Logic uses variables just as mathematics does, but in logic the variable stands for a statement that can be true or false. In mathematics a variable usually stands for a quantity. As in mathematics, there are connectors that can be used to combine statements into more complex statements. Unlike the plus and minus signs, along with asterisks as well as the and/or connectors used in mathematics, the logical connectors "And," "Or" and "Not" are more concerned with validity than with quantity. As with mathematics, there are rules for using logical connectives. For example, if the statement "A And B" is true, then A is true, but "A Or B" can be true, and A can be either true or false.
One of the modern improvements in formal logic is the invention of first order logic. This is an extension of formal logic, which includes functions and quantifiers to make formal logic more expressive. Functions replace the variables in traditional logic with statements that assign an attribute to an object or a variable. For example, "Human (Joe)" means Joe is a human, and "Male (Joe)" means that Joe is male. Functions can be combined like variables, so "Human (Joe) AND Male (Joe)" mean that Joe is a human male. Many of the same rules from formal logic still work for first order logic. For example, if "Human (Joe) Aand Male (Joe)" is true then so is "Human (Joe)."
Quantifiers give first order logic its representational power. The primary quantifiers are reverse-A and reverse-E. Reverse-A means "for all," and "reverse-A x (human x Implies good x)" means all humans are good. Reverse-E means "there exists," and "reverse-E x (human x Implies good x)" means that some humans are good. Notice that "reverse-A x reverse-E y (Human x Implies Father-of y, x)" means that every human has a father, but "reverse-E x reverse-A y (Human x Implies Father-of y, x)" means that there is someone that everybody is the father of.
Reverse-E is called the existential quantifier, and reverse-A is the universal quantifier. There are situations where either of these quantifiers can be eliminated as long as you understand what you are getting into. For example, in "reverse-E x (Even x And Prime x)", the quantifier can be eliminated if we substitute the variable "t" to get "Even(t) And Prime(t)" with the understanding that there is some value of t that makes this true. Getting rid of the reverse-E, in this manner, is called "existential elimination," and it is valid.