# Definition:Language of Predicate Logic/Alphabet

## Definition

The alphabet $\mathcal A$ of the language of predicate logic $\mathcal L_1$ is defined as follows:

### Letters

The letters of $\mathcal L_1$ are separated in three classes:

Each of these three classes is handled differently by the formal grammar of predicate logic.

#### Variables

The variables constitute an infinite set $\mathrm{VAR}$ of arbitrary symbols, for example:

- $\mathrm{VAR} = \left\{{x, y, z, x_0, y_0, z_0, x_1, y_1, z_1, \ldots}\right\}$

#### Predicate Symbols

The predicate symbols are a collection of arbitrary symbols.

Each of these symbols is considered to be endowed with an arity (a natural number $n \in \N$).

We agree to write $\mathcal P$ for the set of predicate symbols, grouped by their arity:

- $\mathcal P = \left\{{\mathcal P_0, \mathcal P_1, \mathcal P_2, \ldots, \mathcal P_k, \ldots}\right\}$

The symbols in $\mathcal P_0$ are inherited from the language of propositional logic.

For example, if $P \in \mathcal P_5$ then $P$ is a quinternary predicate symbol.

#### Function Symbols

The function symbols are a collection (possibly empty) of arbitrary symbols.

Each of these symbols is considered to be endowed with an arity (a natural number $n \in \N$).

We agree to write $\mathcal F$ for the set of function symbols, grouped by their arity:

- $\mathcal F = \left\{{\mathcal F_0, \mathcal F_1, \ldots, \mathcal F_k, \ldots}\right\}$

The symbols in $\mathcal F_0$ are often called **parameters** or **constants**.

Some sources write $\mathcal K$ for the collection of **parameters**.

### Signs

The signs of $\mathcal L_1$ are an extension of the signs of propositional logic.

They split in three classes:

- connectives;
- quantifiers;
- punctuation.

#### Connectives

The **connectives** of $\mathcal L_1$ comprise:

\(\displaystyle \land \) | \(\displaystyle : \) | the conjunction sign | ||||||

\(\displaystyle \lor \) | \(\displaystyle : \) | the disjunction sign | ||||||

\(\displaystyle \implies \) | \(\displaystyle : \) | the conditional sign | ||||||

\(\displaystyle \iff \) | \(\displaystyle : \) | the biconditional sign | ||||||

\(\displaystyle \neg \) | \(\displaystyle : \) | the negation sign | ||||||

\(\displaystyle \top \) | \(\displaystyle : \) | the top sign | ||||||

\(\displaystyle \bot \) | \(\displaystyle : \) | the bottom sign |

The symbols $\land,\lor,\implies$ and $\iff$ are called the **binary connectives**.

The symbols $\neg$ is called a **unary connective**.

The symbols $\top$ and $\bot$ are called the **nullary connectives**.

#### Quantifiers

The **quantifiers** of $\mathcal L_1$ are:

\(\displaystyle \exists \) | \(\displaystyle : \) | the existential quantifier sign | ||||||

\(\displaystyle \forall \) | \(\displaystyle : \) | the universal quantifier sign |

#### Punctuation

The punctuation symbols used in $\mathcal L_1$ are:

\(\displaystyle ( \) | \(\displaystyle : \) | the left parenthesis sign | ||||||

\(\displaystyle ) \) | \(\displaystyle : \) | the right parenthesis sign | ||||||

\(\displaystyle : \) | \(\displaystyle : \) | the colon | ||||||

\(\displaystyle , \) | \(\displaystyle : \) | the comma |

## Sources

- 1996: H. Jerome Keisler and Joel Robbin:
*Mathematical Logic and Computability*: $\S 2.2$ - 2009: Kenneth Kunen:
*The Foundations of Mathematics*... (previous) ... (next): $\mathrm{II}.5$ First-Order Logic Syntax: Definition $\mathrm{II.5.1}$ - 2009: Kenneth Kunen:
*The Foundations of Mathematics*... (previous) ... (next): $\mathrm{II}.5$ First-Order Logic Syntax: Definition $\mathrm{II.5.2}$