Inmathematical logic,formation rules are rules for describingwell-formed words over thealphabet of aformal language.^[1] These rules only address the location and manipulation of the strings of the language. It does not describe anything else about a language, such as itssemantics (i.e. what the strings mean). (See alsoformal grammar).

Aformal language is an organizedset ofsymbols the essential feature being that it can be precisely defined in terms of just the shapes and locations of those symbols. Such a language can be defined, then, without anyreference to anymeanings of any of its expressions; it can exist before anyinterpretation is assigned to it—that is, before it has any meaning. Aformal grammar determines which symbols and sets of symbols areformulas in a formal language.

Aformal system (also called alogical calculus, or alogical system) consists of a formal language together with adeductive apparatus (also called adeductive system). The deductive apparatus may consist of a set oftransformation rules (also calledinference rules) or a set ofaxioms, or have both. A formal system is used toderive one expression from one or more other expressions. Propositional and predicate calculi are examples of formal systems.

The formation rules of apropositional calculus may, for instance, take a form such that;

• if we take Φ to be a propositional formula we can also take${\displaystyle \mathrm{\neg }}$Φ to be a formula;
• if we take Φ and Ψ to be a propositional formulas we can also take (Φ${\displaystyle \wedge }$ Ψ), (Φ${\displaystyle \to }$ Ψ), (Φ${\displaystyle \vee }$ Ψ) and (Φ${\displaystyle \leftrightarrow }$ Ψ) to also be formulas.

Apredicate calculus will usually include all the same rules as a propositional calculus, with the addition ofquantifiers such that if we take Φ to be a formula of propositional logic and α as avariable then we can take (${\displaystyle \mathrm{\forall }}$α)Φ and (${\displaystyle \mathrm{\exists }}$α)Φ each to be formulas of our predicate calculus.

• finite-state automaton

• Formal languages
• Propositional calculus
• Predicate logic
• Rules
• Syntax (logic)
• Logical truth

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