Definition:Classical Propositional Logic

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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 Jan Łukasiewicz. It consists of three axioms and an inference rule: modus ponens. The axioms are as follows:

$\vdash \phi \implies \left({\psi \implies \phi}\right)$
$\vdash \left({\phi \implies \left({\psi \implies \chi}\right)}\right) \implies \left({\left({\phi \implies \psi}\right) \implies \left({\phi \implies \chi}\right)}\right)$
$\vdash \left({\not \psi \implies \not \phi}\right) \implies \left({\phi \implies \psi}\right)$


A Nice List of Hilbert-style Deductive Systems