Логикалық формулалардың синтаксистік құрылымы

Syntactically correct logical formula

In mathematical logic, propositional logic and predicate logic, a well formed formula, abbreviated WFF or wff, often simply formula, is a finite sequence of symbols from a given alphabet that is part of a formal language. A formal language can be identified with the set of formulas in the language. A formula is a syntactic object that can be given a semantic meaning by means of an interpretation. Two key uses of formulas are in propositional logic and predicate logic.

A key use of formulas is in propositional logic and predicate logic such as first order logic. In those contexts, a formula is a string of symbols φ for which it makes sense to ask "is φ true? ", once any free variables in φ have been instantiated. In formal logic, proofs can be represented by sequences of formulas with certain properties, and the final formula in the sequence is what is proven. Although the term "formula" may be used for written marks (for instance, on a piece of paper or chalkboard), it is more precisely understood as the sequence of symbols being expressed, with the marks being a token instance of formula. This distinction between the vague notion of "property" and the inductively defined notion of well formed formula has roots in Weyl's 1910 paper "Uber die Definitionen der mathematischen Grundbegriffe". Thus the same formula may be written more than once, and a formula might in principle be so long that it cannot be written at all within the physical universe. Formulas themselves are syntactic objects. They are given meanings by interpretations. For example, in a propositional formula, each propositional variable may be interpreted as a concrete proposition, so that the overall formula expresses a relationship between these propositions. A formula need not be interpreted, however, to be considered solely as a formula.

An atomic formula is a formula that contains no logical connectives nor quantifiers, or equivalently a formula that has no strict subformulas. The precise form of atomic formulas depends on the formal system 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. According to some terminology, an open formula is formed by combining atomic formulas using only logical connectives, to the exclusion of quantifiers. This is not to be confused with a formula which is not closed.

A closed formula, also ground formula or sentence, is a formula in which there are no free occurrences of any variable. If A is a formula of a first order language in which the variables v1, , vn have free occurrences, then A preceded by ∀v1 ⋯ ∀vn is a closure of A.

A formula A in a language is valid if it is true for every interpretation of A formula A in a language is satisfiable if it is true for some interpretation of A formula A of the language of arithmetic is decidable if it represents a decidable set, i. e. if there is an effective method which, given a substitution of the free variables of A, says that either the resulting instance of A is provable or its negation is.

In earlier works on mathematical logic (e. g. by Church), formulas referred to any strings of symbols and among these strings, well formed formulas were the strings that followed the formation rules of (correct) formulas. Several authors simply say formula. Modern usages (especially in the context of computer science with mathematical software such as model checkers, automated theorem provers, interactive theorem provers) tend to retain of the notion of formula only the algebraic concept and to leave the question of well formedness, i. e. of the concrete string representation of formulas (using this or that symbol for connectives and quantifiers, using this or that parenthesizing convention, using Polish or infix notation, etc.) as a mere notational problem. While the expression well formed formula is still in use, these authors do not necessarily use it in contradistinction to the old sense of formula, which is no longer common in mathematical logic. The expression "well formed formulas" (WFF) also crept into popular culture. WFF is part of an esoteric pun used in the name of the academic game "WFF 'N PROOF: The Game of Modern Logic", by Layman Allen, developed while he was at Yale Law School (he was later a professor at the University of Michigan). The suite of games is designed to teach the principles of symbolic logic to children (in Polish notation). Its name is an echo of whiffenpoof, a nonsense word used as a cheer at Yale University made popular in The Whiffenpoof Song and The Whiffenpoofs.