For the purposes of this hypertext essay, I shall be using some terms in a specific sense, and I define those senses here. Please don't take me to task for using the terms in a manner not quite the same as you would expect; my definitions apply only to this hypertext essay.
This is the general capability to assemble, recognize, and process a sequence of items. The most common form of sequential thinking in animals is path planning. This shows up when animals figure out how to get around an obstacle or remember the path to a waterhole, a feeding ground, or a safe place. This capability is well-developed and generalized (to varying degrees) with mammals. It exists in birds, but in more specialized fashion. Many birds can remember long complex songs (sequences of notes). Some birds can remember long migratory routes. Some birds can figure their way around obstacles. Language is based on sequential thinking, but goes much further.
The attempt to use language in a sequence of statements that purports to reach a reliable conclusion. Unfortunately, words are poor tools for such work, so broadly are they defined, and so reason is sometimes unable to provide us with reliable conclusions. Nonetheless, it often works, and certainly provides us with lots to argue about.
I shall be using this long-disputed term to refer to the belief that the problems of life are best solved by the application of reason and not by emotion. Although emotion is the common whipping-boy of rationalists, in practice rationalism is more precisely described as the rejection of any thought process that does not rely on reason. This applies to pattern-based reasoning. Social reasoning, which is pattern-based, is seldom amenable to reason, and so is often the bane of rationalists.
This term takes a specific meaning here: it is the use of a sequence of syllogisms to arrive at a conclusion. It is a formal process, in which each step is subject to rules of logic that are widely accepted. Such a line of reasoning can be broken if a single step of the process is invalidated. Logic cannot be carried out with regular language; it requires specialized languages such as mathematics, boolean terminology, or legal language.