Editing Rule of inference (section)

=== Classical ===
==== 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>
|}

==== First-order logic ====
{{main|First-order logic}}
[[File:Wismar Marienkirche Bronzebüste Gottlob Frege (01-1).JPG|thumb|upright=.8|alt=Photo of a bronze bust of a bearded man|As one of the founding fathers of modern logic, [[Gottlob Frege]] (1848–1925) explored some of the foundational concepts of first-order logic.<ref>{{multiref | {{harvnb|O'Regan|2017|pp=101–103}} | {{harvnb|Zalta|2024|loc=Lead section}} }}</ref>]]

First-order logic also employs the logical operators from propositional logic but includes additional devices to articulate the internal structure of propositions. Basic propositions in first-order logic consist of a [[Predicate (logic)|predicate]], symbolized with uppercase letters like <math>P</math> and <math>Q</math>, which is applied to [[singular term]]s, symbolized with lowercase letters like <math>a</math> and <math>b</math>. For example, if <math>a</math> stands for "Aristotle" and <math>P</math> stands for "is a philosopher", the formula <math>P(a)</math> means that "Aristotle is a philosopher". Another innovation of first-order logic is the use of the [[Quantifier (logic)|quantifiers]] <math>\exists</math> and <math>\forall</math>, which express that a predicate applies to some or all individuals. For instance, the formula <math>\exists x P(x)</math> expresses that philosophers exist while <math>\forall x P(x)</math> expresses that everyone is a philosopher. The rules of inference from propositional logic are also valid in first-order logic.<ref>{{multiref | {{harvnb|Shapiro|Kouri Kissel|2024|loc=Lead section, § 2. Language}} | {{harvnb|Sider|2010|pp=[https://books.google.com/books?id=-KkPEAAAQBAJ&pg=PA115 115–118]}} | {{harvnb|Cook|2009|pp=119–120}} }}</ref> Additionally, first-order logic introduces new rules of inference that govern the role of singular terms, predicates, and quantifiers in arguments. Key rules of inference are [[universal instantiation]] and [[existential generalization]]. Other rules of inference include [[universal generalization]] and [[existential instantiation]].<ref name="auto">{{multiref | {{harvnb|Hurley|2016|pp=374–377}} | {{harvnb|Shapiro|Kouri Kissel|2024|loc=§ 3. Deduction}} }}</ref>

{|class="wikitable"
|+ Notable rules of inference<ref name="auto"/>
|-
! style="text-align:center;" |Rule of inference !! style="text-align:center;" |Form !! style="text-align:center;" |Example
|-
|style="text-align:center;" |Universal instantiation ||<math>\begin{array}{l}
\forall x P(x) \\ \hline
P(a)
\end{array}</math>{{efn|This example assumes that <math>a</math> refers to an individual in the [[domain of discourse]].}} ||<math>\begin{array}{l}
\text{Everyone must pay taxes.} \\ \hline
\text{Therefore, Wesley must pay taxes.}
\end{array}</math>
|-
|style="text-align:center;" |Existential generalization ||<math>\begin{array}{l}
P(a) \\ \hline
\exists x P(x)
\end{array}</math> ||<math>\begin{array}{l}
\text{Socrates is mortal.} \\ \hline
\text{Therefore, someone is mortal.}
\end{array}</math>
|}