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In
mathematical logic Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of formal ...
, propositional logic and predicate logic, a well-formed formula, abbreviated WFF or wff, often simply formula, is a finite
sequence In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters. Like a set, it contains members (also called ''elements'', or ''terms''). The number of elements (possibly infinite) is call ...
of symbols from a given
alphabet An alphabet is a standardized set of basic written graphemes (called letters) that represent the phonemes of certain spoken languages. Not all writing systems represent language in this way; in a syllabary, each character represents a sylla ...
that is part of a
formal language In logic, mathematics, computer science, and linguistics, a formal language consists of words whose letters are taken from an alphabet and are well-formed according to a specific set of rules. The alphabet of a formal language consists of sym ...
. A formal language can be identified with the set of formulas in the language. A formula is a
syntactic In linguistics, syntax () is the study of how words and morphemes combine to form larger units such as phrases and sentences. Central concerns of syntax include word order, grammatical relations, hierarchical sentence structure (constituency), ...
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.

# Introduction

A key use of formulas is in propositional logic and predicate logic such as
first-order logic First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantifie ...
. In those contexts, a formula is a string of symbols φ for which it makes sense to ask "is φ true?", once any
free variable In mathematics, and in other disciplines involving formal languages, including mathematical logic and computer science, a free variable is a notation (symbol) that specifies places in an expression where substitution may take place and is not ...
s 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.

# Propositional calculus

The formulas of
propositional calculus Propositional calculus is a branch of logic. It is also called propositional logic, statement logic, sentential calculus, sentential logic, or sometimes zeroth-order logic. It deals with propositions (which can be true or false) and relations ...
, also called propositional formulas, are expressions such as $\left(A \land \left(B \lor C\right)\right)$. Their definition begins with the arbitrary choice of a set ''V'' of propositional variables. The alphabet consists of the letters in ''V'' along with the symbols for the propositional connectives and parentheses "(" and ")", all of which are assumed to not be in ''V''. The formulas will be certain expressions (that is, strings of symbols) over this alphabet. The formulas are inductively defined as follows: * Each propositional variable is, on its own, a formula. * If φ is a formula, then ¬φ is a formula. * If φ and ψ are formulas, and • is any binary connective, then ( φ • ψ) is a formula. Here • could be (but is not limited to) the usual operators ∨, ∧, →, or ↔. This definition can also be written as a formal grammar in Backus–Naur form, provided the set of variables is finite: Using this grammar, the sequence of symbols :(((''p'' → ''q'') ∧ (''r'' → ''s'')) ∨ (¬''q'' ∧ ¬''s'')) is a formula, because it is grammatically correct. The sequence of symbols :((''p'' → ''q'')→(''qq''))''p'')) is not a formula, because it does not conform to the grammar. A complex formula may be difficult to read, owing to, for example, the proliferation of parentheses. To alleviate this last phenomenon, precedence rules (akin to the standard mathematical order of operations) are assumed among the operators, making some operators more binding than others. For example, assuming the precedence (from most binding to least binding) 1. ¬   2. →  3. ∧  4. ∨. Then the formula :(((''p'' → ''q'') ∧ (''r'' → ''s'')) ∨ (¬''q'' ∧ ¬''s'')) may be abbreviated as :''p'' → ''q'' ∧ ''r'' → ''s'' ∨ ¬''q'' ∧ ¬''s'' This is, however, only a convention used to simplify the written representation of a formula. If the precedence was assumed, for example, to be left-right associative, in following order: 1. ¬   2. ∧  3. ∨  4. →, then the same formula above (without parentheses) would be rewritten as :(''p'' → (''q'' ∧ ''r'')) → (''s'' ∨ ((¬''q'') ∧ (¬''s'')))

# Predicate logic

The definition of a formula in
first-order logic First-order logic—also known as predicate logic, quantificational logic, and first-order predicate calculus—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantifie ...
$\mathcal$ is relative to the
signature A signature (; from la, signare, "to sign") is a handwritten (and often stylized) depiction of someone's name, nickname, or even a simple "X" or other mark that a person writes on documents as a proof of identity and intent. The writer of a ...
of the theory at hand. This signature specifies the constant symbols, predicate symbols, and function symbols of the theory at hand, along with the arities of the function and predicate symbols. The definition of a formula comes in several parts. First, the set of terms is defined recursively. Terms, informally, are expressions that represent objects from the domain of discourse. #Any variable is a term. #Any constant symbol from the signature is a term #an expression of the form ''f''(''t''1,…,''t''''n''), where ''f'' is an ''n''-ary function symbol, and ''t''1,…,''t''''n'' are terms, is again a term. The next step is to define the
atomic formula In mathematical logic, an atomic formula (also known as an atom or a prime formula) 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 subformu ...
s. #If ''t''1 and ''t''2 are terms then ''t''1=''t''2 is an atomic formula #If ''R'' is an ''n''-ary predicate symbol, and ''t''1,…,''t''''n'' are terms, then ''R''(''t''1,…,''t''''n'') is an atomic formula Finally, the set of formulas is defined to be the smallest set containing the set of atomic formulas such that the following holds: #$\neg\phi$ is a formula when $\phi$ is a formula #$\left(\phi \land \psi\right)$ and $\left(\phi \lor \psi\right)$ are formulas when $\phi$ and $\psi$ are formulas; #$\exists x\, \phi$ is a formula when $x$ is a variable and $\phi$ is a formula; #$\forall x\, \phi$ is a formula when $x$ is a variable and $\phi$ is a formula (alternatively, $\forall x\, \phi$ could be defined as an abbreviation for $\neg\exists x\, \neg\phi$). If a formula has no occurrences of $\exists x$ or $\forall x$, for any variable $x$, then it is called quantifier-free. An ''existential formula'' is a formula starting with a sequence of existential quantification followed by a quantifier-free formula.

# Atomic and open formulas

An ''atomic formula'' is a formula that contains no
logical connective In logic, a logical connective (also called a logical operator, sentential connective, or sentential operator) is a logical constant. They can be used to connect logical formulas. For instance in the syntax of propositional logic, the binary ...
s 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.

# Closed formulas

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 have free occurrences, then A preceded by is a closure of A.

# Properties applicable to formulas

* A formula A in a language $\mathcal$ is '' valid'' if it is true for every interpretation of $\mathcal$. * A formula A in a language $\mathcal$ is '' satisfiable'' if it is true for some interpretation of $\mathcal$. * A formula A of the language of
arithmetic Arithmetic () is an elementary part of mathematics that consists of the study of the properties of the traditional operations on numbers— addition, subtraction, multiplication, division, exponentiation, and extraction of roots. In the 19t ...
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.

# Usage of the terminology

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 prover Automated theorem proving (also known as ATP or automated deduction) is a subfield of automated reasoning and mathematical logic dealing with proving mathematical theorems by computer programs. Automated reasoning over mathematical proof was a m ...
s, 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 Yale Law School (Yale Law or YLS) is the law school of Yale University, a private research university in New Haven, Connecticut. It was established in 1824 and has been ranked as the best law school in the United States by '' U.S. News & World R ...
(he was later a professor at the
University of Michigan , mottoeng = "Arts, Knowledge, Truth" , former_names = Catholepistemiad, or University of Michigania (1817–1821) , budget = $10.3 billion (2021) , endowment =$17 billion (2021)As o ...
). 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 Yale University is a private research university in New Haven, Connecticut. Established in 1701 as the Collegiate School, it is the third-oldest institution of higher education in the United States and among the most prestigious in the wor ...
made popular in ''The Whiffenpoof Song'' and The Whiffenpoofs.Allen (1965) acknowledges the pun.

* Ground expression * Well-defined expression

# References

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