Syntax of PL0 - Propositions as Boolean Terms

Definition: Syntax of PL0 - Propositions as Boolean Terms

The variables and functions defined in signature of propositional logic $PL0$ are called Boolean, named after the English mathematician George Boole.

First, we agree that the formal language $L\subseteq \Sigma^* $ of $PL0$ is defined over an alphabet $\Sigma$ containing the following letters:

space character “ “,
Boolean constants: the characters “$1$” and “$0$”,
Boolean variables: small Latin letters $“a”,“b”,“c”,\ldots,“x”,“y”,“z”$, and letters indexed with natural numbers, e.g. $“a_0”,“a_1”,“a_2”,\ldots,“b_0”,“b_1”,“b_2”,\ldots,$,
the parentheses “$($” and “$)$”, and
the following additional letters “$\neg$”, “$\wedge$”, “$\vee$”, “$\Rightarrow$”, “$\Leftrightarrow$”.

We will now specify the syntax of $PL0$:

A Boolean constant is a proposition (we also will use the notion Boolean term synonymously for “proposition”).
Boolean variables are propositions.
If “$\phi$” and “$\psi$” are propositions, then “$\neg \phi$,” “$\phi\wedge\psi$”, “$\phi\vee\psi$”, “$\phi\Rightarrow\psi$”, “$\phi\Leftrightarrow\psi$” are propositions.
If “$\phi$” is a proposition, then the concatenation “$(\phi)$” is a proposition.