Propositional Logic

Definition: The Formal Language of Propositional Logic

The formal language of propositional logic is the formal language ${\mathcal{L}}_{\text{PL}}$ whose countable alphabet ${\mathcal{A}}_{\text{PL}}$ and syntax ${\mathcal{S}}_{\text{PL}}$ are defined as follows.

The alphabet ${\mathcal{A}}_{\text{PL}}$ contains:

• the parentheses symbols "$($" and "$)$";
• the sentential connective symbols "$\neg$", "$\wedge$", "$\vee$", "$\to$", "$\leftrightarrow$";
• atomic formula symbols $A_1,A_2,\dots$

The syntax ${\mathcal{S}}_{\text{PL}}$ is comprised of the following rules:

• Every atomic formula is a well-formed formula.

• If $P$ is a wff, then $(P)$ is also a wff.

• Any negation of a wff is also a wff.

• Any conjunction of two wffs ia also a wff.

• Any disjunction of two wffs is also a wff.

• Any conditional of two wffs is also a wff.

• Any biconditional of two wffs is also a wff.

• All other expressions are not wffs.

INTUITION

Each well-formed formula $P$ can be thought of as a translation of an English proposition into the language ${\mathcal{L}}_{\text{PL}}$. Atomic formulas then represent the most basic and fundamental propositions we have chosen to examine.

NOTE

The language ${\mathcal{L}}_{\text{PL}}$ is also known as the zeroth order language.