Nyayaikas are adherent of the orthodox Hindu philosophical system Nyaya. This is the school that devoted particular attention to logic, the means-of-knowing called inference. Much of this development is informal (fallacies, how to interpret passages, and the like). The formal portion centers on the "syllogism"
- There is fire on the mountain
- because there is smoke
- as in an oven, as not in a lake
- so (i.e.,smoke present)
- therefore thus (i.e., fire present) here.
The problem then is provide an explication of what conditions have to be met for the argument to be valid. We would say "Wherever there is smoke there is fire" and be done with it, but India did not develop quantifier logic and preferred (for metaphysical reasons) to work with individuals only. So, the solution had to be in terms of the individuals, smokeness and fireness, properties or generalities.
There was a long period of development on this problem in the hands of Buddhist logicians (Madhyamika, more or less -- Nagarjuna, Dignaga, Dharmakirti)and then the problem returned to Nyaya, in the form of Navya-Nyaya (Neo-Nyaya). The problem is presented as defining vyapti (pervasion) with a certain limited set of terms: the generalities, absence, locus of.... In the end, they came up with a couple dozen acceptable (though not obviously equivalent) definitions and a larger number of near misses. The essential point -- though not the tricks to get around quantification -- is that every locus-by-contact of smokeness is a locus-by-contact of fireness. An argument is valid, then, if the subject term (fire) pervades the object term (smoke). In the process a good deal of intentional logic gets sketched.
The Buddhist connection continues in Tibet, where monks are trained to spot vyaptis (or, more often, their lack)in a curious dance (the only choreographed logic I know of), one darting forward with a syllogism, the then coming forward with hand clapping (both hands) to admit it is valid or to deny it and give a counterexample.
Further information in the works of D.H.H. Ingalls and B.K. Matilal.