Well-formed Formula

logic

Definition

Well-formed Formula

A well formed formula is a formula that is syntactically correct according to the formation rules of a formal language.

A string that does not obey the formation rules is not a formula at all.

Propositional Logic

Definition

Well-formed Propositional Formula

In propositional logic, a well-formed propositional formula is a syntactically correct propositional formula generated by the following rules:

1. Every atom is a propositional formula.
2. $\top$ and $\perp$ are propositional formulas.
3. If $P_1,\dots ,P_n$ are propositional formulas with $n\ge 2$, then $(P_1\wedge \cdots \wedge P_n)$ and $(P_1\vee \cdots \vee P_n)$ are propositional formulas.
4. If $P$ is a propositional formula, then $(\neg P)$ is a propositional formula.
5. If $P_1$ and $P_2$ are propositional formulas, then $(P_1\to P_2)$ and $(P_1\leftrightarrow P_2)$ are propositional formulas.
6. Nothing else is a well-formed propositional formula.

Here, $\wedge$, $\vee$, $\neg$, $\to$, and $\leftrightarrow$ are connectives, namely conjunction, disjunction, negation, implication, and equivalence. The symbols $\top$ and $\perp$ are propositional constants.

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First-Order Logic

A well formed formula is built from terms, predicates, logical connectives, and quantifiers.

Examples

• $P\wedge Q$ is a well formed formula.
• $\neg (P\to Q)$ is a well formed formula.
• $P\wedge$ is not a well formed formula.