Editing Rule of inference (section)

== Definition ==
{{Transformation rules}}
A rule of [[inference]] is a way of drawing a conclusion from a set of [[premise]]s.<ref>{{multiref | {{harvnb|Hurley|2016|p=303}} | {{harvnb|Hintikka|Sandu|2006|pp=13–14}} | {{harvnb|Carlson|2017|p=20}} | {{harvnb|Copi|Cohen|Flage|2016|pp=244–245, 447}} }}</ref> Also called ''inference rule'' and ''transformation rule'',<ref>{{multiref | {{harvnb|Shanker|2003|p=[https://books.google.com/books?id=jIzT7AT3ILIC&pg=PA442 442]}} | {{harvnb|Cook|2009|p=152}} }}</ref> it is a norm of correct inferences that can be used to guide [[logical reasoning|reasoning]], justify conclusions, and criticize [[argument]]s. As part of [[deductive logic]], rules of inference are [[argumentation scheme|argument forms]] that preserve the [[truth]] of the premises, meaning that the conclusion is always true if the premises are true.{{efn|Non-deductive arguments, by contrast, support the conclusion without ensuring that it is true, such as [[Inductive reasoning|inductive]] and [[abductive reasoning]].<ref>{{harvnb|Hintikka|Sandu|2006|pp=13–14}}</ref>}} An inference is deductively correct or [[Validity (logic)|valid]] if it follows a valid rule of inference. Whether this is the case depends only on the [[Logical form|form or syntactical structure]] of the premises and the conclusion. As a result, the actual content or concrete meaning of the statements does not affect validity. For instance, ''[[modus ponens]]'' is a rule of inference that connects two premises of the form "if <math>P</math> then <math>Q</math>" and "<math>P</math>" to the conclusion "<math>Q</math>", where <math>P</math> and <math>Q</math> stand for [[Proposition|statements]]. Any argument with this form is valid, independent of the specific meanings of <math>P</math> and <math>Q</math>, such as the argument "If it rains, then the ground is wet. It rains. Therefore, the ground is wet". In addition to ''modus ponens'', there are many other rules of inference, such as ''[[modus tollens]]'', [[disjunctive syllogism]], [[hypothetical syllogism]], [[constructive dilemma]], and [[destructive dilemma]].<ref>{{multiref | {{harvnb|Hurley|2016|pp=54–55, 283–287}} | {{harvnb|Arthur|2016|p=165}} | {{harvnb|Hintikka|Sandu|2006|pp=13–14}} | {{harvnb|Carlson|2017|p=20}} | {{harvnb|Copi|Cohen|Flage|2016|pp=244–245}} | {{harvnb|Baker|Hacker|2014|pp=[https://books.google.com/books?id=vEFVAgAAQBAJ&pg=PA88 88–90]}} }}</ref>

There are different formats to represent rules of inference. A common approach is to use a new line for each premise and separate the premises from the conclusion using a horizontal line. With this format, ''modus ponens'' is written as:<ref>{{harvnb|Hurley|2016|p=303}}</ref>{{efn|The symbol <math>\to</math> in this formula means ''if ... then ...'', expressing [[Material conditional|material implication]].<ref>{{harvnb|Magnus|Button|2021|p=32}}</ref>}}

<math>\begin{array}{l}
P \to Q \\
P \\ \hline
Q
\end{array}</math>

Some logicians employ the [[therefore sign]] (<math>\therefore</math>) together or instead of the horizontal line to indicate where the conclusion begins.<ref>{{multiref | {{harvnb|Copi|Cohen|Flage|2016|pp=137, 245–246}} | {{harvnb|Magnus|Button|2021|p=109}} }}</ref> The [[sequent]] notation, a different approach, uses a single line in which the premises are separated by commas and connected to the conclusion with the [[turnstile symbol]] (<math>\vdash</math>), as in <math>P \to Q, P \vdash Q</math>.<ref>{{harvnb|Sørensen|Urzyczyn|2006|pp=[https://books.google.com/books?id=_mtnm-9KtbEC&pg=PA161 161–162]}}</ref> The letters <math>P</math> and <math>Q</math> in these formulas are so-called [[metavariable]]s: they stand for any simple or compound proposition.<ref>{{harvnb|Reynolds|1998|p=[https://books.google.com/books?id=X_ToAwAAQBAJ&pg=PA12 12]}}</ref>

Rules of inference belong to [[Formal system|logical systems]] and distinct logical systems may use different rules of inference. For example, [[universal instantiation]] is a rule of inference in the system of [[first-order logic]] but not in [[propositional logic]].<ref>{{multiref | {{harvnb|Copi|Cohen|Flage|2016|pp=295–299}} | {{harvnb|Cook|2009|pp=124, 251–252}} | {{harvnb|Hurley|2016|pp=374–375}} }}</ref> Rules of inference play a central role in [[Formal proof|proofs]] as explicit procedures for arriving at a new line of a proof based on the preceding lines. Proofs involve a series of inferential steps and often use various rules of inference to establish the [[theorem]] they intend to demonstrate.<ref>{{multiref | {{harvnb|Cook|2009|pp=124, 230, 251–252}} | {{harvnb|Magnus|Button|2021|pp=112–113}} | {{harvnb|Copi|Cohen|Flage|2016|pp=244–245}} }}</ref> Rules of inference are definitory rules{{em dash}}rules about which inferences are allowed. They contrast with strategic rules, which govern the inferential steps needed to prove a certain theorem from a specific set of premises. Mastering definitory rules by itself is not sufficient for effective reasoning since they provide little guidance on how to reach the intended conclusion.<ref>{{multiref | {{harvnb|Hintikka|Sandu|2006|pp=13–14}} | {{harvnb|Hintikka|2013|p=[https://books.google.com/books?id=rUDsCAAAQBAJ&pg=PA98 98]}} }}</ref> As standards or procedures governing the transformation of symbolic expressions, rules of inference are similar to [[mathematical function]]s taking premises as input and producing a conclusion as output. According to one interpretation, rules of inference are inherent in [[logical operator]]s{{efn|Logical operators or constants are expressions used to form and connect propositions, such as ''not'', ''or'', and ''if...then...''.<ref>{{harvnb|Hurley|2016|pp=238–239}}</ref>}} found in statements, making the meaning and function of these operators explicit without adding any additional information.<ref>{{multiref | {{harvnb|Baker|Hacker|2014|pp=[https://books.google.com/books?id=vEFVAgAAQBAJ&pg=PA88 88–90]}} | {{harvnb|Tourlakis|2011|p=[https://books.google.com/books?id=8jAwgCTgnycC&pg=PA40 40]}} | {{harvnb|Hintikka|Sandu|2006|pp=13–14}} | {{harvnb|McKeon|2010|pp=[https://books.google.com/books?id=76SH_DI3yyoC&pg=PA128 128–129]}}}}</ref>

[[File:PSM V17 D740 George Boole.jpg|thumb|alt=Black-and-white drawing of a man with sideburns, dressed in a dark formal attire with a white high-collared shirt|[[George Boole]] (1815–1864) made key contributions to symbolic logic in general and propositional logic in particular.<ref>{{multiref | {{harvnb|Burris|2024|loc=Lead section}} | {{harvnb|O'Regan|2017|pp=95–96, 103}} }}</ref>]]

Logicians distinguish two types of rules of inference: rules of implication and [[rules of replacement]].{{efn|According to a narrow definition, rules of inference only encompass rules of implication but do not include rules of replacement.<ref>{{harvnb|Arthur|2016|pp=165–166}}</ref>}} Rules of implication, like ''modus ponens'', operate only in one direction, meaning that the conclusion can be deduced from the premises but the premises cannot be deduced from the conclusion. Rules of replacement, by contrast, operate in both directions, stating that two expressions are equivalent and can be freely replaced with each other. In classical logic, for example, a proposition (<math>P</math>) is equivalent to the negation{{efn|Logicians use the symbols <math>\lnot</math> or <math>\sim</math> to express negation.<ref>{{multiref | {{harvnb|Copi|Cohen|Flage|2016|p=446}} | {{harvnb|Magnus|Button|2021|p=32}} }}</ref>}} of its negation (<math>\lnot \lnot P</math>).{{efn|Rules of replacement are sometimes expressed using a double semi-colon. For instance, the double negation rule can be written as <math>P :: \lnot \lnot P</math>.<ref>{{harvnb|Hurley|2016|pp=323–252}}</ref>}} As a result, one can infer one from the other in either direction, making it a rule of replacement. Other rules of replacement include [[De Morgan's laws]] as well as the [[Commutative property|commutative]] and [[Associative property|associative properties]] of [[Logical conjunction|conjunction]] and [[Logical disjunction|disjunction]]. While rules of implication apply only to complete statements, rules of replacement can be applied to any part of a compound statement.<ref>{{multiref | {{harvnb|Arthur|2016|pp=165–166}} | {{harvnb|Hurley|2016|pp=302–303, 323–252}} | {{harvnb|Copi|Cohen|Flage|2016|pp=257–258}} | {{harvnb|Hurley|Watson|2018|pp=403–404, 426–428}} }}</ref>

One of the earliest discussions of formal rules of inference is found in [[Ancient history|antiquity]] in [[Aristotle's logic]]. His explanations of valid and invalid [[syllogisms]] were further refined in [[Medieval philosophy|medieval]] and [[early modern philosophy]]. The development of [[symbolic logic]] in the 19th century led to the formulation of many additional rules of inference belonging to [[Classical logic|classical]] propositional and first-order logic. In the 20th and 21st centuries, logicians developed various [[Non-classical logic|non-classical]] systems of logic with alternative rules of inference.<ref>{{multiref | {{harvnb|Hintikka|Spade|2020|loc=§ Aristotle, § Medieval Logic, § Boole and De Morgan, § Gottlob Frege}} | {{harvnb|O'Regan|2017|p=103}} | {{harvnb|Gensler|2012|p=[https://books.google.com/books?id=jpteBwAAQBAJ&pg=PA362 362]}} }}</ref>