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:

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

::= | $neg$ | ($wedge$) | ($vee$) | ($rightarrow$) | ($leftrightarrow$)

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

See also

• Woodhull Freedom Foundation & Federation (acronym WFF)

Notes