Syntax of Propositional Logic

Formulas of $PL^0$ are made up from:

• countably infinite propositional variables
  • $\mathcal{V}_0:=\left\{P, Q, R, P^1, P^2, \ldots\right\}$
• connectives
  • $\Sigma_0:=\{T, F, \neg, \vee, \wedge, \Rightarrow, \Leftrightarrow, \ldots\}$
• well-formed propositional formulas (wffs)
  • propositional variables
  • $T$ true and $F$ false
  • negations
  • Conjunction
  • Disjunktion
  • Implication
  • equivalences / biimplications
  • where the variables used for the formulas are again wffs

A single formula without any connectives is called an Atom.

The list above can be formulated in terms of a Grammar: