# Propositional Logic Syntax

Let be a countable set of *propositional variables*, denoted by non-terminal *V*. The context-free grammar of propositional logic formulas is the following:

We denote the set of all propositional formulas given by the above context-free grammar by . Each propositional formula is a finite sequence of symbols, given by the above context-free grammar. The set is a countable set: we can order all formulas in this set in a sequence (for example, by writing them down in binary alphabet and sorting the resulting strings alphabetically).

Omitting parentheses:

- , associative
- priorities, from strongest-binding:

When in doubt, use parentheses.

Notation: when we write this means that and are identical formulas (with identical syntax trees). For example, , but it is not the case that .

In Isabelle theorem prover we use this

ASCII notation for Propositional Logic

Usually we work with syntax trees, as in Problem 3 in Homework 1.

denotes the set of free variables in the given propositional formula and can be defined recursively as follows:

If , we call a *ground formula*.