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Udayana, also known as Udayanācārya (Udyanacharya, or Master Udayana), (circa 975 - 1050 CE) was a very important Nyaya Philosopher & Logician of the tenth century who attempted to devise a rational theology to prove the existense of God using logic and counter the attack on the existense of God at the hands of Buddhist philosophers such as Dharmakīrti, Jñānaśrī and against the Indian school of materialism (Chārvaka). 
He worked to reconcile the views held by the two major schools of logic (Nyaya and Vaisheshika). This became the root of the Navya-Nyāya ("New Nyāya") school of the thirteenth century, established by the Gangesha Upadhyaya school of "right" reasoning, which is still recognized and followed in some regions of India today. He lived in Kariyan village in Mithila, near present-day Darbhanga, Bihar state, India.
Udayana wrote a sub-gloss on Vachaspati's work called the Nyaya-vaartika-taatparya-tiikaa-parishuddhi. He wrote several other works such as the Kusumanjali, Atma-tattva-viveka, Kiranaavali and Nyaya-parishishhta (also called Bodha siddhi or Bodha shuddhi).
He is given credit by Naiyâyikas for having demolished in a final fashion the claims of the Buddhist logicians. All his known works are thought to have been preserved, attesting to the importance given to him in Indian philosophy.
It is accepted by most scholars now that he was from Mithila, Bihar. Scholars are almost unanimous in declaring Udayana to be one of the greatest of Indian philosophers. In the Nyāya-Vaiśeṣika school itself, to which he belongs, he occupies a singular position of authority and renown. Flourishing at the period of transition from the Older Nyāya to the New (Navya-Nyāya), he shines as an unrivalled master of the former and an inspiring herald of the latter. For example, Gaṇgeśa Upādhyāya the 14th century Indian Philosopher & Mathematician who established the Navya-Nyāya school refers to Udayana as "Acāryaḥ" (lit. Master/Teacher). D. C. Bhattacharya observes: "From the 12th century onwards he [= Udayana] was looked upon as the greatest exponent of the Nyāya-Vaiśeṣika doctrines and was the greatest target of all scholars of the opposing camps"
One Indian writer of the sixteenth century, Sāyana Mādhava, the author of the Sarvadarsanasamgraha, speaks of him not only as "one whose fame had spread everywhere" (viśvavikhyātakīrtih), but also as "one who has seen the opposite shore of the ocean of the principles of logic" (nyāyanayapārāvārapāradṛk) an epithet which shows his fame as a logician. 
The controversy about Udayana's lifetime seemed to have been settled by the discovery of the Lakṣaṇāvalī, the concluding verse of which states that it was written in Saka era 906 (984—985 A. D.). Hence, he was active as a philosopher from the late 10th century to the beginning of the 11th century. 
Nyayakusumanjali and the existence of God
- Kāryāt (lit. "from effect"): The world is an effect, all effects have efficient cause, hence the world must have an efficient cause. That efficient cause is God.
- Āyojanāt (lit., from combination): Atoms are inactive. To form a substance, they must combine. To combine, they must move. Nothing moves without intelligence and a source of motion. Since we perceive substance, some intelligent source must have moved the inactive atoms. That intelligent source is God.
- Dhŗtyādéḥ (lit., from support): Something sustains this world. Something destroys this world. Unintelligent Adrsta (unseen principles of nature) cannot do this. We must infer that something intelligent is behind it. That is God.
- Padāt (lit., from word): Each word has meaning and represents an object. This representational power of words has a cause. That cause is God.
- Pratyayataḥ (lit, from faith): Vedas are infallible. Human beings are fallible. Infallible Vedas cannot have been authored by fallible human beings. Someone authored the infallible Vedas. That author is God.
- Shrutéḥ (lit., from scriptures): The infallible Vedas testify to the existence of God. Thus God exists.
- Vākyāt (lit., from precepts): Vedas deal with moral laws, the rights and the wrongs. These are divine. Divine injunctions and prohibitions can only come from a divine creator of laws. That divine creator is God.
- Samkhyāviśeşāt (lit., from the specialty of numbers): By rules of perception, only the number "one" can ever be directly perceived. All numbers other than one are inferences and concepts created by consciousness. When man is born, his mind is incapable of inferences and concepts. He develops consciousness as he develops. The consciousness development is self-evident and proven because of man's ability with perfect numerical conception. This ability to conceive numerically perfect concepts must depend on something. That something is divine consciousness. So God must exist.
- Adŗşţāt (lit., from the unforeseen): Everybody reaps the fruits of his own actions. Merits and demerits accrue from his own actions. An unseen power keeps a balance sheet of the merit and demerit. But since this unseen power is unintelligent, it needs intelligent guidance to work. That intelligent guide is God.
Other works by Udayana
Seven works have been ascribed to Udayana. The following are the titles of the works in the chronological order in which they are believed to have been composed.
- Lakṣaṇamālā: This is a short manual giving very clear definitions and the main divisions of the sixteen categories of the Nyaya and the six categories of the Vaisesika system.
- Lakṣaṇāvalī: This work resembles the Laksanamala in form, purpose and method of treatment. It differs from the latter in that it deals exclusively with the Vaisesika categories and their subdivisions.
- Atmatattvaviveka: This is an independent work in four chapters of varying lengths. Each of these chapters is devoted to refute different theories put forward to establish the non-existence of a permanent soul (atma) such as conceived by the Nyäya-Vaisesikas.
- Nyāyakusumañjali: (See outline above)
- Nyāyapariṣista: This deals exclusively with the Jatis and Nigrahasthanas, which form the subject matter of the fifth chapter of the Nyayasutras.
- Nyāyavarttikatātparyapariśuddhi: This is an elaborate commentary on the Nyäyavarttikatätparyatika of Vacaspatimisra.
- Kiraṇāvalī: This is a commentary on the Padarthadharmasamgraha of Prasastapada.
- Dalal, Roshen (2014), Hinduism: An Alphabetical Guide, UK: Penguin
- Dasgupta, Surendranath (1975). A History of Indian Philosophy. Motilal Banarsidass Publ. ISBN 978-81-208-0412-8.
- Matilal, Bimal Krishna; Ganeri, Jonardon; Tiwari, Heeraman (1 January 1998). The Character of Logic in India. SUNY Press. ISBN 978-0-7914-3739-1.
- Subodh, Kapoor (2002), Companion Encyclopaedia of Hindu Philosophy: An Exposition of the Principle [sic Religio-philosophical Systems and an Examination of Different Schools of Thought. Genesis Publishing]
- Tachikawa, M. (2012), The Structure of the World in Udayana’s Realism: A Study of the Lakṣaṇāvalī and theKiraṇāvalī. Springer
- Vidyabhushana S.C. (1988). A History of Indian Logic: Ancient, Mediaeval and Modern Schools. Motilal Banarsidass.
- Potter, Karl; Bhattacharya, Sibajiban (1970). The Encyclopedia of Indian Philosophies, Vol.2. Delhi: Motilal Banarsidass. p. 521. ISBN 9788120803091. Retrieved 8 July 2020.
- Madhavacharya (Vidyaranya Swami), E. B. Cowell (1882). Sarva-Darsana-Samgraha by Madhavacharya (Vidyaranya Swami) - tr by E.B.Cowell. Sabyasachi Mishra.
- George Chemparathy (1972). An Indian rational Theology: Introduction to Udayana's Nyakusumanjali. Druck: Ernst Becvar, A-1130 Wien: Indologisches Institut der Universität Wien, Sammlung De Nobili. pp. 19–25.CS1 maint: location (link)
- Sharma, C. (1997). A Critical Survey of Indian Philosophy, Delhi: Motilal Banarsidass, ISBN 81-208-0365-5, pp.209-10
- Mohanty, J.N. & Gupta, Bina (2000). Classical Indian Philosophy, Rowman and Littlefield
- Majumdar, A.K. (1977). Concise History of Ancient India: Hinduism: society, religion & philosophy