Well-formed formula

In mathematical logic, a well-formed formula (often abbreviated WFF, pronounced "wiff") is a symbol or string of symbols (a formula) that is generated by the formal grammar of a formal language. To say that a string $\ S$ is a WFF with respect to a given formal grammar $\ G$ is equivalent to saying that $\ S$ belongs to the language generated by $\ G$, i.e. $S\in {\boldsymbol {L}}(G)$. A formal language can be identified with the set of its WFFs.

In formal logic, proofs are sequences of WFFs with certain properties, and the final WFF in the sequence is what is proven. This final WFF is called a theorem when it plays a significant role in the theory being developed, or a lemma when it plays an accessory role in the proof of a theorem.

Example

The well-formed formulae of the propositional calculus ${\mathcal {L}}$ 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> | $\neg$<wff> | (<wff>$\wedge$<wff>) | (<wff>$\vee$<wff>) | (<wff>$\rightarrow$<wff>) | (<wff>$\leftrightarrow$<wff>)

The sequence of symbols

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

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

((p $\rightarrow$ q)$\rightarrow$(qq))p))

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

Note that sometimes WFF may become very hard to read, owing to, for example, the proliferation of parentheses. To alleviate this last phenomenon, precedence rules are assumed among the operators, making some operators more binding than others. For example, assuming the precedence (from most binding to least binding) 1. $\neg$   2. $\rightarrow$  3. $\wedge$  4. $\vee$, the above correct expression may be written as:

p $\rightarrow$ q $\wedge$ r $\rightarrow$ s $\vee$ $\neg$q $\wedge$ $\neg$s

This is, however, only a convention used to simplify the written representation of a WFF (commonly used in programming languages).

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.

See also

• 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