Topzle Topzle

Rule of inference

Updated: Wikipedia source

Rule of inference

Rules of inference are ways of deriving conclusions from premises. They are integral parts of formal logic, serving as norms of the logical structure of valid arguments. If an argument with true premises follows a rule of inference then the conclusion cannot be false. Modus ponens, an influential rule of inference, connects two premises of the form "if P {\displaystyle P} then Q {\displaystyle Q} " and " P {\displaystyle P} " to the conclusion " Q {\displaystyle Q} ", as in the argument "If it rains, then the ground is wet. It rains. Therefore, the ground is wet." There are many other rules of inference for different patterns of valid arguments, such as modus tollens, disjunctive syllogism, constructive dilemma, and existential generalization. Rules of inference include rules of implication, which operate only in one direction from premises to conclusions, and rules of replacement, which state that two expressions are equivalent and can be freely swapped. Rules of inference contrast with formal fallacies—invalid argument forms involving logical errors. Rules of inference belong to logical systems, and distinct logical systems use different rules of inference. Propositional logic examines the inferential patterns of simple and compound statements or propositions. First-order logic extends propositional logic by analyzing the components or internal structure of propositions. It introduces new rules of inference governing how this internal structure affects valid arguments. Modal logics explore concepts like possibility and necessity, examining the inferential structure of these concepts. Intuitionistic, paraconsistent, and many-valued logics propose alternative inferential patterns that differ from the traditionally dominant approach associated with classical logic. Various formalisms are used to express logical systems. Natural deduction systems employ many intuitive rules of inference to reflect how people naturally reason while Hilbert systems provide minimalistic frameworks to represent foundational principles without redundancy. Rules of inference are relevant to many areas, such as proofs in mathematics and automated reasoning in computer science. Their conceptual and psychological underpinnings are studied by philosophers of logic and cognitive psychologists.

Tables

Notable rules of inference[40] · Systems of logic › Classical › Propositional logic
Modus ponens
Modus ponens
Rule of inference
Modus ponens
Form
P → Q P Q {\displaystyle {\begin{array}{l}P\to Q\\P\\\hline Q\end{array}}}
Example
If Kim is in Seoul, then Kim is in South Korea. Kim is in Seoul. Therefore, Kim is in South Korea. {\displaystyle {\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}}}
Modus tollens
Modus tollens
Rule of inference
Modus tollens
Form
P → Q ¬ Q ¬ P {\displaystyle {\begin{array}{l}P\to Q\\\lnot Q\\\hline \lnot P\end{array}}}
Example
If Koko is a koala, then Koko is cuddly. Koko is not cuddly. Therefore, Koko is not a koala. {\displaystyle {\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}}}
Hypothetical syllogism
Hypothetical syllogism
Rule of inference
Hypothetical syllogism
Form
P → Q Q → R P → R {\displaystyle {\begin{array}{l}P\to Q\\Q\to R\\\hline P\to R\end{array}}}
Example
If Leo is a lion, then Leo roars. If Leo roars, then Leo is fierce. Therefore, if Leo is a lion, then Leo is fierce. {\displaystyle {\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}}}
Disjunctive syllogism
Disjunctive syllogism
Rule of inference
Disjunctive syllogism
Form
P ∨ Q ¬ P Q {\displaystyle {\begin{array}{l}P\lor Q\\\lnot P\\\hline Q\end{array}}}
Example
The book is on the shelf or on the table. The book is not on the shelf. Therefore, the book is on the table.  {\displaystyle {\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}}}
Double negation elimination
Double negation elimination
Rule of inference
Double negation elimination
Form
¬ ¬ P P {\displaystyle {\begin{array}{l}\lnot \lnot P\\\hline P\end{array}}}
Example
We were not unable to meet the deadline. We were able to meet the deadline.  {\displaystyle {\begin{array}{l}{\text{We were not unable to meet the deadline.}}\\\hline {\text{We were able to meet the deadline. }}\end{array}}}
Rule of inference
Form
Example
Modus ponens
P → {l}P\to Q\\P\\\hline Q\end{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}}}
Modus tollens
P → Q ¬ Q ¬ {l}P\to Q\\\lnot Q\\\hline \lnot P\end{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}}}
Hypothetical syllogism
P → Q Q → R P → {l}P\to Q\\Q\to R\\\hline P\to R\end{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}}}
Disjunctive syllogism
P ∨ Q ¬ {l}P\lor Q\\\lnot P\\\hline Q\end{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}}}
Double negation elimination
¬ ¬ {l}\lnot \lnot P\\\hline P\end{array}}}
{l}{\text{We were not unable to meet the deadline.}}\\\hline {\text{We were able to meet the deadline. }}\end{array}}}
Notable rules of inference[9] · Systems of logic › Classical › First-order logic
Universal instantiation
Universal instantiation
Rule of inference
Universal instantiation
Form
∀ x P ( x ) P ( a ) {\displaystyle {\begin{array}{l}\forall xP(x)\\\hline P(a)\end{array}}}
Example
Everyone must pay taxes. Therefore, Wesley must pay taxes. {\displaystyle {\begin{array}{l}{\text{Everyone must pay taxes.}}\\\hline {\text{Therefore, Wesley must pay taxes.}}\end{array}}}
Existential generalization
Existential generalization
Rule of inference
Existential generalization
Form
P ( a ) ∃ x P ( x ) {\displaystyle {\begin{array}{l}P(a)\\\hline \exists xP(x)\end{array}}}
Example
Socrates is mortal. Therefore, someone is mortal. {\displaystyle {\begin{array}{l}{\text{Socrates is mortal.}}\\\hline {\text{Therefore, someone is mortal.}}\end{array}}}
Rule of inference
Form
Example
Universal instantiation
∀ x P ( x ) P ( a ) {\displaystyle {\begin{array}{l}\forall xP(x)\\\hline P(a)\end{array}}}
{l}{\text{Everyone must pay taxes.}}\\\hline {\text{Therefore, Wesley must pay taxes.}}\end{array}}}
Existential generalization
P ( a ) ∃ x P ( x ) {\displaystyle {\begin{array}{l}P(a)\\\hline \exists xP(x)\end{array}}}
{l}{\text{Socrates is mortal.}}\\\hline {\text{Therefore, someone is mortal.}}\end{array}}}

References

  1. Non-deductive arguments, by contrast, support the conclusion without ensuring that it is true, such as inductive and abd
  2. The symbol → {\displaystyle \to } in this formula means if ... then ..., exp
  3. Universal instantiation infers a statement about a specific individual from a universal claim, as in the argument "Every
  4. Logical operators or constants are expressions used to form and connect propositions, such as not, or, and if...then....
  5. According to a narrow definition, rules of inference only encompass rules of implication but do not include rules of rep
  6. Logicians use the symbols ¬ {\displaystyle \lnot } or
  7. Rules of replacement are sometimes expressed using a double semi-colon. For instance, the double negation rule can be wr
  8. Additionally, formal systems may also define axioms or axiom schemas.
  9. This example assumes that a {\displaystyle a} refers to an individual in the
  10. An important difference between first-order and second-order logic is that second-order logic is incomplete, meaning tha
  11. This situation is also known as a deductive explosion.
  12. The Fitch notation is an influential way of presenting proofs in natural deduction systems.
  13. Hurley 2016, p. 303Hintikka & Sandu 2006, pp. 13–14Carlson 2017, p. 20Copi, Cohen & Flage 2016, pp. 244–245, 447
  14. Shanker 2003, p. 442Cook 2009, p. 152
    https://books.google.com/books?id=jIzT7AT3ILIC&pg=PA442
  15. Hintikka & Sandu 2006, pp. 13–14
  16. Hurley 2016, pp. 54–55, 283–287Arthur 2016, p. 165Hintikka & Sandu 2006, pp. 13–14Carlson 2017, p. 20Copi, Cohen & Flage
    https://books.google.com/books?id=vEFVAgAAQBAJ&pg=PA88
  17. Hurley 2016, p. 303
  18. Magnus & Button 2021, p. 32
  19. Copi, Cohen & Flage 2016, pp. 137, 245–246Magnus & Button 2021, p. 109
  20. Sørensen & Urzyczyn 2006, pp. 161–162
    https://books.google.com/books?id=_mtnm-9KtbEC&pg=PA161
  21. Hurley 2016, pp. 374–377Shapiro & Kouri Kissel 2024, § 3. Deduction
  22. Copi, Cohen & Flage 2016, pp. 295–299Cook 2009, pp. 124, 251–252Hurley 2016, pp. 374–375
  23. Cook 2009, pp. 124, 230, 251–252Magnus & Button 2021, pp. 112–113Copi, Cohen & Flage 2016, pp. 244–245
  24. Hintikka & Sandu 2006, pp. 13–14Hintikka 2013, p. 98
    https://books.google.com/books?id=rUDsCAAAQBAJ&pg=PA98
  25. Hurley 2016, pp. 238–239
  26. Baker & Hacker 2014, pp. 88–90Tourlakis 2011, p. 40Hintikka & Sandu 2006, pp. 13–14McKeon 2010, pp. 128–129
    https://books.google.com/books?id=vEFVAgAAQBAJ&pg=PA88
  27. Burris 2024, Lead sectionO'Regan 2017, pp. 95–96, 103
  28. Arthur 2016, pp. 165–166
  29. Copi, Cohen & Flage 2016, p. 446Magnus & Button 2021, p. 32
  30. Hurley 2016, pp. 323–252
  31. Arthur 2016, pp. 165–166Hurley 2016, pp. 302–303, 323–252Copi, Cohen & Flage 2016, pp. 257–258Hurley & Watson 2018, pp.
  32. Walton 2013, pp. 5–7, 265
  33. Epstein & King 2002, pp. 1–2, 38
  34. Hintikka & Spade 2020, § Aristotle, § Medieval Logic, § Boole and De Morgan, § Gottlob FregeO'Regan 2017, p. 95–96, 103G
    https://books.google.com/books?id=jpteBwAAQBAJ&pg=PA362
  35. Hurley 2016, pp. 303, 429–430Hintikka & Sandu 2006, pp. 13–14Carlson 2017, p. 20Copi, Cohen & Flage 2016, pp. 244–245, 4
  36. Audi 1999, pp. 679–681Lowe 2005, pp. 699–701Dowden 2020, p. 24Copi, Cohen & Rodych 2019, p. 4
    https://books.google.com/books?id=38bADwAAQBAJ&pg=PA4
  37. Hintikka 2019, § Nature and Varieties of LogicHaack 1978, pp. 1–10Schlesinger, Keren-Portnoy & Parush 2001, p. 220
  38. Hintikka 2019, Lead section, § Nature and Varieties of LogicAudi 1999, p. 679
  39. Hintikka & Sandu 2006, pp. 13–14Audi 1999, pp. 679–681Cannon 2002, pp. 14–15
    https://books.google.com/books?id=79qb93CQE2AC&pg=PA14
  40. Tully 2005, pp. 532–533Hodges 2005, pp. 533–536Walton 1996Johnson 1999, pp. 265–268
  41. Hodel 2013, p. 7
    https://books.google.com/books?id=SxRYdzWio84C&pg=PA7
  42. Cook 2009, p. 124Jacquette 2006, pp. 2–4Hodel 2013, p. 7
    https://books.google.com/books?id=SxRYdzWio84C&pg=PA7
  43. Cook 2009, pp. 9–10Fitting & Mendelsohn 2012, pp. 68–69Boyer & Moore 2014, pp. 144–146
    https://books.google.com/books?id=5IxqCQAAQBAJ&pg=PA68
  44. Cook 2009, p. 287
  45. Asprino 2020, p. 4Hodges 2005, pp. 533–536Audi 1999, pp. 679–681
    https://books.google.com/books?id=H6EGEAAAQBAJ&pg=PA4
  46. Copi, Cohen & Rodych 2019, p. 30Hurley 2016, pp. 42–43, 434–435
    https://books.google.com/books?id=38bADwAAQBAJ&pg=PA30
  47. Gossett 2009, pp. 50–51Carlson 2017, p. 20Hintikka & Sandu 2006, p. 16
    https://books.google.com/books?id=NuFeW8N2hlkC&pg=PA50
  48. Kirwan 2005Corcoran 2007, 146
    https://books.google.com/books?id=9IMEm4dYQXUC&pg=PA146
  49. Copi, Cohen & Flage 2016, pp. 244–245, 447Hurley 2016, pp. 267–270
  50. Klement, Lead section, § 1. Introduction, § 3. The Language of Propositional LogicSider 2010, pp. 25–30
    https://books.google.com/books?id=-KkPEAAAQBAJ&pg=PA25
  51. Hurley 2016, pp. 303, 315Copi, Cohen & Flage 2016, p. 247Klement, § Deduction: Rules of Inference and Replacement
  52. Hurley 2016, pp. 303, 315Copi, Cohen & Flage 2016, p. 247
  53. O'Regan 2017, pp. 101–103Zalta 2024, Lead section
  54. Shapiro & Kouri Kissel 2024, Lead section, § 2. LanguageSider 2010, pp. 90–95Cook 2009, pp. 119–120
    https://books.google.com/books?id=-KkPEAAAQBAJ&pg=PA90
  55. Garson 2024, Lead section, § 2. Modal LogicsSider 2010, pp. 133–136, 227–228
    https://books.google.com/books?id=-KkPEAAAQBAJ&pg=PA133
  56. Garson 2024, § 3. Deontic Logics
  57. Garson 2024, § 1. What is Modal Logic?, § 4. Temporal LogicsSider 2010, pp. 183–187
    https://books.google.com/books?id=-KkPEAAAQBAJ&pg=PA183
  58. O'Regan 2017, pp. 90–91, 103
  59. Smith 2022, Lead section, § 3. The Subject of Logic: “Syllogisms”Groarke, Lead section, § 3. From Words into Proposition
  60. Sharma 2000, pp. 199–200Dasti, § 1.b.ii. The Structure of Inference
  61. Väänänen 2024, Lead section, § 1. Introduction
  62. Väänänen 2024, § 1. IntroductionGrandy 1979, p. 122Linnebo 2014, p. 123
    https://books.google.com/books?id=ItgJhsGE-RAC&pg=PA122
  63. Pollard 2015, p. 98
    https://books.google.com/books?id=6cY-CgAAQBAJ&pg=PA98
  64. Moschovakis 2024, Lead section, § 1. Rejection of Tertium Non DaturSider 2010, pp. 86–89Kleene 2000, p. 81
    https://books.google.com/books?id=-KkPEAAAQBAJ&pg=PA86
  65. Shapiro & Kouri Kissel 2024, § 3. DeductionSider 2010, pp. 80–82Priest, Tanaka & Weber 2025, Lead section
    https://books.google.com/books?id=-KkPEAAAQBAJ&pg=PA80
  66. Carnielli & Coniglio 2016, p. ix
    https://books.google.com/books?id=VApkDAAAQBAJ&pg=PR9
  67. Weber, Lead section, § 2. Logical BackgroundSider 2010, pp. 80–82Priest, Tanaka & Weber 2025, Lead section
    https://books.google.com/books?id=-KkPEAAAQBAJ&pg=PA80
  68. Sider 2010, pp. 73–74, 77–79Gottwald 2022, Lead section, § 3.4 Three-valued systems
    https://books.google.com/books?id=-KkPEAAAQBAJ&pg=PA73
  69. Egré & Rott 2021, § 2. Three-Valued ConditionalsGottwald 2022, Lead section, § 2. Proof Theory
  70. Nederpelt & Geuvers 2014, pp. 159–162Sørensen & Urzyczyn 2006, pp. 161–162
    https://books.google.com/books?id=orsrBQAAQBAJ&pg=PA159
  71. Pelletier & Hazen 2024, Lead section, § 2.2 Modern Versions of Jaśkowski's Method, § 5.1 Normalization of Intuitionistic
    https://books.google.com/books?id=orsrBQAAQBAJ&pg=PA159
  72. Akiba 2024, p. 7
    https://books.google.com/books?id=ftksEQAAQBAJ&pg=PA7
  73. Bacon 2023, pp. 423–424Nederpelt & Geuvers 2014, pp. 159–162Sørensen & Urzyczyn 2006, pp. 161–162
    https://books.google.com/books?id=qa3WEAAAQBAJ&pg=PA423
  74. Reynolds 1998, p. 12Cook 2009, p. 26
    https://books.google.com/books?id=X_ToAwAAQBAJ&pg=PA12
  75. Smullyan 2014, pp. 102–103
    https://books.google.com/books?id=n6S-AwAAQBAJ&pg=PA102
  76. Metcalfe, Paoli & Tsinakis 2023, pp. 36–37Nederpelt & Geuvers 2014, pp. 159–162Sørensen & Urzyczyn 2006, pp. 161–162
    https://books.google.com/books?id=CkPsEAAAQBAJ&pg=PA36
  77. Rathjen & Sieg 2024, § 2.2 Sequent CalculiSørensen & Urzyczyn 2006, pp. 161–165
    https://books.google.com/books?id=_mtnm-9KtbEC&pg=PA161
  78. Copi, Cohen & Flage 2016, pp. 46–47, 227Cook 2009, p. 123Hurley & Watson 2018, pp. 125–126, 723
  79. Copi, Cohen & Flage 2016, pp. 224, 439Hurley & Watson 2018, pp. 385–386, 720
  80. Copi, Cohen & Flage 2016, pp. 46, 228, 442Hurley & Watson 2018, pp. 385–386, 722
  81. Copi, Cohen & Flage 2016, pp. 443, 449Hurley & Watson 2018, pp. 723, 728Cohen 2009, p. 254
    https://books.google.com/books?id=4aGIsKmkODgC&pg=PA254
  82. Fetzer 1996, pp. 241–243Dent 2024, p. 36
    https://books.google.com/books?id=JbNI3b-j0mkC&pg=PA241
  83. Horsten 2023, Lead section, § 5.4 Mathematical ProofPolkinghorne 2011, p. 65
    https://books.google.com/books?id=AJV0P1pPBNoC&pg=PA65
  84. Cook 2009, pp. 174, 185Porta et al. 2011, p. 237
    https://books.google.com/books?id=NDMID6mWcZcC&pg=PA237
  85. Butterfield & Ngondi 2016, § Computer ScienceCook 2009, p. 174Dent 2024, p. 36
    https://books.google.com/books?id=NbA0EQAAQBAJ&pg=PA36
  86. Butterfield & Ngondi 2016, § Logic Programming Languages, § PrologWilliamson & Russo 2010, p. 45
    https://books.google.com/books?id=SmYu9-1HGtYC&pg=PA45
  87. Butterfield & Ngondi 2016, § Theorem proving, § Mechanical Verifier
  88. Butterfield & Ngondi 2016, § Expert System, § Knowledge Base, § Inference EngineFetzer 1996, pp. 241–243
    https://books.google.com/books?id=JbNI3b-j0mkC&pg=PA241
  89. McKeon, Lead section, § 1. Introduction, § 2b. Logical and Non-Logical TerminologyMcKeon 2010, pp. 24–25, 126–128Hintikk
    https://books.google.com/books?id=76SH_DI3yyoC&pg=PA24
  90. Schechter 2013, p. 227Evans 2005, pp. 171–174
  91. Minghui 2009, pp. 58–59, 72Kelsen 1973, pp. 228–229
Image
Source: —
Tip: Wheel or +/− to zoom, drag to pan, Esc to close.