It would be difficult to discuss logical reasoning without mentioning Aristotle. A Greek philosopher who lived from 384-322 B.C., Aristotle was a student of Plato and a founder of what we know today as formal logic.
Most of Aristotle's writings are lost, but his students and subsequent scholars wrote many commentaries based on his teachings. A group of his treatises were collected under the title Organon, meaning "instrument." Rather than presenting logic as a discipline unto itself, Aristotle considered it an instrument for philosophical reasoning. The Organon was composed of six parts: Categories, On Interpretation, Prior Analytics, Posterior Analytics, Topics, and On Sophistical Refutations.
Aristotle contended that all knowledge must be derived from what is already known, and so his logic was centered around a type of thinking called deduction. When using deduction, you start from a given set of rules and conditions and determine what must be true as a consequence. To quote Prior Analytics, "a deduction is speech in which, certain things having been supposed, something different from those [things] supposed results of necessity because of their being so." The "things supposed" are premises, and the "something different from those supposed" are conclusions.
The basic form of this method of reasoning has become known as the syllogism. Syllogisms are structured sentences (called assertions) that are either true or false; they must contain a subject and predicate, and must either affirm or deny the predicate of the subject. In other words, they are sentences stating that one concept must, or must not, follow from another. You might even be familiar with this form:
If A is predicated of all B,Aristotle's famous example reads,
and B is predicated of all C,
then A is predicated of all C.
If all men are mortal,
and all Greeks are men,
then all Greeks are mortal.
You can often find this type of "If...then" statement in mathematical proofs, and that is due to the far-reaching influence of Aristotelian methods. They changed the face of scientific thought in their time, and for almost 2000 years after, allowed deductions of new truths to be made from established facts or principles. If you could translate arguments into syllogisms, then you could predict new outcomes or consequences, whether in math, science, or philosophy. Only during the past century have Aristotle's methods been questioned by scholars such as Gottlob Frege and Bertrand Russell. New types of logic have been developed that provide a more accurate foundation for mathematical and scientific inquiry. The importance of Aristotle's work, however, will never be forgotten.