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| [[File:Formal languages.svg|thumb|300px|right|This diagram shows the [[Syntax (logic)|syntactic entities]] which may be constructed from [[formal language]]s. The [[symbol (formal)|symbols]] and [[string (computer science)|strings of symbols]] may be broadly divided into [[nonsense]] and well-formed formulas. A formal language can be thought of as identical to the set of its well-formed formulas. The set of well-formed formulas may be broadly divided into [[theorem]]s and non-theorems.]]
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| In [[mathematical logic]], a '''well-formed formula''', shortly '''wff''', often simply '''formula''', is a [[string (computer science)|word]] (i.e. a finite [[sequence]] of [[symbol (formal)|symbols]] from a given [[alphabet (computer science)|alphabet]]) that is part of a [[formal language]].<ref>Formulas are a standard topic in introductory logic, and are covered by all introductory textbooks, including Enderton (2001), Gamut (1990), and Kleene (1967)</ref> A formal language can be considered to be identical to the set containing all and only its formulas.
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| A formula is a [[syntax (logic)|syntactic]] formal object that can be informally given a semantic [[Formal semantics (logic)|meaning]].
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| ==Introduction==
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| A key use of formulas is in [[propositional logic]] and [[predicate logic]]s 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 variable]]s in φ have been instantiated. In formal logic, [[Mathematical proof|proof]]s can be represented by sequences of formulas with certain properties, and the final formula in the sequence is what is proven.
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| 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 being expressed, with the marks being a [[type-token distinction|token]] instance of formula. It is not necessary for the existence of a formula that there be any actual tokens of it. A formal language may thus have an infinite number of formulas regardless whether each formula has a token instance. Moreover, a single formula may have more than one token instance, if it is written more than once.
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| Formulas are quite often [[interpretation (logic)|interpreted]] as [[proposition]]s (as, for instance, in [[propositional logic]]). However formulas are [[syntax (logic)|syntactic entities]], and as such must be specified in a formal language without regard to any [[interpretation (logic)|interpretation]] of them. An interpreted formula may be the [[name]] of something, an [[adjective]], an [[adverb]], a [[preposition]], a [[phrase]], a [[clause]], an [[imperative sentence]], a [[string (computer science)|string]] of sentences, a string of names, etc.{{Citation broken|date=March 2011}}. A formula may even turn out to be [[nonsense]], if the symbols of the language are specified so that it does. Furthermore, a formula need not be given ''any'' interpretation.
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| ==Propositional calculus==
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| The formulas of [[propositional calculus]], also called [[propositional formula]]s,<ref>First-order logic and automated theorem proving, Melvin Fitting, Springer, 1996 [http://books.google.com/books?id=T8OMqvXvWWQC&lpg=PA11&dq=%22propositional%20formula%22&hl=fr&pg=PA11#v=onepage&q=%22propositional%20formula%22&f=false]</ref> are expressions such as <math>(A \land (B \lor C))</math>. Their definition begins with the arbitrary choice of a set ''V'' of [[propositional variable]]s. The alphabet consists of the letters in ''V'' along with the symbols for the [[logical connective|propositional connective]]s 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.
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| The formulas are [[inductive definition|inductively]] defined as follows:
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| * Each propositional variable is, on its own, a formula.
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| * If φ is a formula, then <math>\lnot</math>φ is a formula.
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| * 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 ↔.
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| This definition can also be written as a formal grammar in [[Backus–Naur form]], provided the set of variables is finite:
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| :<alpha set> ::= p | q | r | s | t | u | ... (the arbitrary finite set of propositional variables)
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| :<form> ::= <alpha set> | <math>\neg</math><form> | (<form><math>\wedge</math><form>) | (<form><math>\vee</math><form>) | (<form><math>\rightarrow</math><form>) | (<form><math>\leftrightarrow</math><form>)
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| Using this grammar, the sequence of symbols
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| :(((''p'' <math>\rightarrow</math> ''q'') <math>\wedge</math> (''r'' <math>\rightarrow</math> ''s'')) <math>\vee</math> (<math>\neg</math>''q'' <math>\wedge</math> <math>\neg</math>''s''))
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| is a formula, because it is grammatically correct. The sequence of symbols
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| :((''p'' <math>\rightarrow</math> ''q'')<math>\rightarrow</math>(''qq''))''p''))
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| is not a formula, because it does not conform to the grammar.
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| 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 [[Order of operations|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. <math>\neg</math> 2. <math>\rightarrow</math> 3. <math>\wedge</math> 4. <math>\vee</math>. Then the formula
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| :(((''p'' <math>\rightarrow</math> ''q'') <math>\wedge</math> (''r'' <math>\rightarrow</math> ''s'')) <math>\vee</math> (<math>\neg</math>''q'' <math>\wedge</math> <math>\neg</math>''s''))
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| may be abbreviated as
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| :''p'' <math>\rightarrow</math> ''q'' <math>\wedge</math> ''r'' <math>\rightarrow</math> ''s'' <math>\vee</math> <math>\neg</math>''q'' <math>\wedge</math> <math>\neg</math>''s''
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| 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. <math>\neg</math> 2. <math>\wedge</math> 3. <math>\vee</math> 4. <math>\rightarrow</math>, then the same formula above (without parentheses) would be rewritten as
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| :(''p'' <math>\rightarrow</math> (''q'' <math>\wedge</math> ''r'')) <math>\rightarrow</math> (''s'' <math>\vee</math> ((<math>\neg</math>''q'') <math>\wedge</math> (<math>\neg</math>''s'')))
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| ==Predicate logic==
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| The definition of a formula in [[first-order logic]] <math>\mathcal{QS}</math> is relative to the [[Signature (mathematical logic)|signature]] of the theory at hand. This signature specifies the constant symbols, relation symbols, and function symbols of the theory at hand, along with the [[Arity|arities]] of the function and relation symbols.
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| 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.
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| #Any variable is a term.
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| #Any constant symbol from the signature is a term
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| #an expression of the form ''f''(''t''<sub>1</sub>,...,''t''<sub>''n''</sub>), where ''f'' is an ''n''-ary function symbol, and ''t''<sub>1</sub>,...,''t''<sub>''n''</sub> are terms, is again a term.
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| The next step is to define the [[atomic formula]]s.
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| #If ''t''<sub>1</sub> and ''t''<sub>2</sub> are terms then ''t''<sub>1</sub>=''t''<sub>2</sub> is an atomic formula
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| #If ''R'' is an ''n''-ary relation symbol, and ''t''<sub>1</sub>,...,''t''<sub>''n''</sub> are terms, then ''R''(''t''<sub>1</sub>,...,''t''<sub>''n''</sub>) is an atomic formula
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| Finally, the set of formulas is defined to be the smallest set containing the set of atomic formulas such that the following holds:
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| #<math>\neg\phi</math> is a formula when <math>\ \phi</math> is a formula
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| #<math>(\phi \land \psi)</math> and <math>(\phi \lor \psi)</math> are formulas when <math>\ \phi</math> and <math>\ \psi</math> are formulas;
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| #<math>\exists x\, \phi</math> is a formula when <math>\ x</math> is a variable and <math>\ \phi</math> is a formula;
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| #<math>\forall x\, \phi</math> is a formula when <math>\ x</math> is a variable and <math>\ \phi</math> is a formula (alternatively, <math>\forall x\, \phi</math> could be defined as an abbreviation for <math>\neg\exists x\, \neg\phi</math>).
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| If a formula has no occurrences of <math>\exists x</math> or <math>\forall x</math>, for any variable <math>\ x</math>, 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.
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| ==Atomic and open formulas==
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| {{Main|Atomic formula}}
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| An ''atomic formula'' is a formula that contains no [[logical connective]]s nor [[Quantification|quantifiers]], or equivalently a formula that has no strict subformulas.
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| The precise form of atomic formulas depends on the formal system under consideration; for [[propositional logic]], for example, the atomic formulas are the [[propositional variable]]s. For [[predicate logic]], the atoms are predicate symbols together with their arguments, each argument being a [[Formation rules|term]].
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| According to some terminology, an ''open formula'' is formed by combining atomic formulas using only logical connectives, to the exclusion of quantifiers.<ref>Handbook of the history of logic, (Vol 5, Logic from Russell to Church), Tarski's logic by Keith Simmons, D. Gabbay and J. Woods Eds, p568 [http://books.google.fr/books?id=K5dU9bEKencC&lpg=PA568&dq=%22open%20formula%22&pg=PA568#v=onepage&q=open%20formula&f=false].</ref> This has not to be confused with a formula which is not closed.
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| ==Closed formulas==
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| {{Main|Sentence (mathematical logic)}}
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| A ''closed formula'', also ''[[ground expression|ground]] formula'' or ''sentence'', is a formula in which there are no [[Free and bound variables|free occurrences]] of any [[variable (mathematics)|variable]]. If '''A''' is a formula of a first-order language in which the variables ''v<sub>1</sub>'', ..., ''v<sub>n</sub>'' have free occurrences, then '''A''' preceded by {{all}}''v<sub>1</sub>'' ... {{all}}''v<sub>n</sub>'' is a closure of '''A'''.
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| ==Properties applicable to formulas==
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| * A formula '''A''' in a language <math>\mathcal{Q}</math> is ''[[satisfiability and validity|valid]]'' if it is true for every [[interpretation (logic)|interpretation]] of <math>\mathcal{Q}</math>.
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| * A formula '''A''' in a language <math>\mathcal{Q}</math> is ''[[satisfiability and validity|satisfiable]]'' if it is true for some [[interpretation (logic)|interpretation]] of <math>\mathcal{Q}</math>.
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| * A formula '''A''' of the language of [[Peano arithmetic|arithmetic]] is ''decidable'' if it represents a [[decidable set]], i.e. if there is an [[effective method]] which, given a [[substitution of variables|substitution]] of the free variables of '''A''', says that either the resulting instance of '''A''' is provable or its negation is.
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| ==Usage of the terminology==
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| In earlier works on mathematical logic (e.g. by Church<ref>Alonzo Church, [1996] (1944), Introduction to mathematical logic, page 49</ref>), 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.
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| Several authors simply say formula.<ref>Hilbert, David; Ackermann, Wilhelm (1950) [1937], Principles of Mathematical Logic, New York: Chelsea</ref><ref>Hodges, Wilfrid (1997), A shorter model theory, Cambridge University Press, ISBN 978-0-521-58713-6</ref><ref>Barwise, Jon, ed. (1982), Handbook of Mathematical Logic, Studies in Logic and the Foundations of Mathematics, Amsterdam: North-Holland, ISBN 978-0-444-86388-1</ref><ref>Cori, Rene; Lascar, Daniel (2000), Mathematical Logic: A Course with Exercises, Oxford University Press, ISBN 978-0-19-850048-3</ref> Modern usages (especially in the context of computer science with mathematical software such as [[List of model checking tools|model checkers]], [[automated theorem prover]]s, [[Interactive theorem proving|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 [[order of operations|parenthesizing convention]], using [[Polish notation|Polish]] or [[infix notation|infix]] notation, etc.) as a mere notational problem.
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| However, the expression well-formed formulas can still be found in various works,<ref>Enderton, Herbert [2001] (1972), A mathematical introduction to logic (2nd ed.), Boston, MA: Academic Press, ISBN 978-0-12-238452-3</ref><ref>R. L. Simpson (1999), Essentials of Symbolic Logic, page 12</ref><ref>Mendelson, Elliott [2010] (1964), An Introduction to Mathematical Logic (5th ed.), London: Chapman & Hall</ref> these authors using the name well-formed formula without necessarily opposing it to the old sense of formula as arbitrary string of symbols so that it is no longer common in mathematical logic to refer to arbitrary strings of symbols in the old sense of formulas.
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| The expression "well-formed formulas" (WFF) also pervaded in popular culture. Indeed, ''WFF'' is part of an esoteric pun used in the name of the academic game "[[Academic_Games#WFF_.27N_Proof|WFF 'N PROOF: The Game of Modern Logic]]," by Layman Allen,<ref>Ehrenburg 2002</ref> 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]]).<ref>More technically, [[Propositional calculus|propositional logic]] using the [[Fitch-style calculus]].</ref> Its name is an echo of ''[[whiffenpoof]]'', a [[nonsense word]] used as a [[Cheering|cheer]] at [[Yale University]] made popular in ''The Whiffenpoof Song'' and [[The Whiffenpoofs]].<ref>Allen (1965) acknowledges the pun.</ref>
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| ==See also==
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| {{Portal|Logic}}
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| * [[Ground expression]]
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| ==Notes==
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| {{Reflist}}
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| ==References==
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| *{{citation
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| |first1=Layman E.
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| |last1= Allen
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| |title=Toward Autotelic Learning of Mathematical Logic by the WFF 'N PROOF Games
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| | journal= Mathematical Learning: Report of a Conference Sponsored by the Committee on Intellective Processes Research of the Social Science Research Council
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| |series= Monographs of the Society for Research in Child Development
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| |volume=30
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| |issue=1
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| |year=1965
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| |pages=29–41
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| }}
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| * {{Citation
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| | last1=Boolos
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| | first1=George
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| | author1-link=George Boolos
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| | last2=Burgess
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| | first2=John
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| | last3=Jeffrey
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| | first3=Richard
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| | author3-link=Richard Jeffrey
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| | title=Computability and Logic
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| | publisher=[[Cambridge University Press]]
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| | edition=4th
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| | isbn=978-0-521-00758-0 (pb.)
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| | year=2002}}
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| * {{cite news
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| | first=Rachel
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| | last=Ehrenberg
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| | title=He's Positively Logical
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| | date=Spring 2002
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| | publisher=University of Michigan
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| | url=http://www.umich.edu/~newsinfo/MT/02/Spr02/mt9s02.html
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| | work=Michigan Today
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| | accessdate=2007-08-19
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| }}
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| * {{Citation
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| | last1=Enderton
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| | first1=Herbert
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| | title=A mathematical introduction to logic
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| | publisher=[[Academic Press]]
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| | location=Boston, MA
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| | edition=2nd
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| | isbn=978-0-12-238452-3
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| | year=2001
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| }}
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| * {{Citation
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| | last1=Gamut
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| | first1=L.T.F.
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| | title=Logic, Language, and Meaning, Volume 1: Introduction to Logic
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| |publisher= University Of Chicago Press
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| | year= 1990
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| | isbn=0-226-28085-3
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| }}
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| *{{Citation
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| | author-last=Hodges
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| | author-first=Wilfrid
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| | section=Classical Logic I: First-Order Logic
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| | editor1-last=Goble
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| | editor1-first=Lou
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| | title=The Blackwell Guide to Philosophical Logic
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| | publisher=Blackwell
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| | isbn=978-0-631-20692-7
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| | year=2001
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| }}
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| *{{Citation
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| | last1=Hofstadter
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| | first1=Douglas
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| | author1-link=Douglas Hofstadter
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| | title=Gödel, Escher, Bach: An Eternal Golden Braid
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| | publisher=[[Penguin Books]]
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| | isbn=978-0-14-005579-5
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| | year=1980
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| }}
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| * {{Citation
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| | last1=Kleene
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| | first1=Stephen Cole
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| | author1-link=Stephen Kleene
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| | title=Mathematical logic
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| | origyear=1967
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| | publisher=[[Dover Publications]]
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| | location=New York
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| | isbn=978-0-486-42533-7
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| | mr=1950307
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| | year=2002
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| }}
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| * {{Citation|last=Rautenberg|first=Wolfgang|authorlink=Wolfgang Rautenberg|doi=10.1007/978-1-4419-1221-3|title=A Concise Introduction to Mathematical Logic|url=http://www.springerlink.com/content/978-1-4419-1220-6/|publisher=[[Springer Science+Business Media]]|location=[[New York City|New York]]|edition=3rd|isbn=978-1-4419-1220-6|year=2010}}
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| ==External links==
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| *[http://www.cs.odu.edu/~toida/nerzic/content/logic/pred_logic/construction/wff_intro.html Well-Formed Formula for First Order Predicate Logic] - includes a short [[Java (programming language)|Java]] quiz.
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| *[http://www.apronus.com/provenmath/formulas.htm Well-Formed Formula at ProvenMath]
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| *[http://wffnproof.com/ WFF N PROOF game site]
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| {{logic}}
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| {{DEFAULTSORT:Well-Formed Formula}}
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| [[Category:Formal languages]]
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| [[Category:Metalogic]]
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| [[Category:Syntax (logic)]]
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| [[Category:Mathematical logic]]
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| [[Category:Logical expressions]]
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