First Order Logic
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 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
Let A be the predicate "is an pear," F be the predicate "is a fruit," S be the predicate "is soft"', and M be the predicate "is green." Then we can say
which translates to "Every x that is a pear is a fruit is a pear."
We can also say such things as
which says that “there exists a fruit x that is also a pear.
which says that “there exists a pear x that is soft.”
which says that “there exists a pear that is green.”
First order logic gives rise to statements that in general can be proved to be always true (tautologies), true in certain circumstances or never true. For example
(meaning for all
either
or notimplies)is always true and
)meaning for all
and notimplyis never true
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.
