Firstorder predicate calculus
Firstorder predicate calculus or firstorder logic (FOL) is a theory in symbolic logic that permits the formulation of quantified statements such as "there is at least one X such that..." or "for any X, it is the case that...", where X is an element of a set called the domain of discourse.
Firstorder logic is distinguished from higherorder logic in that it does not allow statements such as "for every property, it is the case that..." or "there exists a set of objects such that..."
Nevertheless, firstorder logic is strong enough to formalize all of set theory and thereby virtually all of mathematics. Its restriction to quantification over individuals makes it difficult to use for the purposes of topology, but it is the classical logical theory underlying mathematics. It is a stronger theory than sentential logic, but a weaker theory than arithmetic, set theory, or Secondorder logic.
Like any logical theory, firstorder calculus consists of
 a specification of how to construct syntactically correct statements (the wellformed formulas)
 a set of axioms, each axiom being a wellformed formula itself
 a set of inference rules which allow one to prove theorems from axioms or earlier proven theorems.
While the set of inference rules in firstorder calculus is finite, the set of axioms may very well be and often is infinite. However we require that there is a general algorithm which can decide for a given wellformed formula whether it is an axiom or not. Furthermore, there should be an algorithm which can decide whether a given application of an inference rule is correct or not.
Table of contents 
2 Examples of wellformed formulas 3 Case study: firstorder Peano axioms 4 References 
Wellformed formulas of firstorder logic
The wellformed formulas contain: variables such as x, y, ... which are place holders for objects of the domain under consideration
 object constants such as 0, 1 or the empty set ø which stand for fixed individual objects in our domain
 predicate constants such (lessThan), (isIn), (equals) which stand for fixed relations between or properties of our objects. These are also called firstorder predicates to distinguish them from predicates that talk about predicates.
 function constants such as , which stand for fixed functions taking objects as arguments and returning objects as values
 logical connectives such as (and), (or), (conditional), (biconditional) (not), (thereExists existential quantifier) and (forAll or universal quantifier). All of these except for the last two are also used in propositional logic.
The object, predicate and function constants will typically depend on the particular domain we are talking about.
Examples of wellformed formulas
Instead of giving the lengthy definition of wellformed formulas, we will simply look at some examples from arithmetic together with their ordinary interpretation. Our domain here is the set of natural numbers:
 (i.e.: forAll x, thereExists y suchThat y isGreaterThan x )
 (i.e.: thereExists y suchThat forAll x, y isGreaterThan x )
 (i.e.: forAll x, ( thereExists y suchThat 6*y=x) implies (thereExists y suchThat 3*y=x) )
 (i.e.: thereExists x suchThat (not ThereExists y suchThat y < x ) )
Now one would have to write down 15 logical axioms and 2 inference rules to completely specify the firstorder calculus. How can one be sure that those axioms and rules are enough? This is the subject of Gödel's completeness theorem;: if you start out with some subject matter axioms and you look at a certain statement, then it is possible to prove that statement using only the subject matter axioms, the 15 logical axioms and the two inference rules if and only if the statement is true in every domain in which the subject matter axioms are true. (See also Leon Henkin)
Case study: firstorder Peano axioms
The Peano axioms for the natural numbers are sometimes formulated as secondorder statements (the induction axiom talks about all "properties" or all "sets of numbers"), but this is not necessary if one is willing to allow infinitely many firstorder axioms. A firstorder version of the Peano axioms follows.We are using the object constants 0 and 1, the function constants + and *, and the predicate constant =. Here are the axioms:

 i.e.: forAll x, not (0 = x + 1)
 i.e.: no number has 0 as its successor

 i.e.: forAll x, forAll y, not(x=y) implies not(x + 1 = y + 1)
 i.e.: if x ≠ y, then x+1 ≠ y+1

 i.e.: forAll x, x + 0 = x

 i.e.: forAll x, forAll y, (x + y) + 1 = x + (y + 1)
 i.e.: for all x and y, (x + y) + 1 = x + (y + 1)

 i.e.: forAll x, x * 0 = 0

 i.e.: forAll x, forAll y, x * (y + 1) = x * y + x
 i.e.: for all x and y, x * (y + 1) = x * y + x
 This is an axiom scheme consisting of infinitely many axioms. If P(x) is any formula involving the constants 0, 1, +, *, = and a single free variable x, then the following formula is an axiom:
 i.e.: ( P(0) and (forAll x, ( P(x) implies P(x + 1) ) ) ) implies (forAll x, P(x) )
 i.e.: if something is true for 0, and from its being true for x it follows that it is also true for x + 1, then it is true for all x (induction)
References
 Introduction to mathematical logic
 Metamath: a project to construct mathematics using an axiomatic system based on propositional calculus, predicate calculus, and set theory