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.

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