# 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

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