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:


 * variables;
 * predicate or relation symbols;
 * function symbols.

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:

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:

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