(1) Brief introduction:
1 - The following should be understood within the perspectives outlined in this post. (It would also help to read through some of the other posts in that thread.) See also the following thread, as well as this one.
2 - As mentioned in one of the posts in the above-linked thread, Qiyas is comprised of four components; if either of them is missing, Qiyas cannot take place:
One: The Asl: That issue whose ruling is specifically mentioned in the Qur'an and/or Sunnah
Two: The Far': That issue whose ruling is not specifically mentioned in either the Qur'an or Sunnah --- its lack of ruling (as per a Fiqh-framework in which Qiyas is valid) is the reason why Qiyas is taking place: To seek its ruling
Three: The 'Illah: That which has been deciphered to be common in both the Asl and the Far' (various ways this can take place. Again, per a Fiqh-framework in which Qiyas is valid).
Four: The Hukm: That ruling the Asl provides; the very reason the Asl is considered the Asl, in this case.
(2) Brief mention of its proponents' views, and Dhahiri-responses:
3 - The claim, here, if I am not mistaken, is that there exists an Ijma' that Qiyas is valid, and perhaps even that the Halal and the Haram can be known through Qiyas.
4 - I don't have much to say about this; I believe proponents of Qiyas' validity have much stronger evidences, relatively speaking, so I will move on to those, in shaa' Allah
Two: The Qur'an and Sunnah
5 - Basically, if I am not mistaken, proponents of Qiyas' validity as a means of understanding Allah Ta'aalaa's Rulings point to all the Aayat and Ahadith in which two things are compared to one another.
6 - The Aayat are well-known, as are the Hadith. However, the Text that is the most seemingly-explicit in this regard, would have to be this Hadith:
7 - And I think the reasons for why that is, is obvious. Indeed, "Qiyas" is taking place --- a Qiyas in the linguistic sense: The comparing of one thing to another, for purposes of elucidation, and further clarification. What one is hard-pressed to claim is taking place, is Qiyas in its technical sense, as outlined in point no. 2 above.
8 - The reason for this is because, in this Hadith, the Far' is missing. What is considered by Qiyas' proponents to be the Far' in this Hadith, is the issue the woman Sahabi mentioned in her question; what they consider the Asl, is the question the Prophet, sallaa Allahu 'alaihi wa sallam, asked her in his response to hers.
9 - Indeed, it is the Asl --- they are the words of the Prophet, sallaa Allahu 'alaihi wa sallam, after all. However, the answer to the question the Prophet, sallaa Allahu 'alaihi wa sallam, asked her in his response to her initial question, was also an "Asl", at the time: She responded by acknowledging her prior knowledge of its answer; it is only that she did not recall it at the time. In other words: Two Asl's were being compared to each other: One that was already known, and another that further clarified it.
10 - So there was "Qiyas" taking place, only a more linguistic Qiyas; one that might be better described as an act of Tamthil. Al-Bukhari's chapter-heading prefacing this Hadith makes the nature of this alleged Qiyas more obvious.
Three: The Sahabah and Salaf
11 - There is really a lot that can be said on this particular sub-topic, and it really is a very fascinating one. I am not of Al-Qassab or Ibn Hazm's view that Qiyas and Ta'lil never took place in the era of the Sahabah, radi Allahu 'anhum. However, what I will say for now, is:
12 - Those blaming the Ahl Al-Dhahir for considering Qiyas invalid, on account of this or that example of the Sahabah, must better assess their case. I will eventually write more on this sub-topic in the future, as part of something else, in shaa' Allahu Ta'aalaa.
"[I fight] only for something deep in my heart . . .", — Captain Harlock.
Maktabat Madrasat Al-Qur'an Wal-Hadith Li Dirasat Al-Qur'an Wal-Hadith Wal-'Amal Bihima. (Established in my heart: Rajab 9, 1435.)