Aristotle’s Syllogisms

More than 23 hundred years ago, Aristotle noticed that in certain inferences it is impossible for their premises to be true and their conclusions false. An example is the inference from "All men are mortals" and "All mortals fear death" to "All men fear death."

In modern logic, such inferences are said to be deductively valid. Aristotle discovered that the validity of an inference depends not on its subject matter, but only on the form of the premises and conclusion. All inferences of the form "All Fs are Gs, and All Gs are Hs, therefore All Fs are Hs" are valid. He described a number of such forms, which are called "syllogisms."

Until the 19th century, the subject of logic pretty much consisted of Aristotle's syllogisms. But syllogisms are only a small portion of all valid inferences, and do not include many of the patterns of valid inference that are employed in science and mathematics. In 1879 Gottlob Frege devised a much more general characterization of valid inference that is sufficient for representing mathematical and scientific reasoning. A descendant of Frege's system, called "First Order Logic with Identity,“ is now generally thought to be capable of representing mathematical theories and proofs, and is taught to all philosophy students.

Remember: An inference (or argument) is valid when it is impossible for its premises to be true and its conclusion false.

Additionally: In the 20th century two great mathematical results were proved concerning first order logic: it is complete, and it is undecidable. Kurt Godel demonstrated that it is possible to program a computer to list all the valid inferences (completeness), and Alonzo Church demonstrated that it is impossible to program a computer to determine whether or not every inference is valid (undecidability).

source:
30-Second Philosophies by Barry Loewer