number of arguments to support their notion.  For example, they argued learning the shari'ah and teaching it to people is Fard Kifayah and not wajib on each individual as Allah said:

وَمَا كَانَ الْمُؤْمِنُونَ لِيَنفِرُوا كَافَّةً ۚ فَلَوْلَا نَفَرَ مِن كُلِّ فِرْقَةٍ مِّنْهُمْ طَائِفَةٌ لِّيَتَفَقَّهُوا فِي الدِّينِ وَلِيُنذِرُوا قَوْمَهُمْ إِذَا رَجَعُوا إِلَيْهِمْ لَعَلَّهُمْ يَحْذَرُونَ
"And it is not for the believers to go forth [to battle] all at once. For there should separate from every division of them a group [remaining] to obtain understanding in the religion and warn their people when they return to them that they might be cautious." (Qur'an 9:122)
So, the order (khitab) here is only for a specific group of people who are capable and enough to establish each mission i.e. some go to learn and teach and others go and fight.

 They also argued calling to Islam and Hisbah (enjoying the good and forbidding the evil) is a Fard Kifayah prescribed, in the Qur'an, only on a group of Muslims capable of it:

وَلْتَكُن مِّنكُمْ أُمَّةٌ يَدْعُونَ إِلَى الْخَيْرِ وَيَأْمُرُونَ بِالْمَعْرُوفِ وَيَنْهَوْنَ عَنِ الْمُنكَرِ ۚ وَأُولَٰئِكَ هُمُ الْمُفْلِحُونَ
"And let there be [arising] from you a nation inviting to [all that is] good, enjoining what is right and forbidding what is wrong, and those will be successful." (Qur'an 3:104)
So only a group of them is enough and entitled.

They also noted that Allah says:

لَا يُكَلِّفُ اللَّهُ نَفْسًا إِلَّا وُسْعَهَا
"Allah does not charge a soul except (with that which is within) its capacity." (Qur'an 2:286)
Therefore, how can an individual (like someone disabled) who has nothing to do with jihad being entitled and punished for leaving something they are not capable of? There is also no actual benefit or harm that would be removed by making him obliged with action beyond his capacity. (Shatibi, Vol. 1, p. 280, Ameer Badshah, Tayseer al-Tahreer, Vol. 2, p. 213-214)

They also argued there are many Fard Kifayahs (such as the Islamic judges, implementers of hudud, tax collectors, imams ... etc.) that could not be established by anyone, rather by a group of people who enjoy certain capacity and qualification.

The eighth-century Hanbali mujtahid Ibn Taymiyah summarised the arguments:

الْمُخَاطِبُ بِالْفِعْلِ لَا بِدْ أَنْ يَكُونَ قَادِرًا عَلَيهِ وَالْعَاجِزُونَ لَا يَجِبُ عَلَيْهِمْ وَقَدْ عُلْمَ أَنَّ هَذَا فَرْضٌ عَلَى الْكِفَايَةِ ... فَلِهَذَ : وَجَبَ إقامَةُ الْحُدودِ عَلَى ذِي السُّلْطَانَ وَنُوَّابَهُ

"The one who is ordained with an (obligation) of an act must be capable of doing it and those who are not able to, are not required to do it as it is Fard Kifayah … So, the hudud are legally prescribed upon the sultan or his deputy to perform." (Ibn Taymiyah, Majmu' al-Fatawa, Vol. 34, p. 175)

Contextualization

I believe there is an ommited point that is common to both groups. Both groups agree Allah will not punish anyone for a sin he did not commit nor was unable to prevent from happening. So, the better worded principle which each would agree to is

"Everyone is obliged to do that according to his capacity for which he would be accounted for."
For example, one of the key contemporary Fard Kifayah issues of our time is the establishment of the Khilafah Rashidah (Rightly-guided Caliphate). Everyone should work according to his capacity: those who can teach it to children, those who can study how to build it, those who can write, those who can fund the process, those who can support it politically, those who can only pray and wish… etc. are all required to make the best effort in their circle of influence. None of us is beyond the obligation as long as one can do even a little and none of us would be punished for doing his best effort and failing.

Fard Kifayah becoming Fard Ayn

There are potentially some scenarios where a Fard Kifayah becomes Fard Ayn cited by the scholars. They fit into three main categories:

The only one capable of it:

In cases where there is only one person (or a group) capable of doing the Fard Kifayah, the hukm here becomes Fard Ayn upon him (or them). For example, if there is only one scholar who is capable of being a mufti or a judge, it is Fard Ayn upon him to establish it. Or a good swimmer in a case of saving a drowning person when there is no one able of doing it but him. (Juwaini, Ghiyath al-Umam, p. 143, Khalaf, Ilm Usul al-Fiqh, p. 105)

The physical jihad is also Fard Kifayah, but when the enemies occupy a Muslim land it becomes a Fard Ayn upon all of its mature (and above) residents to defend their home. (Al-Badashkhi, Minhaj al-Uqool, Vol. 1, p. 117)

Ibn al-Najar explained the statement of ibn al-Hajib al-Maliki of this issue:

وَيَجِبُ" عَيْنًا "عَلَى مَنْ ظَنَّ أَنَّ غَيْرَهُ لا يَقُومُ بِهِ" أَيْ بِفَرْضِ الْكِفَايَةِ.
"It is an Ayn obligation upon who thinks (knows) no other one could do it but him." (Sharh al-Kawkab al-Muneer, Vol. 1, p. 375)
Nawawi also noted:

وَاِعْلَمْ أَنَّ تَعْلِيمَ الطّالِبِيِّينَ وَإفْتَاءَ المستفتين فَرْضُ كِفَايَةٍ، فَإِنْ لَمْ يَكُنْ مِنْ يَصْلُحُ إلّا وَاحِدًا أَوْ كَانَ هُنَاكَ جَمَاعَةٌ لَا يَحْصُلُ الْغَرَضُ إلّا بِكُلُّهُمْ تُعَيِّنَ عَلَيهُمْ.

"Know that teaching the students and giving fatwas to the questioner is Fard Kifayah when there is only one person qualified to it, or a group can't be established except by them all, it becomes a (Fard) Ayn upon them." (Nawawi, Rawdat al-Talibeen, Vol. 10, p. 225)

Appointed individuals:

If the hakim (Muslim ruler) appointed individuals to carry on the Fard Kifayahs, it became a Fard Ayn upon them to establish it.

For example, Iftaa, Qadaa, Hisbah, Jihad and many other obligations are Fard Kifayahs, but if the ruler appointed an individual as a judge, mufti, guider, soldiers … etc. it becomes Fard Ayn upon him to accomplish the work. (Juwaini, Ghiyath al-Umam, p. 143)

Initiation of Work:

Some scholars (Zarqashi, Nawawi, Suyuti, ibn Al-Hajib, Ibn al-Najar, Ibn al-Laham) argued if a man (or a group of people) initiated four kinds of Fard Kifayah, it becomes a Fard Ayn on them to complete it: the fighting in Jihad (to not to defeat the army), the prayer of Janazah (as a respect to the dead Muslim), al-Hajj (voluntarily) and learning and teaching what you have learnt. (Ibn al-Laham, al-Qawa'id wa al-Fawa'id, p. 188, Ibn al-Najar al-Hanbali, Sharh al-Kawkab al-Muneer, Vol. 1, p. 378, Khalaf, Ilm Usul al-Fiqh, 105, Zarkashi, al-Manthoor, Vol. 3, p. 38, Suyuti, al-Ashbah, Vol. 1, p. 175, Nawawi, Rawdat al-Talibeen, Vol. 10, p. 214)

Conclusion

Fard Ayn is an obligation prescribed an individual must perform themselves, whilst Fard Kifayah is a collective obligation that is done according to each person's capacity otherwise all legally responsible persons will be considered sinful (each one according to his capacity).

Sometimes a Fard Kifayah becomes a Fard Ayn upon a certain person or group if they are the only ones capable of performing the obligation, if appointed by the ruler to perform it or if they initiated it.

References

Aal (the family of) ibn Taymiyah, al-Muswadah
Asfahani, Sharh Mukhtasar ibn al-Hajib
Asnuwi, al-Tamheed
Al-Isnawi, Nihayat al-Sool
Al-Ijji, Sharh Mukhtasar al-Muntaha
Ameer Badshah, Tayseer al-Tahreer
Al-Badashkhi, Minhaj al-Uqool
Gazali, al-Mustasfa min Ilm al-Usul
Ibn al-Hajib, Mukhtasar Muntaha al-Sool
Ibn Qudamah, al-Mughni
Ibn al-Lahaam, al-Qawa'id wa al-Fawa'id al-Usuliyah
Ibn al-Najaar al-Hanbali, al-Kawkab al-Muneer
Ibn Taymiyah, Majmu' al-Fatawa
Juwini, Ghiyath al-Umam
Khalaf, Ilm Usul al-Fiqh
Kudari, Usul al-Fiqh
Nawawi, Rawdat al-Talibeen
Qarafi, Tanqeeh al-Fusul Fi Ilm al-Usul
Shafi'i, al-Risalah
Shafi'i, al-Umm
Shatibi, al-Muwafaqat
Shinqeeti, Usul al-Fiqh
Suyuti, al-Ashbah
Zarkashi, al-Bahr al-Muheet
Zarkashi, al-Manthoor

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