Buddhist logic is a term used in Western scholarship to refer to Buddhist traditions of 'Hetuvidya' (Sanskrit) and 'Pramanavada' (Sanskrit), which arose circa 500CE,which in turn were influenced by 'Indian Logic', from which they seceded. Indian logic, and Buddhist Logic—in main heralded by Dignāga (c. 480 - 540 CE)—are both primarily studies of 'inference'-patterns, where ‘inference’ is a gloss of anumāna (Sanskrit).
Sadhukhan, et al. (1994: p. 7) frames the centrality of 'syllogism' to Buddhist Logic and foregrounds its indivisibility as an investigative, authenticating and proofing tool instituted to establish the valid cognitive insights of the Buddhadharma:
Buddhist logic obviously contains the forms and nature of syllogism, the essence of judgement, etc. for which it deserves the name of logic. But that logic is not only logic it also establishes the doctrines of the Buddhists. Thus the philosophical tenets were the fulcrum and the logic developed as tools to establish those.
Following the work of Tucci (1929) and the critique of Anacker (2005, rev.ed.) upon the collation of Frauwallner (1957), it is now understood that Vasubandhu's Vāda-vidhi ("A Method for Argumentation") refined the five argument logic of the Nyāya-sūtra to a three argument form and not his pupil Dignāga. In addition to pruning the two redundant arguments from the syllogism, Vasubandhu tendered a further qualification: he posited that a sound relationship, a 'logical pervasion' (vyapti) needs to be defined between the first and second arguments, a relationship between the 'Demonstrandum' (pratijna) and the 'Justification' (hetu) that is assumed in the Nyāya-sūtra and other literature of the Nyāya school. This logical pervasion is required to fashion sound arguments. Vasubandhu's Vāda-vidhi was reconstructed by Frauwallner from embedded quotations harvested from the works of Dignāga, amongst others. Dignāga as the oft-cited wellspring of the logical triune in the Buddhadharma is now invalidated.
‘Indian Logic’ should be understood as being a different system of logic than modern classical logic (e.g. modern predicate calculus), but as anumāna-theory, a system in its own right. ‘Indian Logic’ was influenced by the study of grammar, whereas Classical Logic which principally informed modern Western Logic was influenced by the study of mathematics.