A review of some concepts and terms from nyAya is in order before we proceed further. Please see "Materials for the study of Navya Nyaya Logic" by Ingalls or "The Navya-Nyaya doctrine of negation" by Bimal Krishna Matilal for more details.

Process of inference:

Every noneternal entity, according to nyAya, must be a result of  an instrumental cause (karaNa) and an operation (vyApAra).  An inference (anumAna), that is a means to knowledge of man and hence  noneternal, must have an instrumental cause and an operation by which  the cause brings about the inference.

In the process of inference, the operation is called parAmarsha or  consideration, and the instrumental cause is the knowledge of  invariable concomitance (vyApti), also called pervasion. This vyApti  corresponds very roughly to logical implication in Western logic.

In the inference, "the mountain possesses fire because it has smoke",  the instrumental cause, karaNa is the knowledge of the invariable  concomitance, "smoke is the invariable concomitant of fire",  vahni-vyApyo dhUmaH, ie. where there is smoke there is fire.  The operation, vyApAra is the consideration (parAmarsha) that is  a knowledge of the occurrence of the concomitant in the subject (pakshha)  where the inference is being made. In the inference, "the mountain  possesses fire because it has smoke", the parAmarsha will be of the  form, "the mountain possesses smoke which is an invariable concomitant  of fire" - parvato vahni-vyApya-dhUmavAn.h

In polemical treatises and debates, an inference is stated tersely  by listing the inference (conclusion) followed by a single word  representing the application of the consideration and the invariable  concomitance. For example, parvato vahnimAn.h dhUmAt.h, "the mountain  possesses fire because of smoke."

In every inference, there are evident three terms, called 1) sAdhya  or that which is to be proved, 2) the hetu or linga, the concomitant,  and 3) the subject or pakshha, that in which the hetu is known to occur.  In the inference, parvato vahnimAn.h dhUmAt.h, "the mountain has fire  because of smoke", the sAdhya is vahni (fire), the hetu is smoke, and  the pakshha is parvata (mountain).

In the standard form of inference, the three terms are placed as  follows:

For example, parvato vahnimAn.h dhUmAt.h, "the mountain has fire  because of smoke."

The definition of vyApti is very important. It is defined by the  bhAshhA-parichchheda (of VishvanAtha) as:

Invariable concomitance is said to be the absence of relation of  the (hetu) to anything other than what possesses the sAdhya.

Here the relation should be the same as the relation under consideration  for the vyApti. As per this definition of vyApti, smoke is an invariable  concomitant of fire (or smoke is "pervaded" by fire) because it is not  the case that there is something that has smoke but not fire. However,  fire is not an invariable concomitant of smoke. There are things such  as a red-hot iron rod that has fire but no smoke. So fire is not an  invariable concomitant of smoke.

A relation between two entities is often explained as the superimposition  of one entity on the other. Here, the entity that is superimposed is  called the superstratum or Adheya. The entity on which the Adheya is  superimposed is called substratum or locus. Sanskrit names for  "substratum" is  AdhAra or Ashraya or adhikaraNa. In the example,  bhUtale ghaTo vartate, "there is a pot on the ground", the superstratum  is the pot (ghaTa), the ground (bhUtalam.h) is the substratum or locus,  and the relation is "contact", saMyoga.

Another way of looking at relations (saMbandha) is to differentiate  between what Ingalls calls "occurrence-exacting" and "non occurrence  exacting" relations. Relations such as inherence (samavAya) are  always occurrence exacting. samavAya or inherence is the relation  between a whole and its parts, a genus or class (jAti) and a particular  instance of the class, etc. Contact (saMyoga) can sometimes be  occurrence exacting but sometimes not.

In relation between two entities, one of the entities is an adjunct or  pratiyogin, and the other is a subjunct or anuyogin. If a relation is  such that one entity is a locus or substratum (AdhAra) of the other which  must be the superstratum (Adheya), then the AdhAra is the subjunct or  anuyogin. The Adheya is the adjunct or pratiyogin. In the example, bhUtale ghaTaH, "there is a pot on the ground", ghaTa is the pratiyogin  while bhUtalam.h is the anuyogin.

Two types of absences (abhAva) are distinguished in navya-nyAya. One  is called anyonya-abhAva that is essentially a denial of identity  between to entities. The other is saMsarga-abhAva or relational absence.  Here, there are three kinds: 1) prAgabhAva, the absence of a thing before  it is brought into being, 2) dhvamsAbhAva, the absence of a thing after  it is destroyed, and 3) atyanta-abhAva, eternal absence.

The terms pratiyogin and anuyogin are also used in the context of  absences. When we say "bhUtale ghaTo nAsti", "there is no pot on the  ground", the pratiyogin of the absence is "ghaTa", pot, and the  anuyogin is "bhUtalam.h", ground. The pratiyogin of this type should  be called "abhAvIya-pratiyogin" to distinguish it from the relational  adjunct, but naiyAyikas often write "pratiyogin" for both relational  and absential adjuncts.