In the formal languages used in mathematical logic and computer science, a well-formed formula or simply formula[2] (often abbreviated wff, pronounced "wiff" or "wuff") is an idea, abstraction or concept which is expressed using the symbols and formation rules (also called the formal grammar) of a particular formal language. To say that a string of symbols is a wff with respect to a given formal grammar is equivalent to saying that belongs to the language generated by . A formal language can be identified with the set of its wffs.
Although the term "well-formed formula" is commonly used to refer to the written marks, for instance, on a piece of paper or chalkboard which are being used to express an idea; it is more precisely understood as the idea being expressed and the marks as a token instance of the well formed formula. Two different strings of marks may be tokens of the same well-formed formula. This is to say that there may be many different formulations of the same the idea.
It is not necessary for the existence of a well-formed formula that there be any actual tokens of it. Formal languages may have an infinite number of well-formed formula, regardless of whether there actually exist any token instances of them.
Well-formed formulas are quite often interpreted as propositions (as, for instance, in propositional logic). However wffs are syntactic entities, and as such must be specified in a formal language without regard to any interpretation of them. An interpreted well-formed formula may be the name of something, an adjective, an adverb, a preposition, a phrase, a clause, an imperative sentence, a string of sentences, a string of names, etcetera. A well-formed formula may even turn out to be nonsense, if the symbols of the language are specified so that it does. Furthermore, a well-formed formula need not be given any interpretation.
The set of well-formed formulas of a particular formal language is determined by a fiat of its creator, who simply lays down what things are to be wffs of his language. Usually this is done by specifying a set of symbols, and a set of formation rules.
A key use of wffs is in propositional logic and predicate logics 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 variables in φ have been instantiated.
In formal logic, proofs can be represented by 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.
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