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From LogicWiki

The syntax, or grammar, of Propositional Logic determines whether a string of symbols gets to count as potentially meaningful. A symbol that counts as potentially meaningful is called a well-formed formula, or wff for short, of the language. A well-formed formula, is a formula that is potentially meaningful.

In order to specify the wffs of Propositonal Logic, we must specify two things: the vocabulary of the language, and the grammar of Propositional Logic.

Contents 1 The Language of Propositional Logic 2 The Syntax of Propositional Logic 3 Examples 4 Some Additional Syntactic Concepts 4.1 Examples 5 See Also

The Language of Propositional Logic

The Language of Propositional Logic: ${\displaystyle A,B,C,D,...\neg ,\wedge ,\vee ,\to ,\leftrightarrow ,⊢,(,)}$

The Syntax of Propositional Logic

The Syntax of Propositional Logic:

1. Every sentence letter is a wff.
2. If ${\displaystyle P}$ is a wff, then ${\displaystyle \neg P}$ is a wff.
3. If ${\displaystyle P}$ and ${\displaystyle Q}$ are wffs, then ${\displaystyle P\wedge Q}$ is a wff.
4. If ${\displaystyle P}$ and ${\displaystyle Q}$ are wffs, then ${\displaystyle P\vee Q}$ is a wff.
5. If ${\displaystyle P}$ and ${\displaystyle Q}$ are wffs, then ${\displaystyle P\to Q}$ is a wff.
6. If ${\displaystyle P}$ and ${\displaystyle Q}$ are wffs, then ${\displaystyle P\leftrightarrow Q}$ is a wff.
7. No other string of symbols is a wff, unless it is the result of applications of rules 1-6.

Examples

The following strings of symbols are not wffs:

${\displaystyle P\wedge }$ ${\displaystyle \leftrightarrow Q}$

${\displaystyle P}$ ${\displaystyle R\vee Q}$

${\displaystyle P\wedge Q\vee P}$

${\displaystyle \neg (P\vee Q}$

The following strings of symbols are wffs:

${\displaystyle (P\wedge R)\leftrightarrow Q}$

${\displaystyle (P\wedge R)\vee Q}$

${\displaystyle P\wedge (Q\vee P)}$

${\displaystyle \neg (P\vee Q)}$

Some Additional Syntactic Concepts

The main connective of a non-atomic wff is the connective that governs the largest components in the wff.

Examples

In the sentence ${\displaystyle \neg (\neg P\to Q)}$, the first ${\displaystyle \neg }$ is the main connective. It governs the wff ${\displaystyle (\neg P\to Q)}$, the largest component.

In the sentence ${\displaystyle ((P\wedge Q)\to F)}$, the main connective is ${\displaystyle \to }$. It governs ${\displaystyle (P\wedge Q)}$ and ${\displaystyle F}$, whereas ${\displaystyle \wedge }$ only governs ${\displaystyle P}$ and ${\displaystyle Q}$.

See Also

Logical Connectives, Sentence