Rangtong / Prasaṅgika Emptiness
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"Prasaṅgika Emptiness" is the theory that all things and phenomena lack (or are empty of) any type of inherent identity or self-characterizing essence. This lack of inherent existence does not mean that all phenomena are non-existent; inherent non-existence is also negated by Prasaṅgika emptiness. In the philosophy of Mahayana Buddhism, specifically in the Madhyamaka view, Prasaṅgika Emptiness is a category of Madhyamaka viewpoints attributed primarily to Indian scholar Candrakirti, but which is based also on Buddhapalita's commentaries on Nagarjuna. Nagarjuna and later commentators use the method of logical consequence (prasanga in Sanskrit) to refute flawed views. By using this type of reductio ad absurdum, Nagarjuna, Buddhapalita, Candrakirti, and, later on, Lama Tsongkhapa, simultaneously refute incorrect/opposing viewpoints and establish the position of Prasaṅgika.
"Emptiness," in the Prasaṅgika sense, does not mean space, lack of physical obstruction, non-existence, or the simple not-finding of a conventional identity for the object or phenomenon being scrutinized. Prasaṅgika is the conclusive analytic absence of any findable internal quality of an object or phenomenon that is a sufficient condition to provide a name, identity, function, and/or discreteness to that object or phenomenon. The strong implication is that the mind of the observer (technically, a conceptually designating consciousness) provides the name, identity, function, and conventional discreteness to that object, i.e., imposes such a characterization to that seeming object. The relationship between the mind and object is commonly described as "Co-dependent arising, dependent arising, and dependent origination." The mind is also viewed as dependently arising on the basis of the object of which that mind is aware. This relationship explains why, when one analyzes a material object, a mind, or an abstract phenomena (like time, non-composed space, coming, going, causality, etc.) nothing self-characterizing or identity-providing is found.
Zhentong / Shentong
Shentong is a philosophical sub-school found in Tibetan Buddhism. Its adherents generally hold that the nature of mind, the substratum of the mindstream, is "empty" (tong; Wylie: stong) of 'other' (shen or zhän; Wylie: gzhan), i.e., empty of all qualities other than an inherent, ineffable nature. The contrasting Rangtong view of the followers of Prasaṅgika Mādhyamaka is that all phenomena are unequivocally empty of self-nature, without positing anything beyond that. According to a Shentongpa (proponent of Shentong), the emptiness of ultimate reality should not be characterized in the same way as the emptiness of apparent phenomena because it is prabhāsvara-saṃtāna, or "clear light mental continuum," endowed with limitless Buddha qualities.[1] It is empty of all that is false, not empty of the limitless Buddha qualities that are its innate nature.