In Usool-al-Fiqh Shia uses Aql or intellect as a source of legislation as mentioned in this answer-
And the Sunni uses Qiyas or analogy in Fiqh. In my understanding, both are related with reasoning. How do they differ?
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Qiyas is used by some Sunni scholars in ususl Fiqh and is same as Analogy in Logic but Qiyas in usul Fiqh is completely different of Qiyas in Logic and only they have similar names.
Qiyas in Fiqh is transfer of a rule from one subject to another subject by guess. For example God said wine is Haram and so a scholar think the Haram rule of wine is because its befuddle. So he transfer this rule to any other liquid that has befuddle. In this case there is no problem because there are many Hadith saying anything having befuddle is Haram. But if we did not have such Hadith could we have such conclude?
Problem is where we have no verse or Hadith and only use this guess. Can we understand the reason of each rule only using our mind? And use that reason ourself have guessed for making new rules? It is clear we can not find the reason of rules with certainty and it is only at level of guess and we have no certain evidence (like verse or Hadith) about this guess. So in Shia Fiqh this method of making Fitwa is not accepted and even is banned and Haram. Imam Sadiq S.A. was strictly against using this method and had many debates with the leader of this method (Abu Hanifa).
وَمَا يَتَّبِعُ أَكْثَرُهُمْ إِلَّا ظَنًّا إِنَّ الظَّنَّ لَا يُغْنِي مِنَ الْحَقِّ شَيْئًا إِنَّ اللَّـهَ عَلِيمٌ بِمَا يَفْعَلُونَ
[10:36] And most of them follow not except assumption. Indeed, assumption avails not against the truth at all. Indeed, Allah is Knowing of what they do.
This verse prohibited us from following guess.
But about Aql. Not any kind of Aql is accepted in Shia Fiqh. The conclude of Aql is accepted only if is certainly and surely and there is no probability of error. if Aql with this ceriteria reached a conclude then it is accepted as a rule.
The conclude of Aql is sometimes Axiom (for example injustice is bad) which is called independent concludes of Aql and sometimes is theoretical that it divides to two type: certain theoretical and guess theoretical (like Qiyas in usul Fiqh).
So in short the difference of Qiyas and Aql is in the level of certainty.
There is a Hadith saying:
anything is commanded by Islam is commanded by Aql and anything commanded by Aql is commanded by Islam
Using Aql in Fiqh means that Aql (internal reasoning ability of human also known as internal messenger) judges that if an action is good or evil. it means that each act is good or evil and some neutral and Aql can determine it (independent of what Islam says about that act) and God commands to only Good and prohibits only evil. So God commanded to veracity because it is originally good and prohibited lie because it is originally evil. As an example the Aql says veracity is good and lie is evil.
Sometimes Aql is certain about some act and sometimes Aql is not certain and has doubt and only certain Judgments of Aql are accepted in shia Fiqh and Qiyas is not considered as certain judgement of Aql.
As example consider a man is dying if you quickly do not take him to hospital he will die and there is a car which is not yours and the only possible way to reach hospital is using that car. Islam says using properties of others without permission is Haram and also Islam says saving the life of a Muslim is Wajib. Here in Fiqh Aql is used to determine what a Muslim should do.
Using Aql in Fiqh is very wide and very complicated and is part of advanced Islamic studies. And using Aql in Fiqh contains many categories.
This was only a simple answer.
The Role of Reason in Ijtihad (in this article the "reason" is used as translation of Aql)
selections from Shiism in Sunnism (from chapter 4)
My answer is from Sunni point of view.
I do not have much knowledge about fundamentals of "Aql" in Shia Usool-al-fiqh. But as I understand, it is a general principle that can be rationally supported and does not have any basis from Qur'an or Sunnah or it is not restricted by Qur'an and Sunnah. I also understand that the Shia believe that the person who uses aql (reason) realizes a deep connection with God. Hence, most Shia fiqh is derived from Jaʿfar aṣ-Ṣādiq ( who is also one of their divinely appointed Imam).
Qiyas is not the same. Qiyas means analogical deduction. And in Shariah, it is the analogy from the Book and the Sunnah. We call it al-qiyaasus-saheeh (sound and correct analogy), it is derived from the Book and the Sunnah. It is linking a subsidiary branch with its root, due to a common Illah (effective cause) between them. So whenever the Lawgiver (Allah) indicates a matter, or describes it with a particular wasf (characteristic); or the Scholars deduce that the ruling has been legislated because of that particular wasf (characteristic), then if that particular wasf (characteristic) is found to exist in another issue, which the Lawgiver has not legislated any particular ruling for - without their being a difference between it and the texts - then it is obligatory to link the two in their ruling. This is because the All-Wise Lawgiver does not differentiate between matters equivalent in their characteristics, just as He does not join between dissimilar and opposing matters. This sound and correct qiyaas (analogy) is al-Meezaan (the Balance) which Allaah sent down. And it is inclusive of justice, and it is that by which justice is recognised.
Qiyaas is only resorted to when there exists no text. So this asl (fundamental principle) is turned to when there exists no other source. And qiyaas supports the text. Thus, all that the texts that the Lawgiver has given rulings to, then they are in agreement with qiyaas, not in opposition to it.
Qiyaas is mainly used in knowledge of current affairs where there is no direct textual ruling. Qiyas is basically analogical deduction from any of the four things below:
The Book and the Sunnah.
Unanimous agreement of scholars for things without proof from Book and the Sunnah.
When some or one of the companions of the Prophet (salallahu alayhi wassalam) said something and we do not find any objection to it from rest of them.
When companions differed among themselves, the Prophet (salallahu alayhi wassalam) did not object to it.
So, Qiyaas is not something independent of the texts. In fact, a valid Qiyaas should not contradict any of them.
An example of Qiyaas is following:
Scholars cite as an example that the criminal liability for murder with a bludgeon is the same as that for murder with a knife, since in both cases there is “an intentional and hostile act of killing”.
The difference here to the examples above is that the shared reason for the ruling is one that has been deduced by the jurists from the ruling prohibiting murder. The formula “an intentional and hostile act of killing” is a legal construct developed by legal theorists to define when a killing is legally an act of murder. It is not something that is explicitly stated in the texts, but rather something that is deduced from them.
Though I haven't talked much about aql (since I don't know much about it), I think I was able to make some obvious basic comparison between aql and Qiyas, Insha Allah.
Source: Qiyas in Islamic Law
The difference is simple.
Aql (intellect) is deductive logic e.g. you start with a general law from the Quran or Sunnah, then deduce a new ruling on a specific example that is covered by the general law.
Qiyas is inductive logic, this is where you start with a ruling, try to guess the general law, then apply the general law to a new specific case.
Shia reject qiyas as in fiqh the mujtahid (clergyman practicing religious jurisprudence) must have certainty, qiyas can never lead to certainty, therefore useless.
Qiyas and fiqh are more of a rational thing with base and argument and conceptualization rather that experience or wijdan ( I do not mean anything Sufi, I just mean like in the same way that you know justice is good, you know that aiding your parents is good, etc)
Aql can do Qiyas but it’s more of a light... whatever you put it to it shines. It tells what it right and what is wrong so that you may worship Allah.Though it gets damaged by the sins that we do and would heal relative to our service to Allah.
Have you not seen it to be easy for you to persuade some people for more easier and what is good and what is right? It's because they have done far more less of sins. As in Quran 45:23, Allah is telling us that if you no follow me (and commit too many sins then).
أَفَرَأَيْتَ مَنِ اتَّخَذَ إِلَهَهُ هَوَاهُ وَأَضَلَّهُ اللَّهُ عَلَى عِلْمٍ وَخَتَمَ عَلَى سَمْعِهِ وَقَلْبِهِ وَجَعَلَ عَلَى بَصَرِهِ غِشَاوَةً فَمَن يَهْدِيهِ مِن بَعْدِ اللَّهِ أَفَلَا تَذَكَّرُونَ.
have you then considered him who takes his low desire for his god, and Allah has made him err having knowledge and has set a seal upon his ear and his heart and put a covering upon his eye. who can then guide him after Allah? Will you not then be mindful?
From Usul al-Kafi:
العقل ، أنه ما يتميز به الحسن من القبيح ، والخير من الشر ، والحق من الباطل .
By Aql (itself) you distinguish right from wrong, good from evil and truth from falsehood.
العقل ما عبد بالرحمان واکتسب به الجنان .
By 'Aql' you worship Allah and achieve heaven.
For instance if we are to speak of kindness:
Qiyas or fiqh: would bring you numerous reasons and talk to you of a basis...like that if there is no sort of socialization in between humans then this and that...it could go up to a page of explanation
Aql: simply knows that kindness is good its already there, like a pre-installed Operating System...every human being is born with it. Do people have to inform you that kindness is good? No never you already know it...
Again tell me do you have to educate a 2-3 year old child that kindness is good? No!
The most you would do is to give a tadzakkur (تذکر) just as the Prophet (PBUH) was instructed by the Quran فذکر انما انت مذکر.
Do you have to explain to a child that lying/hitting is bad? Or the moment that you say honey don't lie/hit they get it. They are reminded of it... as if for a second they have lost that light (عقل) but then again reminded of it.
Or do you need to remind children of Allah? Or is it that they already know of him, yet not completely... Where did that knowing of them come from? It comes from that already given gift of Allah.
Quran 30:30: فطره الله التي فطر الناس عليها لا تبديل لخلق الله ذلك الدين القيم ولكن اكثر الناس لا يعلمون
the nature made by Allah in which he has made men; there is no altering of Allah’s creation; that is the right religion, but most people do not know
There is a long narration from the book of Kafi (a Shia book) which talks about the army of intelligence and the army of ignorance ( search for army to find the exact sentence in the book) ;more than 70 aspects of each. Basically ignorance happens when you have sinned and darkened your heart.
وَ الْإِیمَانُ وَ ضِدَّهُ الْکُفْرَ-
وَ التَّصْدِیقُ وَ ضِدَّهُ الْجُحُودَ
وَ الرَّجَاءُ وَ ضِدَّهُ الْقُنُوطَ
وَ الْعَدْلُ وَ ضِدَّهُ الْجَوْرَ
وَ الرِّضَا وَ ضِدَّهُ السُّخْطَ
وَ الشُّکْرُ وَ ضِدَّهُ الْکُفْرَانَ
وَ الطَّمَعُ وَ ضِدَّهُ الْیَأْسَ
وَ التَّوَکُّلُ وَ ضِدَّهُ الْحِرْصَ
وَ الرَّأْفَةُ وَ ضِدَّهَا الْقَسْوَةَ
وَ الرَّحْمَةُ وَ ضِدَّهَا الْغَضَبَ
وَ الْعِلْمُ وَ ضِدَّهُ الْجَهْلَ
وَ الْفَهْمُ وَ ضِدَّهُ الْحُمْقَ
وَ بِرُّ الْوَالِدَیْنِ وَ ضِدَّهُ الْعُقُوقَ
وَ الْحَقِیقَةُ وَ ضِدَّهَا الرِّیَاءَ
وَ الْمَعْرُوفُ وَ ضِدَّهُ الْمُنْکَرَ
وَ السَّتْرُ وَ ضِدَّهُ التَّبَرُّجَ
وَ التَّقِیَّةُ وَ ضِدَّهَا الْإِذَاعَةَ
وَ الْإِنْصَافُ وَ ضِدَّهُ الْحَمِیَّةَ
To conclude the difference is that Aql is very much already there... it just needs re-awakening or tadzakkur relative to how much you have damaged it with your sins.
Qiyas doesn't care about that humans have been taught or not before and that they need to be taught through argument.