Definition:Classical Propositional Logic

Definition
Classical propositional logic is the branch of propositional logic based upon Aristotelian logic, whose philosophical tenets state that:


 * $(1): \quad$ A statement must be either true or false, and there can be no middle value.


 * $(2): \quad$ A statement cannot be both true and not true at the same time, that is, it may not contradict itself.

Thus, we proceed by recalling the formal language of propositional logic $\mathcal L_0$.

To make $\mathcal L_0$ into a formal system, we need to endow it with a deductive apparatus.

That is, with axioms and rules of inference.

There are several (equivalent) methods of defining such a deductive apparatus.

One popular and particularly elegant set of axioms for classical logic was devised by. It consists of three axioms and an inference rule: modus ponens. The axioms are as follows:


 * $\vdash \phi \implies ( \psi \implies \phi )$
 * $\vdash ( \phi \implies ( \psi \implies \chi ) ) \implies ( ( \phi \implies \psi ) \implies ( \phi \implies \chi ) )$
 * $\vdash ( \not \psi \implies \not \phi ) \implies ( \phi \implies \psi )$