Trouble viewing the formulas? You need a MathML compatible browser.

# Syntax for Propositional logic

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.

## The Language of Propositional Logic

The Language of Propositional Logic: $A,B,C,D,...¬,\wedge ,\vee ,\to ,↔,⊢,\left(,\right)$

## The Syntax of Propositional Logic

The Syntax of Propositional Logic:

1. Every sentence letter is a wff.
2. If $P$ is a wff, then $¬P$ is a wff.
3. If $P$ and $Q$ are wffs, then $P\wedge Q$ is a wff.
4. If $P$ and $Q$ are wffs, then $P\vee Q$ is a wff.
5. If $P$ and $Q$ are wffs, then $P\to Q$ is a wff.
6. If $P$ and $Q$ are wffs, then $P↔Q$ 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:

$P\wedge$ $↔Q$

$P$ $R\vee Q$

$P\wedge Q\vee P$

$¬\left(P\vee Q$

The following strings of symbols are wffs:

$\left(P\wedge R\right)↔Q$

$\left(P\wedge R\right)\vee Q$

$P\wedge \left(Q\vee P\right)$

$¬\left(P\vee Q\right)$

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

### Examples

In the sentence $¬\left(¬P\to Q\right)$, the first $¬$ is the main connective. It governs the wff $\left(¬P\to Q\right)$, the largest component.

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