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Well-formed formula

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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.

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 L {\displaystyle {\mathcal {L}}} are defined by the following formal grammar, written in BNF:

<alpha set> ::= p | q | r | s | t | u | ... (a 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 L {\displaystyle {\mathcal {L}}} .

Trivia

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

Notes

  2. More technically, propositional logic using the Fitch-style calculus.

See also

External links

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