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The pramanas other than "pratyaksa" and "anumana" are "upamana" and "sabda". What is "upamana"? It is knowing what is not known by means of comparison with the known. There is an animal called "gavaya". We do not know what it looks like. It is like a wild buffalo: to look at it is like a cow, so it is said. We go to the neighbourhood of the forest and there we spy an animal resembling a cow, so we conclude that it must be a gavaya. Here we have recourse to upamana.
"Sabda-pramana" is verbally testimony, the pronouncements of the Vedas and the words of great men. When the scriptures speak of things that we do not know, their words must be accepted as authority. The naiyayikas, or exponents of Nyaya, believe that the Vedas are the words of Isvara. The words of great men who are wedded to truth are also verbal testimony.
These four pramanas are accepted in Kumarilabhatta's school of Mimamsa. To them he has added two more: "arthapatti" and "anupalabdhi". Thus there are six pramanas in all and they are part of the non-dualistic doctrine also.
Our Sastras give a clear idea of arthapatti through an illustration. "Pino Devadatto diva na bhunkte". What does the statement mean? "The fat Devadatta doesn't eat during daytime". Though Devadatta does not eat during daytime, he still remains a fat fellow. How? We guess that he must be eating at night. There is something contradictory about an individual not eating and still not being thin. Here arthapatti helps us to discover the cause of Devadatta being fat. Our guess that he eats at night does not belong to the category of anumana. To make an inference there must be a hint or clue in the original statement itself. There must be a "linga" like smoke from fire, thunder from clouds. Here there is no such linga.
It is the same with upamana. When we come to the conclusion that the animal we have seen is the beast called "gavaya", it does not mean that we made an inference or anumana. We did not recognise the animal by means of any sign but from the fact that its appearance tallied with the description we had been given.
The last pramana is anupalabdhi. It is the means by which we come to know a non-existent object. I spoke about "abhava", the last of the seven padarthas according to Nyaya. Anupalabdhi is the means by which we know abhava. Suppose someone tells us, "Go and see if the elephant is in the stable". We go to the stable to see for ourselves whether or not the elephant is there. We find that there is no elephant in the stable: to recognise such absence (non-existence) is anupalabdhi.
Arthapatti and anupalabdhi are part of Mimamsa and Vedanta, not of Nyaya. (However, anupalabdhi is mentioned only in the Kumarilabhatta school of Mimamsa, not in the Prabhakara school. )
For a general background, please see here