Propositional Logic

definitionlogic

The Language of Propositional Logic

Consists of:

• Variables: $v_0,v_1,v_2,\dots ,v_n,\dots$ for $n\in \mathbb{N}$.
• Logical symbols: $\neg ,\wedge ,\vee ,\to ,\leftrightarrow$.
• Parentheses: $(,)$.

Propositional Formulas

The set of propositional formulas is defined as follows:

1. Each proposition variable is a propositional formula.
2. If $\phi$ is a propositional formula, then so is $\neg \phi$.
3. If $\phi$ and $\psi$ are propositional formulas, then so are $(\phi \wedge \psi ),(\phi \vee \psi ),(\phi \to \psi ),(\phi \leftrightarrow \psi )$.
4. A string of symbols is a propositional formulas if and only if it is generated from 1) by finitely many applications of 2) and 3).