The Logical Operators

© 2003 by Teed Rockwell

Categorical logic always divides a simple sentence into two parts — a subject and a predicate — and deals with the relationships between them, and between predicates in different sentences. There is another kind of logic which treats a simple subject-predicate sentence as the basic unit (called a proposition) and strings those propositions together by means of certain logical connectors. This form of logic is called propositional logic. In categorical logic, we would translate the sentence “All dogs are mammals” with a separate symbol for the subject (dog) and the predicate (mammal). In other words, we would say “All D are M”. In propositional logic, a single symbol, such as “D” would refer to the entire sentence, which must include both a subject and a predicate. The inferences in propositional logic make connections between these complete sentences that enable us to derive one proposition (called a conclusion) from other propositions (called premises).

One of the most important discoveries of the twentieth century was the relationship between logic and mathematics, which was not obvious in Aristotle’s day. In the twentieth century, predicate logic and propositional logic were brought together by means of two logical operators called quantifiers. These are essentially the mathematical equivalent to Aristotle’s quantifier words, and they can fully translate into modern logic anything that can be expressed in Aristotelian syllogisms. This was an important achievement, because it revealed that logic was a branch of…