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In logic, WFF (pronounced "wiff") is an abbreviation for well-formed formula. Given a formal grammar, a WFF is any string that is generated by that grammar. To say that a string ${\displaystyle S}$ is a WFF with respect to a given formal grammar ${\displaystyle G}$ is equivalent to saying that ${\displaystyle S}$ belongs to the language generated by ${\displaystyle G}$, i.e. ${\displaystyle S\in {\boldsymbol {L}}(G)}$.

In formal logic, proofs are sequences of WFFs with certain properties, and the final WFF in the sequence is what is proven.

Example

The well-formed formulae of the propositional calculus ${\displaystyle {\mathcal {L}}}$ are defined by the following formal grammar, written in BNF:

<alpha set> ::= p | q | r | s | t | u | ... (arbitrary finite set of propositional variables)

<wff> ::= <alpha set> | ${\displaystyle \neg }$<wff> | (<wff>${\displaystyle \wedge }$<wff>) | (<wff>${\displaystyle \vee }$<wff>) | (<wff>${\displaystyle \rightarrow }$<wff>) | (<wff>${\displaystyle \leftrightarrow }$<wff>)

The sequence of symbols

(((p ${\displaystyle \rightarrow }$ q) ${\displaystyle \wedge }$ (r ${\displaystyle \rightarrow }$ s)) ${\displaystyle \wedge }$ (${\displaystyle \neg }$q ${\displaystyle \vee }$ ${\displaystyle \neg }$s))

is a WFF because it is grammatically correct. The sequence of symbols

((p ${\displaystyle \rightarrow }$ q)${\displaystyle \rightarrow }$(qq))p))

is not a WFF, because it does not conform to the grammar of ${\displaystyle {\mathcal {L}}}$.

Trivia

WFF is part of an esoteric pun used in the name of "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 a pun on whiffenpoof, a nonsense word used as a cheer at Yale University made popular in The Whiffenpoof Song and The Whiffenpoofs.

Notes

1. Ehrenberg, Rachel (Spring 2002). "He's Positively Logical". Michigan Today. University of Michigan. Retrieved 2007-08-19. {{cite news}}: Cite has empty unknown parameter: |coauthors= (help)
2. More technically, propositional logic using the Fitch-style calculus.

See also

• Formula (mathematical logic)
• Woodhull Freedom Foundation & Federation (acronym WFF)

External links

• Well-Formed Formula for First Order Predicate Logic - includes a short Java quiz.
• Well-Formed Formula at ProvenMath
• WFF N PROOF game site

• Logic