Editing Rule of inference (section)

=== 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>