Editing Rule of inference (section)

== Systems of logic ==
=== 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>
|}

=== Modal logics ===
{{main|Modal logic}}
Modal logics are formal systems that extend propositional logic and first-order logic with additional logical operators. Alethic modal logic introduces the operator <math>\Diamond</math> to express that something is possible and the operator <math>\Box</math> to express that something is necessary. For example, if the <math>P</math> means that "Parvati works", then <math>\Diamond P</math> means that "It is possible that Parvati works" while <math>\Box P</math> means that "It is necessary that Parvati works". These two operators are related by a rule of replacement stating that <math>\Box P</math> is equivalent to <math>\lnot \Diamond \lnot P</math>. In other words: if something is necessarily true then it is not possible that it is not true. Further rules of inference include the necessitation rule, which asserts that a statement is necessarily true if it is provable in a formal system without any additional premises, and the distribution axiom, which allows one to derive <math>\Diamond P \to \Diamond Q </math> from <math>\Diamond (P \to Q)</math>. These rules of inference belong to system K, a weak form of modal logic with only the most basic rules of inference. Many formal systems of alethic modal logic include additional rules of inference, such as system T, which allows one to deduce <math>P</math> from <math>\Box P</math>.<ref>{{multiref | {{harvnb|Garson|2024|loc=Lead section, § 2. Modal Logics}} | {{harvnb|Sider|2010|pp=[https://books.google.com/books?id=-KkPEAAAQBAJ&pg=PA171 171–176, 286–287]}} }}</ref>

Non-alethic systems of modal logic introduce operators that behave like <math>\Diamond</math> and <math>\Box</math> in alethic modal logic, following similar rules of inference but with different meanings. [[Deontic logic]] is one type of non-alethic logic. It uses the operator <math>P</math> to express that an action is permitted and the operator <math>O</math> to express that an action is required, where <math>P</math> behaves similarly to <math>\Diamond</math> and <math>O</math> behaves similarly to <math>\Box</math>. For instance, the rule of replacement in alethic modal logic asserting that <math>\Box Q</math> is equivalent to <math>\lnot \Diamond \lnot Q</math> also applies to deontic logic. As a result, one can deduce from <math>O Q</math> (e.g. Quinn has an obligation to help) that <math>\lnot P \lnot Q</math> (e.g. Quinn is not permitted not to help).<ref>{{harvnb|Garson|2024|loc=§ 3. Deontic Logics}}</ref> Other systems of modal logic include [[Temporal logic|temporal modal logic]], which has operators for what is always or sometimes the case, as well as [[Doxastic logic|doxastic]] and [[epistemic modal logic]]s, which have operators for what people believe and know.<ref>{{multiref | {{harvnb|Garson|2024|loc=§ 1. What is Modal Logic?, § 4. Temporal Logics}} | {{harvnb|Sider|2010|pp=[https://books.google.com/books?id=-KkPEAAAQBAJ&pg=PA234 234–242]}} }}</ref>

=== Others ===
[[File:Aristotle Altemps Inv8575.jpg|thumb|upright=.8|alt=Photo of a marble bust of a bearded man|The rules of inference in [[Aristotle]]'s (384–322 BCE) logic have the form of syllogisms.<ref>{{harvnb|O'Regan|2017|pp=90–91, 103}}</ref>]]

Many other systems of logic have been proposed. One of the earliest systems is [[Aristotelian logic]], according to which each statement is made up of two [[Term logic|terms]], a subject and a predicate, connected by a [[Copula (linguistics)|copula]]. For example, the statement "all humans are mortal" has the subject "all humans", the predicate "mortal", and the copula "is". All rules of inference in Aristotelian logic have the form of [[syllogism]]s, which consist of two premises and a conclusion. For instance, the ''Barbara'' rule of inference describes the validity of arguments of the form "All men are mortal. All Greeks are men. Therefore, all Greeks are mortal."<ref>{{multiref | {{harvnb|Smith|2022|loc=Lead section, § 3. The Subject of Logic: “Syllogisms”}} | {{harvnb|Groarke|loc=Lead section, § 3. From Words into Propositions, § 4. Kinds of Propositions, § 9. The Syllogism}} }}</ref>

[[Second-order logic]] extends first-order logic by allowing quantifiers to apply to predicates in addition to singular terms. For example, to express that the individuals Adam (<math>a</math>) and Bianca (<math>b</math>) share a property, one can use the formula <math>\exists X (X(a) \land X(b))</math>.<ref>{{harvnb|Väänänen|2024|loc=Lead section, § 1. Introduction}}</ref> Second-order logic also comes with new rules of inference.{{efn|An important difference between first-order and second-order logic is that second-order logic is [[Completeness (logic)|incomplete]], meaning that it is not possible to provide a finite set of rules of inference with which every theorem can be deduced.<ref>{{multiref | {{harvnb|Väänänen|2024|loc=§ 1. Introduction}} | {{harvnb|Grandy|1979|p=[https://books.google.com/books?id=ItgJhsGE-RAC&pg=PA122 122]}} | {{harvnb|Linnebo|2014|p=[https://books.google.com/books?id=EKZOBAAAQBAJ&pg=PA123 123]}} }}</ref>}} For instance, one can infer <math>P(a)</math> (Adam is a philosopher) from <math>\forall X X(a)</math> (every property applies to Adam).<ref>{{harvnb|Pollard|2015|p=[https://books.google.com/books?id=6cY-CgAAQBAJ&pg=PA98 98]}}</ref>

[[Intuitionistic logic]] is a non-classical variant of propositional and first-order logic. It shares with them many rules of inference, such as ''modus ponens'', but excludes certain rules. For example, in classical logic, one can infer <math>P</math> from <math>\lnot \lnot P</math> using the rule of double negation elimination. However, in intuitionistic logic, this inference is invalid. As a result, every theorem that can be deduced in intuitionistic logic can also be deduced in classical logic, but some theorems provable in classical logic cannot be proven in intuitionistic logic.<ref>{{multiref | {{harvnb|Moschovakis|2024|loc=Lead section, § 1. Rejection of ''Tertium Non Datur''}} | {{harvnb|Sider|2010|pp=[https://books.google.com/books?id=-KkPEAAAQBAJ&pg=PA110 110–114, 264–265]}} | {{harvnb|Kleene|2000|p=[https://books.google.com/books?id=q-LG8Ep7WFcC&pg=PA81 81]}} }}</ref>

[[Paraconsistent logics]] revise classical logic to allow the existence of [[Contradiction (logic)|contradictions]]. In logic, a contradiction happens if the same proposition is both affirmed and denied, meaning that a formal system contains both <math>P</math> and <math>\lnot P</math> as theorems. Classical logic prohibits contradictions because classical rules of inference lead to the [[principle of explosion]], an admissible rule of inference that makes it possible to infer <math>Q</math> from the premises <math>P</math> and <math>\lnot P</math>. Since <math>Q</math> is unrelated to <math>P</math>, any arbitrary statement can be deduced from a contradiction, making the affected systems useless for deciding what is true and false.<ref>{{multiref | {{harvnb|Shapiro|Kouri Kissel|2024|loc=§ 3. Deduction}} | {{harvnb|Sider|2010|pp=[https://books.google.com/books?id=-KkPEAAAQBAJ&pg=PA102 102–104]}} | {{harvnb|Priest|Tanaka|Weber|2025|loc=Lead section}} }}</ref> Paraconsistent logics solve this problem by modifying the rules of inference in such a way that the principle of explosion is not an admissible rule of inference. As a result, it is possible to reason about inconsistent information without deriving absurd conclusions.<ref>{{multiref | {{harvnb|Weber|loc=Lead section, § 2. Logical Background}} | {{harvnb|Sider|2010|pp=[https://books.google.com/books?id=-KkPEAAAQBAJ&pg=PA102 102–104]}} | {{harvnb|Priest|Tanaka|Weber|2025|loc=Lead section}} }}</ref>

[[Many-valued logics]] modify classical logic by introducing additional truth values. In classical logic, a proposition is either true or false with nothing in between. In many-valued logics, some propositions are neither true nor false. [[Kleene logic]], for example, is a [[three-valued logic]] that introduces the additional truth value ''undefined'' to describe situations where information is incomplete or uncertain.<ref>{{multiref | {{harvnb|Sider|2010|pp=[https://books.google.com/books?id=-KkPEAAAQBAJ&pg=PA93 93–94, 98–100]}} | {{harvnb|Gottwald|2022|loc=Lead section, § 3.4 Three-valued systems}} }}</ref> Many-valued logics have adjusted rules of inference to accommodate the additional truth values. For instance, the classical rule of replacement stating that <math>P \to Q</math> is equivalent to <math>\lnot P \lor Q</math> is invalid in many three-valued systems.<ref>{{multiref | {{harvnb|Egré|Rott|2021|loc=§ 2. Three-Valued Conditionals}} | {{harvnb|Gottwald|2022|loc=Lead section, § 2. Proof Theory}} }}</ref>