Pujya Swamiji explained that Vikalpa means any differentiating factor that differentiates one from the other. It can be anything like Jaati, Guna, and Kriya etc.
In Nyaya School of philosophy, the padaarthas (the whole world is reduced into) are divided into seven categories.
1.Dravya or substance or material.
2.Guna or quality. (subdivided into)
a. Saamaanya (species) or common quality.
b. Visesha or specific quality.
3.Kriya or action.
4.Samavaaya or inherence.
5.Abhaava or absence.
In the verse 51 Pujya Vidyaranyaji says that the same logical fallacies (which we discussed in the previous verses) can be shown in any object having dravya, kriya, guna etc. (in all the seven padaarthas mentioned above). For example if we talk about kriya or action, then the question comes is it in the action less (attribute less) or is it in the one with action (with attributes). Then again all the five defects mentioned above will come into the picture. You cannot logically establish their reality in Brahman. All these seven padarthas are of the nature of the world; they do not pertain to the ultimate Reality. World is Mithya in the sense that it can be seen but it does not stand the test of reality. It is the nature of the world to be inexplicable or illogical or unreasonable.
In the verse 52 Pujya Vidyaranyaji explains that the Brahman is untouched by the words like savikalpa or nirvikalpa, and the relationship of modification (either savikalpa or nirvikalpa), the Mahavakyas indicating Brahman through Lakshyartha, Brahman as jagatkaaranam are all superimposed or alleged on Brahman.
Now again the question comes, it has been told in the earlier verse (verse 51) that all the vikalpas pertain to the world and Supreme reality is untouched by vikalpas. Now, in the verse 52 it has been told that Brahman is neither savikalpa nor vikalpa.
It has been explained as follows.
With respect to savikalpa, Supreme Reality is nirvikalpa. But, when vikalpa itself has been established false, then how can one posit the absence of vikalpa.
It has been explained with Rope&snake example. When one is deluded to think rope as snake (as the snake is full of poison), then to explain to that person one says that in reality it is rope and rope is free of poison. Once the person understands that it is not snake but rope, then how can one say that the rope is free of poison!
Similarly, to differentiate Brahman from the savikalpa world it has been told in the scriptures that Brahman is nirvikalpa, when vikalpa itself is false, how can we say Brahman is nirvikalpa.
In the same way, the Mahavaakyas indicate Brahman at the teaching level, but when Maya &Avidya are themselves false, where is the dropping of them in Lakshyaartha! In order to take the person to the understanding of Brahman lakshanaa vritti is explained. When one attains the Brahman, at that highest level, the lakshyartha of Tat&Tvam is not there. That Mahavaakyas through which one attains the Brahman is based on the acceptance of Jiva&Iswara is true at the level of teaching. It helps us to get the clarity of Jiva Brahma Aikyam.But, at the ultimate level of reality Brahman is not even indicated by the Mahavaakyas. The supreme Reality is beyond words.
For teaching purposes the words have been employed and once the Truth is understood, the words are dropped. So, one should not argue that why the words are used in the first place as they are going to be dropped at the ultimate level.