Editing Rule of inference (section)

==== Propositional logic ====
{{main|Propositional logic}}
Propositional logic examines the inferential patterns of simple and compound [[proposition]]s. It uses letters, such as <math>P</math> and <math>Q</math>, to represent simple propositions. Compound propositions are formed by modifying or combining simple propositions with [[logical operator]]s, such as <math>\lnot</math> (''not''), <math>\land</math> (''and''), <math>\lor</math> (''or''), and <math>\to</math> (''if ... then ...''). For example, if <math>P</math> stands for the statement "it is raining" and <math>Q</math> stands for the statement "the streets are wet", then <math>\lnot P</math> expresses "it is not raining" and <math>P \to Q</math> expresses "if it is raining then the streets are wet". These logical operators are [[truth-functional]], meaning that the truth value of a compound proposition depends only on the truth values of the simple propositions composing it. For instance, the compound proposition  <math>P \land Q</math> is only true if both <math>P</math> and <math>Q</math> are true; in all other cases, it is false. Propositional logic is not concerned with the concrete meaning of propositions other than their truth values.<ref>{{multiref | {{harvnb|Klement|loc=Lead section, § 1. Introduction, § 3. The Language of Propositional Logic}} | {{harvnb|Sider|2010|pp=[https://books.google.com/books?id=-KkPEAAAQBAJ&pg=PA30 30–35]}} }}</ref> Key rules of inference in propositional logic are [[modus ponens]], [[modus tollens]], [[hypothetical syllogism]], [[disjunctive syllogism]], and [[double negation elimination]]. Further rules include [[conjunction introduction]], [[conjunction elimination]], [[disjunction introduction]], [[disjunction elimination]], [[constructive dilemma]], [[destructive dilemma]], [[Absorption (logic)|absorption]], and [[De Morgan's laws]].<ref>{{multiref | {{harvnb|Hurley|2016|pp=303, 315}} | {{harvnb|Copi|Cohen|Flage|2016|p=247}} | {{harvnb|Klement|loc=§ Deduction: Rules of Inference and Replacement}} }}</ref>

{|class="wikitable"
|+ Notable rules of inference<ref>{{multiref | {{harvnb|Hurley|2016|pp=303, 315}} | {{harvnb|Copi|Cohen|Flage|2016|p=247}} }}</ref>
|-
! style="text-align:center;" |Rule of inference !! style="text-align:center;" |Form !! style="text-align:center;" |Example
|-
|style="text-align:center;" |Modus ponens ||<math>\begin{array}{l}
P \to Q \\
P \\ \hline
Q
\end{array}</math> ||<math>\begin{array}{l}
\text{If Kim is in Seoul, then Kim is in South Korea.} \\
\text{Kim is in Seoul.} \\ \hline
\text{Therefore, Kim is in South Korea.}
\end{array}</math>
|-
|style="text-align:center;" |Modus tollens ||<math>\begin{array}{l}
P \to Q \\
\lnot Q \\ \hline
\lnot P
\end{array}</math> ||<math>\begin{array}{l}
\text{If Koko is a koala, then Koko is cuddly.} \\
\text{Koko is not cuddly.} \\ \hline
\text{Therefore, Koko is not a koala.}
\end{array}</math>
|-
|style="text-align:center;" |Hypothetical syllogism ||<math>\begin{array}{l}
P \to Q \\
Q \to R \\ \hline
P \to R
\end{array}</math> ||<math>\begin{array}{l}
\text{If Leo is a lion, then Leo roars.} \\
\text{If Leo roars, then Leo is fierce.} \\ \hline
\text{Therefore, if Leo is a lion, then Leo is fierce.}
\end{array}</math>
|-
|style="text-align:center;" |Disjunctive syllogism ||<math>\begin{array}{l}
P \lor Q \\
\lnot P \\ \hline
Q
\end{array}</math> ||<math>\begin{array}{l}
\text{The book is on the shelf or on the table.} \\
\text{The book is not on the shelf.} \\ \hline
\text{Therefore, the book is on the table. }
\end{array}</math>
|-
|style="text-align:center;" |Double negation elimination ||<math>\begin{array}{l}
\lnot \lnot P \\ \hline
P
\end{array}</math> ||<math>\begin{array}{l}
\text{We were not unable to meet the deadline.} \\ \hline
\text{We were able to meet the deadline. }
\end{array}</math>
|}