Our religious system is such that if we go to the root of all padarthas (categories)and understand their source, the Truth will become illumined. We must make use of all pramanas (source or instruments of knowledge) for this purpose. (That by which we perceive objects is a pramana).
Objects that are apprehended by the senses, that is by the eyes, ears, etc, are not many. Others have to be known by inference. And inference helps us in understanding the truths of the truths of the Vedas. That is why Nyaya is called an Upanga (an auxiliary "limb") of the Vedas.
In Nyaya the padarthas are divided into seven categories. Of them there are two divisions: "existent" and "non-existent" -- "bhavo abhavasca", the latter being the seventh padartha. "Bhava" or the existent is further divided into six sub-catagories.
How does that which does not exist become a padartha? What does "padartha" mean? In a literal sense, it is the meaning of a pada or word. Is there not a word which means No? There is non-existence of certain objects in some places, and not in some others. Here there are no flowers. There, in the pavilion where the puja is performed, there are flowers which means that the non-existence of flowers does not apply to the pavilion. So there is non-existence [of objects] in some places and on certain occasions. Thus the fact of non-existence [of a thing] in certain places and at certain times is also to be known as padartha.
The seven padarthas are: dravya(substance), guna(quality), karma(action), samanya(association), visesa(difference), samavaya(inheritance) and abhava(non-existence). Dravya, guna and karma are padarthas that belong to the category of "sat" or being. We can demonstrate their existence but not of the other four padarthas. Dravya can also be shown in its gross form. But qualities like jnana, desire, happiness, sorrow, etc, cannot be shown as independent entities. Redness is the quality of, say, the lotus and it cannot be separated from that flower. That on which it is dependent is dravya. And, though qualities like happiness and sorrow cannot be "shown", we can know whether a person is happy or sad: we "see" in him happiness or sorrow. When we see a red lotus we know what is red. Karma is work, activity. Such "work" as movement, running, is karma and it is also dependent on dravya. When a man runs, his "running" cannot be separated from him. But we do see him running and know that he is not sitting or lying down. That means we " see" the running. Samanya is the fourth padartha and it means "jati" ("species"). We see a number of cows. They have the common quality of being cows. This common quality is of jati. Among objects or individuals that have a common quality there may still be differnces. This is what is called "visesa". Suppose there is a herd of cows(they belong to the same jati): among them we will be able to tell apart individual cows because of their distinctive characteristics.
What is "samavaya"? The quality of a substance cannot be separated from it(the substance), nor the work associated with it. The parts of a whole object cannot be separated if it is still to remain the object that we know it to be. Here we have samavaya, the quality inhering in something. Fire has a radiant form. But this radiance cannot be separated from it. Here again is an example of samavaya. When one dravya or substance combines with another substance we have "samyoga". The two can remain independently without combining. There is samavaya when a substance combines with guna or quality and there is samavaya again when dravya and karma combine. The quality and the karma cannot be separated from the substance.
I have already spoken about "abhava".
Each of the seven padarthas is now further subdivided. Dravya or substance is divided into nine: prithvi(earth), ap(water), tejas(fire), vayu(air), akasa(space), kala(time), dik(direction), the Atman(Self), manas(mind). The first five are called "pancabhutas". Corresponding to them in the body are the five sense organs, the eyes that see, the ears that hear, the tongue that tastes, the organ of touch that feels warmth and cold, the nose that smells. The organ of touch is not skin alone; the entire body possesses the sense of touch. It is because it exists within the body too that we feel stomach ache, chest pain, etc.
These faculties are associated with individual parts of the body. Sight is in the eyes; the ears cannot see. Music is heard by the ears; the eyes and the nose cannot hear it. If an object comes into contact with your tongue, you know its taste but not its smell-- the nose does not know that sugarcane is sweet. So these five qualities can be recognized individually by the five sense organs. The eye recognizes the quality called tangible form, "rupa", which means colour, size, shape, etc. White, yellow, green, red, brown are some of the colours. The nose perceives pleasant and unpleasant smells. Heat and cold are known by the skin. The tongue apprehends the six different flavours(rasas). Thus there are five different sense organs for five different qualities and they are called jnanendriyas.
Without the sense organs or indriyas the quality of an object will not be recognised. If we had six organs we could perhaps know six gunas or qualities and if we had a thousand sense organs we could perhaps appreciate a thousand qualities. We have no knowledge of all objects of the universe. If we did not possess the sense organ of touch we would not be able to feel heat and cold. We cannot claim that we have knowledge of cold or heat (that is we can feel heat and cold) because they exist. We recognise qualities only be means of our sense organs. The blind and the deaf do not perceive form or sound though form and sound do exist in the world. All the five qualities, form, flavor, smell, touch, and sound, are known respectively with the eye, tongue, nose, skin and ear. The Lord had invested the pancabhutas, the five elements, with these five qualities. The earth has all the five gunas or qualities. It has form and flavour. Our body, aubergines, jaggery- all are earth. Earth has smell. The fragrant flower is indeed earth. Earth has qualities like cold and heat known by the sense of touch and it has also sound. If you drop one end of a string to earth and keep the other end of it to your ear you will hear sound. Water has four qualities but not smell. It smells only when we mix perfume in it. If we beat its surface it sounds. Though earth has all the five qualities its special quality is smell which is absent in the other four elements. The special quality of water is flavour. Without water there is no rasa. That is why the sense organ of taste, the tongue, is always wet. If the tongue becomes dry you will not be able to appreciate any taste. As a matter of fact, the word "rasa" itself also means water. Fire has neither smell nor flavour but it has form, sound and touch, form being its special quality. Vayu or air has no form, but it has sound and touch- the last-mentioned is its special quality. That is how we know when the wind blows on us. Akasa or space has only one quality, sound.
To sum up, akasa or space has only one quality; vayu or air has, in addition to sound, the quality of touch; agni or fire has the three qualities of sound, touch and form; water has the qualities of sound, touch, form and rasa; but earth has all the five qualities. Such are the pancabhutas or five elements.
The remaining four subdivisions of dravya(substance) are time, dik, the Atman and manas. Terms like "hour", "yesterday", "today", "year", "yuga", indicate time. "Dik" means direction or area, the points of the compass, what we mean by "here" or "there". In short it denotes "space", "akasa". The Atman is the entity that knows all this. He or It is of two types, the intelligent and the unintelligent, the Paramatman and the jivatman. The Paramatman is a mere witness to all that passes in the world while the jivatman or the individual self is trapped in it(the world) and given to sorrow. The individual souls are many while the Paramatman is one and only one. Both the jivatman and the Paramatman are spiritual entities of jnana.
According to Vedanta, knowledge itself is the Atman; the Atman is jnana in a plenary sense. Apart from it, and outside it, there is nothing to be known. Indeed we cannot speak of different jivatmans. According to Nyaya, the Atman is a dravya or substance, knowledge(jnana) being its quality.
Nyaya describes the Paramatman alone as jnana that is full since there is nothing that is not known to him. The individual self possesses only a little knowledge. So we are called "kinjijnas", "kinjit" meaning little. The Paramatman is "Sarvajna", the One who knows all. We are in a mixed state of being dependent both on jnana and ajnana. The Paramatman is dependent on (or is) jnana alone. The Atman is "vibhu", all-pervading. Nyaya also says that the Paramatman is all-pervading, but it does not speak of the to being the same, the Atman and the Paramatman. The reason for this is that, according to Nyaya, knowledge exists independently in each individual as a separate factor. The place where it dwells is the mind- and it is the mind that causes sorrow and happiness.
In Nyaya guna is divided into 24 categories and karma into five. The Truth will be known, says Nyaya, if we have knowledge of the padarthas and develop detachment that will lead to release. Liberation is a state in which we know neither sorrow nor happiness. Even if we adhere to the Vedantic concept of liberation, Nyaya affords a method to reflect upon the instruction received from our guru. We are able to know the pancabhutas or the five elements, the individual self and the mind. But how are we to know the Paramatman? He alone is not known. It is to know him that we must employ anumana, the method of inference. To know the rest "pratyaksa pramanas" or direct sources of knowledge are sufficient. The Vedas proclaim the existence of Isvara; Nyaya establishes it with anumana or inference.
Let us now see a small example of inference. We know that the throne on which I am seated must have been made by someone. Because we don't know him, can we describe the fact of its having been made itself to be false? We have seen other thrones being made and from that we deduce that there must be somebody who must have made this one also. Similarly, there must be someone who must have created this universe. He is omniscient, omnipotent and compassionate- and he is the protector of all. Such matters are dealt with in Nyaya: a proposition is stated, objections raised and answered.
For a general background, please see here