First-order logic is symbolized reasoning in which each sentence, or statement, is broken down into a subject and a predicate. The predicate modifies or defines the properties of the subject. In first-order logic, a predicate can only refer to a single subject. First-order logic is also known as first-order predicate calculus or first-order functional calculus.
A sentence in first-order logic is written in the form Px or P(x), where P is the predicate and x is the subject, represented as a variable. Complete sentences are logically combined and manipulated according to the same rules as those used in Boolean algebra.
In first-order logic, a sentence can be structured using the universal quantifier (symbolized ) or the existential quantifier ( ). Consider a subject that is a variable represented by x. Let A be a predicate "is an apple," F be a predicate "is a fruit," S be a predicate "is sour"', and M be a predicate "is mushy." Then we can say
x : Ax Fx
which translates to "For all x, if x is an apple, then x is a fruit." We can also say such things as
x : Fx Ax
x : Ax Sx
x : Ax Mx
where the existential quantifier translates as "For some."
First-order logic can be useful in the creation of computer programs. It is also of interest to researchers in artificial intelligence ( AI ). There are more powerful forms of logic, but first-order logic is adequate for most everyday reasoning. The Incompleteness Theorem , proven in 1930, demonstrates that first-order logic is in general undecidable. That means there exist statements in this logic form that, under certain conditions, cannot be proven either true or false.
Also see Mathematical Symbols .