Predicate Logic Formula
A Predicate Logic Formula is a Well-Formed Formula of the Predicate Logic System in which a predicate variable is followed by any number of individual variables.
- AKA: Atomic Formula.
- Example(s):
- Counter-Example(s):
- See: Satisfiable, Mathematical Logic, Formula (Mathematical Logic), Proposition, Logical Connective, Subformula, Well-Formed Formula, Propositional Logic, Propositional Variable, Predicate Logic, First-Order Logic#Formation Rules, Model Theory.
References
2019a
- (Wikipedia, 2019) ⇒ https://en.wikipedia.org/wiki/Atomic_formula Retrieved:2019-6-9.
- In mathematical logic, an atomic formula (also known simply as an atom) is a formula with no deeper propositional structure, that is, a formula that contains no logical connectives or equivalently a formula that has no strict subformulas. Atoms are thus the simplest well-formed formulas of the logic. Compound formulas are formed by combining the atomic formulas using the logical connectives.
The precise form of atomic formulas depends on the logic under consideration; for propositional logic, for example, the atomic formulas are the propositional variables. For predicate logic, the atoms are predicate symbols together with their arguments, each argument being a term. In model theory, atomic formula are merely strings of symbols with a given signature, which may or may not be satisfiable with respect to a given model.
- In mathematical logic, an atomic formula (also known simply as an atom) is a formula with no deeper propositional structure, that is, a formula that contains no logical connectives or equivalently a formula that has no strict subformulas. Atoms are thus the simplest well-formed formulas of the logic. Compound formulas are formed by combining the atomic formulas using the logical connectives.
2019b
- (Nelson, 2019) ⇒ Randal C. Nelson (2019). https://www.cs.rochester.edu/~nelson/courses/csc_173/predlogic/expressions.html Retrieved:2019-6-9.
- QUOTE: A logical expression in predicate logic has much the same form as a logical expression in propositional logic, with the addition of atomic formulae (ie., predicates), and the universal and existential quantifiers.
An atomic formula is a logical expression.
- 1. A predicate with all constant arguments is a ground atomic formula.
- A proposition is a predicate with no arguments, and therefore is a ground atomic formula.
- A predicate with at least one variable argument is a nonground atomic formula.
- A literal is either an atomic formula or its negation.
- 2. If L1 and L2 are logical expressions, then L1 AND L2, L1 OR L2, NOT L1, L1 -> L2, and L1 == L2 are logical expressions.
- 3. If L1 is a logical expressions, then (A X) L1 is a logical expression.
- 4. If L1 is a logical expressions, then (E X) L1 is a logical expression.
- QUOTE: A logical expression in predicate logic has much the same form as a logical expression in propositional logic, with the addition of atomic formulae (ie., predicates), and the universal and existential quantifiers.
- Quantifiers have the highest precedence in logical expressions.
2018
- (Schagrin & Hughes, 2018) ⇒ Morton L. Schagrin, and G.E. Hughes (November 02, 2018). Formal logic: https://www.britannica.com/topic/formal-logic/The-predicate-calculus Retrieved: 2019-06-09. In: Encyclopaedia Britannica.
- QUOTE: In general, a predicate variable followed by any number of individual variables is a wff of the predicate calculus. Such a wff is known as an atomic formula, and the predicate variable in it is said to be of degree n, if n is the number of individual variables following it. The degree of a predicate variable is sometimes indicated by a superscript—e.g., ϕxyz may be written as ϕ3xyz; ϕ3xy would then be regarded as not [[well formed][. This practice is theoretically more accurate, but the superscripts are commonly omitted for ease of reading when no confusion is likely to arise.