First order logic
First order logic is a type of logic which is used in certain branches of mathematics and philosophy. First order logic enables the definition of a syntax which is independent of the mathematical or logical terms.
In first order logic, reasoning can be done from two points of view: either using syntax alone, or including semantic terms. First order logic is different from propositional logic: In first order logic, there are quantifiers, called for all (written as ) and there is at least one (written as ). Negation, conjunction, inclusive disjunction, exclusive disjunction and implication are all defined the same way as in propositional logic. Because of that, first order logic can be thought of as an extension of propositional logic.
The completeness of first order logic, the result which asserts the equivalence between valid formulas and formal theorems, is established by Gödel. Together with Zermelo–Fraenkel set theory, first order logic is the foundation of many branches of modern mathematics.