Editing Propositional calculus (section)

===Semantic proof systems===
{{Image frame|content=<math>\begin{array}{|c|c|c|c|}
x_0 & x_1 & \bar{x_1} & x_0 \& \bar{x_1} \\
\hline
0 & 0 & 1 & 0 \\
0 & 1 & 0 & 0 \\
1 & 0 & 1 & 1 \\
1 & 1 & 0 & 0
\end{array}</math>
|width=200|align=right|caption=Example of a [[truth table]]}}
[[File:Partially built tableau.svg|thumb|200px|A graphical representation of a partially built [[Method of analytic tableaux|propositional tableau]]]]
Semantic proof systems rely on the concept of semantic consequence, symbolized as <math>\varphi \models \psi</math>, which indicates that if <math>\varphi</math> is true, then <math>\psi</math> must also be true in every possible interpretation.<ref name=":16" />

====Truth tables====
{{Main article|Truth table}}
A [[truth table]] is a semantic proof method used to determine the truth value of a propositional logic expression in every possible scenario.<ref name="ms41"/> By exhaustively listing the truth values of its constituent atoms, a truth table can show whether a proposition is true, false, tautological, or contradictory.<ref name=":27"/> See {{section link||Semantic proof via truth tables}}.

====Semantic tableaux====
{{Main article|Method of analytic tableaux}}
A [[semantic tableau]] is another semantic proof technique that systematically explores the truth of a proposition.<ref name="ms42"/> It constructs a tree where each branch represents a possible interpretation of the propositions involved.<ref name="ms43"/> If every branch leads to a contradiction, the original proposition is considered to be a contradiction, and its negation is considered a [[Tautology (logic)|tautology]].<ref name=":13"/> See {{section link||Semantic proof via tableaux}}.