Editing Rule of inference (section)

== Basic concepts ==
Rules of inference describe the structure of arguments, which consist of premises that support a conclusion.<ref>{{multiref | {{harvnb|Hurley|2016|pp=303, 429–430}} | {{harvnb|Hintikka|Sandu|2006|pp=13–14}} | {{harvnb|Carlson|2017|p=20}} | {{harvnb|Copi|Cohen|Flage|2016|pp=244–245, 447}} }}</ref> Premises and conclusions are statements or propositions about what is true. For instance, the assertion "The door is open." is a statement that is either true or false, while the question "Is the door open?" and the command "Open the door!" are not statements and have no truth value.<ref>{{multiref | {{harvnb|Audi|1999|pp=679–681}} | {{harvnb|Lowe|2005|pp=699–701}} | {{harvnb|Dowden|2020|p=24}} | {{harvnb|Copi|Cohen|Rodych|2019|p=[https://books.google.com/books?id=38bADwAAQBAJ&pg=PA4 4]}} }}</ref> An inference is a step of reasoning from premises to a conclusion while an argument is the outward expression of an inference.<ref>{{multiref | {{harvnb|Hintikka|2019|loc=§ Nature and Varieties of Logic}} | {{harvnb|Haack|1978|pp=1–10}} | {{harvnb|Schlesinger|Keren-Portnoy|Parush|2001|p=220}} }}</ref>

[[Logic]] is the study of correct reasoning and examines how to distinguish good from bad arguments.<ref>{{multiref | {{harvnb|Hintikka|2019|loc=Lead section, § Nature and Varieties of Logic}} | {{harvnb|Audi|1999|p=679}} }}</ref> Deductive logic is the branch of logic that investigates the strongest arguments, called deductively valid arguments, for which the conclusion cannot be false if all the premises are true. This is expressed by saying that the conclusion is a [[logical consequence]] of the premises. Rules of inference belong to deductive logic and describe argument forms that fulfill this requirement.<ref>{{multiref | {{harvnb|Hintikka|Sandu|2006|pp=13–14}} | {{harvnb|Audi|1999|pp=679–681}} | {{harvnb|Cannon|2002|pp=[https://books.google.com/books?id=79qb93CQE2AC&pg=PA14 14–15]}} }}</ref> In order to precisely assess whether an argument follows a rule of inference, logicians use [[formal languages]] to express statements in a rigorous manner, similar to [[mathematical formula]]s.<ref>{{multiref | {{harvnb|Tully|2005|pp=532–533}} | {{harvnb|Hodges|2005|pp=533–536}} | {{harvnb|Walton|1996}} | {{harvnb|Johnson|1999|pp=265–268}} }}</ref> They combine formal languages with rules of inference to construct [[formal systems]]—frameworks for formulating propositions and drawing conclusions.{{efn|Additionally, formal systems may also define [[axiom]]s or [[axiom schema]]s.<ref>{{harvnb|Hodel|2013|p=[https://books.google.com/books?id=SxRYdzWio84C&pg=PA7 7]}}</ref>}} Different formal systems may employ different formal languages or different rules of inference.<ref>{{multiref | {{harvnb|Cook|2009|p=124}} | {{harvnb|Jacquette|2006|pp=2–4}} | {{harvnb|Hodel|2013|p=[https://books.google.com/books?id=SxRYdzWio84C&pg=PA7 7]}} }}</ref> The basic rules of inference within a formal system can often be expanded by introducing new rules of inference, known as ''[[admissible rule]]s''. Admissible rules do not change which arguments in a formal system are valid but can simplify proofs. If an admissible rule can be expressed through a combination of the system's basic rules, it is called a ''derived'' or ''derivable rule''.<ref>{{multiref | {{harvnb|Cook|2009|pp=9–10}} | {{harvnb|Fitting|Mendelsohn|2012|pp=[https://books.google.com/books?id=5IxqCQAAQBAJ&pg=PA68 68–69]}} | {{harvnb|Boyer|Moore|2014|pp=[https://books.google.com/books?id=eMHSBQAAQBAJ&pg=PA144 144–146]}} }}</ref> Statements that can be deduced in a formal system are called ''[[theorem]]s'' of this formal system.<ref>{{harvnb|Cook|2009|p=287}}</ref> Widely-used systems of logic include [[propositional logic]], [[first-order logic]], and [[modal logic]].<ref>{{multiref | {{harvnb|Asprino|2020|p=[https://books.google.com/books?id=H6EGEAAAQBAJ&pg=PA4 4]}} | {{harvnb|Hodges|2005|pp=533–536}} | {{harvnb|Audi|1999|pp=679–681}} }}</ref>

Rules of inference only ensure that the conclusion is true if the premises are true. An argument with false premises can still be valid, but its conclusion could be false. For example, the argument "If pigs can fly, then the sky is purple. Pigs can fly. Therefore, the sky is purple." is valid because it follows ''modus ponens'', even though it contains false premises. A valid argument is called ''[[Soundness|sound argument]]'' if all premises are true.<ref>{{multiref | {{harvnb|Copi|Cohen|Rodych|2019|p=[https://books.google.com/books?id=38bADwAAQBAJ&pg=PA30 30]}} | {{harvnb|Hurley|2016|pp=42–43, 434–435}} }}</ref>

Rules of inference are closely related to [[Tautology (logic)|tautologies]]. In logic, a tautology is a statement that is true only because of the logical vocabulary it uses, independent of the meanings of its non-logical vocabulary. For example, the statement "if the tree is green and the sky is blue then the tree is green" is true independently of the meanings of terms like ''tree'' and ''green'', making it a tautology. Every argument following a rule of inference can be transformed into a tautology. This is achieved by forming a [[Logical conjunction|conjunction]] (''and'') of all premises and connecting it through [[Material conditional|implication]] (''if ... then ...'') to the conclusion, thereby combining all the individual statements of the argument into a single statement. For example, the valid argument "The tree is green and the sky is blue. Therefore, the tree is green." can be transformed into the tautology "if the tree is green and the sky is blue then the tree is green".<ref>{{multiref | {{harvnb|Gossett|2009|pp=[https://books.google.com/books?id=NuFeW8N2hlkC&pg=PA50 50–51]}} | {{harvnb|Carlson|2017|p=20}} | {{harvnb|Hintikka|Sandu|2006|p=16}} }}</ref> Rules of inference are also closely related to [[laws of thought]], which are basic principles of logic that can take the form tautologies. For example, the [[law of identity]] asserts that each entity is [[Identity (philosophy)|identical]] to itself. Other traditional laws of thought include the [[law of non-contradiction]] and the [[law of excluded middle]].<ref>{{multiref | {{harvnb|Kirwan|2005}} | {{harvnb|Corcoran|2007|loc=[https://books.google.com/books?id=9IMEm4dYQXUC&pg=PA146 146]}} }}</ref>

Rules of inference are not the only way to demonstrate that an argument is valid. Alternative methods include the use of [[truth table]]s, which applies to propositional logic, and [[truth tree]]s, which can also be employed in first-order logic.<ref>{{multiref | {{harvnb|Copi|Cohen|Flage|2016|pp=244–245, 447}} | {{harvnb|Hurley|2016|pp=267–270}} }}</ref>