First-Order Logic

The set of terms of first-order logic (also known as first-order predicate calculus) is defined by the following rules:

1. A variable is a term.

2. If  is an -place function symbol (with ) and , ...,  are terms, then  is a term.

If  is an -place predicate symbol (again with ) and , ...,  are terms, then  is an atomic statement.

Consider the sentential formulas  and , where  is a sentential formula,  is the universal quantifier ("for all"), and  is the existential quantifier ("there exists").  is called the scope of the respective quantifier, and any occurrence of variable  in the scope of a quantifier is bound by the closest  or . The variable  is free in the formula  if at least one of its occurrences in  is not bound by any quantifier within .

The set of sentential formulas of first-order predicate calculus is defined by the following rules:

1. Any atomic statement is a sentential formula.

2. If  and  are sentential formulas, then  (NOT ),  ( AND ),  ( OR ), and  ( implies ) are sentential formulas (cf. propositional calculus).

3. If  is a sentential formula in which  is a free variable, then  and  are sentential formulas.

In formulas of first-order predicate calculus, all variables are object variables serving as arguments of functions and predicates. (In second-order predicate calculus, variables may denote predicates, and quantifiers may apply to variables standing for predicates.) The set of axiom schemata of first-order predicate calculus is comprised of the axiom schemata of propositional calculus together with the two following axiom schemata:

	(1)
	(2)

where  is any sentential formula in which  occurs free,  is a term,  is the result of substituting  for the free occurrences of  in sentential formula , and all occurrences of all variables in  are free in .

Rules of inference in first-order predicate calculus are the Modus Ponens and the two following rules:

	(3)
	(4)

where  is any sentential formula in which  occurs as a free variable,  does not occur as a free variable in formula , and the notation means that if the formula above the line is a theorem formally deducted from axioms by application of inference rules, then the sentential formula below the line is also a formal theorem.

Similarly to propositional calculus, rules for introduction and elimination of  and  can be derived in first-order predicate calculus. For example, the following rule holds provided that  is the result of substituting variable  for the free occurrences of  in sentential formula  and all occurrences of  resulting from this substitution are free in ,

(5)

Gödel's completeness theorem established equivalence between valid formulas of first-order predicate calculus and formal theorems of first-order predicate calculus. In contrast to propositional calculus, use of truth tables does not work for finding valid sentential formulas in first-order predicate calculus because its truth tables are infinite. However, Gödel's completeness theorem opens a way to determine validity, namely by proof.

REFERENCES

Chang, C.-L. and Lee, R. C.-T. Symbolic Logic and Mechanical Theorem Proving. New York: Academic Press, 1997.

Kleene, S. C. Mathematical Logic. New York: Dover, 2002.Mendelson, E. Introduction to Mathematical Logic, 4th ed. London: Chapman & Hall, p. 12, 1997.