Truth table semantics for classical propositional logic

A classical semantics of the Formal language of propositional logic is in truth tables. Namely, each atomic proposition $𝑝$ is assigned to a truth value in $Ω = {⊤, ⊥}$ and then we inductively define the truth value of connectives. For the unary connective we have

$𝜙$	$\neg 𝜙$
$⊤$	$⊥$
$⊥$	$⊤$

and for the binary connectives we have

$𝜙$	$𝜓$	$𝜙 \land 𝜓$	$𝜙 \lor 𝜓$	$𝜙 \Rightarrow 𝜓$	$𝜙 \Leftrightarrow 𝜓$
$⊤$	$⊤$	$⊤$	$⊤$	$⊤$	$⊤$
$⊤$	$⊥$	$⊥$	$⊤$	$⊥$	$⊥$
$⊥$	$⊤$	$⊥$	$⊤$	$⊤$	$⊥$
$⊥$	$⊥$	$⊥$	$⊥$	$⊤$	$⊤$

Since every proposition evaluates to $⊤$ or $⊥$ , these semantics validate the Law of excluded middle and are thus classical.

Classification of propositions

Under truth table semantics, we classify a proposition $𝜙$ as a

tautology iff it is $⊤$ under any assignment of its atomic propositions;
contradiction iff it is $⊥$ under any assignment of its atomic propositions;
contingent iff there are assignments of its atomic propositions yielding distinct truth values;
satisfiable iff it is $⊤$ under some assignment of its atomic propositions, i.e. either contingent or a tautology.

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