Definition:Alphabet of Predicate Calculus
Contents |
Definition
Vocabulary
The vocabulary of $\mathcal L_1$ can be defined as the following set $\mathcal P$ of more-or-less arbitrary symbols:
- $\mathcal P = \left\{{\mathcal P_0, \mathcal P_1, \ldots, \mathcal P_k, \ldots}\right\}$
such that each $\mathcal P_n \in \mathcal P$ is a set of $n$-ary predicate symbols for each natural number $n \in \N$.
At least one of the sets $\mathcal P_n$ is non-empty.
The set $\mathcal P_0$ is a set of propositional symbols as in the Alphabet of Propositional Calculus.
In addition to the propositional symbols and signs of propositional calculus, the following are primitive symbols of predicate calculus:
Predicate Symbols
The predicate symbols, from $\mathcal P_0, \mathcal P_1, \mathcal P_2, \ldots$.
Variable Symbols
An infinite set $VAR = \left\{{x, y, z, x_0, y_0, z_0, x_1, y_1, z_1, \ldots}\right\}$ of symbols called variables.
Parameters
A set $\mathcal K$ of symbols called parameters.
Quantifiers
The quantifiers are:
- The universal quantifier $\forall$;
- The existential quantifier $\exists$.
Punctuation
The punctuation symbols used are:
- The left parenthesis, right parenthesis, colon and comma: $( \ , \ : \ )$.
Each of the symbols of $\mathcal L_1$ is considered to have length of $1$.
The variables and parameters are referred to collectively as individual symbols.
WFF with Parameters
The set of all WFFs formed from the vocabulary $\mathcal P$ with parameters from $\mathcal K$ can be denoted $WFF \left({\mathcal P, \mathcal K}\right)$.
Relations
Let $X^n$ be the set of all length $n$ sequences: $\left({x_1, x_2, \ldots, x_n}\right)$ of elements from $X$.
The set of all $n$-ary relations on $X$ can be denoted $REL_n \left({X}\right)$.
Thus:
- The set $X^1$ is the same as $X$, and a unary relation on $X$ is a subset of $X$.
- The set $X^2$ is the same as $ X \times X$, and a binary relation on $X^2$ is a subset of $X^2$
The $0$-ary relations on $X$ correspond to truth values, thus:
The only sequence of length $0$ is the null sequence $\left({}\right)$.
So $X_0 = \left\{{\left({}\right)}\right\}$.
There are two $0$-ary relations on $X$:
- The empty set $\varnothing$ corresponding to the truth value $F$;
- The set $X_0 = \left\{{\left({}\right)}\right\}$ corresponding to the truth value $T$.
