In antiquity, two rival theories of the syllogism existed&colon; Aristotelian syllogistic and Stoic syllogistic. One approach is to reject disjunction introduction but keep disjunctive syllogism and transitivity. One allows disjunction introduction, and one allows disjunctive syllogism. The third Idea is based on the disjunctive syllogism. Below is an example of equivocation in a syllogism (a logical chain of reasoning). Aristotelian logic was the first type of logic to attempt to categorize every valid syllogism. The first Idea is based on the categorical syllogism. The disputant sets out to break down the dialectical syllogism. With this machine he could analyze a "syllogism or any other simple logical argument". The whole doctrine of judgment, syllogism and method mixes Aristotelian and Ramist notions. Conferring notability is related to transitivity and the syllogism. The Aristotelian syllogism dominated Western philosophical thought for many centuries. The second Idea is based on the hypothetical syllogism. From the Middle Ages onwards, "categorical syllogism" and "syllogism" were usually used interchangeably. Thomas Aquinas presented it in his fifth proof of God's existence as a syllogism.