How do logical connectives such as “or” and “not” build complex sentences in a formal language?

In formal logic, logical connectives like "not" (negation) and "or" (disjunction) are used to build complex sentences from simpler ones through a set of formation rules. These rules specify how to construct well-formed formulas (wffs) mechanically, ensuring that the resulting expressions are unambiguous.12

The construction process typically follows these inductive principles:

• Negation ("not"): Symbolized as $~$, this is a monadic operator. If a formula $\alpha$ is a wff, then $~\alpha$ is also a wff. It functions by reversing the truth value of the original proposition.324
• Disjunction ("or"): Symbolized as $\vee$, this is a dyadic operator. If $\alpha$ and $\beta$ are wffs, then $(\alpha \vee \beta )$ is a wff. It asserts that at least one of the component propositions is true.234

These connectives are considered truth-functional, meaning the truth value of the complex sentence they create is determined entirely by the truth values of their component parts. By applying these rules recursively, logicians can build increasingly complex structures from atomic sentences.341