Deductive reasoning Warning: You are not logged in. Your IP address will be publicly visible if you make any edits. If you log in or create an account, your edits will be attributed to your username, along with other benefits.Anti-spam check. Do not fill this in! == Rules of inference == Deductive reasoning usually happens by applying [[rules of inference]]. A rule of inference is a way or schema of drawing a conclusion from a set of premises.<ref name="Macmillan">{{cite book |first=Sanford |last=Shieh |editor1-last=Borchert |editor1-first=Donald |title=Macmillan Encyclopedia of Philosophy, 2nd Edition |date=2006 |publisher=Macmillan |url=https://www.encyclopedia.com/humanities/encyclopedias-almanacs-transcripts-and-maps/logical-knowledge |chapter=LOGICAL KNOWLEDGE}}</ref> This happens usually based only on the [[logical form]] of the premises. A rule of inference is valid if, when applied to true premises, the conclusion cannot be false. A particular argument is valid if it follows a valid rule of inference. Deductive arguments that do not follow a valid rule of inference are called [[formal fallacies]]: the truth of their premises does not ensure the truth of their conclusion.<ref name="IEPFallacies"/><ref name="Stump"/> In some cases, whether a rule of inference is valid depends on the logical system one is using. The dominant logical system is [[classical logic]] and the rules of inference listed here are all valid in classical logic. But so-called [[deviant logic]]s provide a different account of which inferences are valid. For example, the rule of inference known as [[double negation elimination]], i.e. that if a proposition is ''not not true'' then it is also ''true'', is accepted in classical logic but rejected in [[intuitionistic logic]].<ref name="Moschovakis">{{cite web |last1=Moschovakis |first1=Joan |title=Intuitionistic Logic: 1. Rejection of Tertium Non Datur |url=https://plato.stanford.edu/entries/logic-intuitionistic/#RejTerNonDat |website=The Stanford Encyclopedia of Philosophy |publisher=Metaphysics Research Lab, Stanford University |access-date=11 December 2021 |date=2021}}</ref><ref name="MacMillanNonClassical">{{cite book |last1=Borchert |first1=Donald |title=Macmillan Encyclopedia of Philosophy, 2nd Edition |date=2006 |publisher=Macmillan |url=https://philpapers.org/rec/BORMEO |chapter=Logic, Non-Classical}}</ref> === Prominent rules of inference === ==== Modus ponens ==== {{Main|Modus ponens|selfref = None}} Modus ponens (also known as "affirming the antecedent" or "the law of detachment") is the primary deductive [[rule of inference]]. It applies to arguments that have as first premise a [[Material conditional|conditional statement]] (<math>P \rightarrow Q</math>) and as second premise the antecedent (<math>P</math>) of the conditional statement. It obtains the consequent (<math>Q</math>) of the conditional statement as its conclusion. The argument form is listed below: # <code><math>P \rightarrow Q</math></code> (First premise is a conditional statement) # <math>P</math> (Second premise is the antecedent) # <math>Q</math> (Conclusion deduced is the consequent) In this form of deductive reasoning, the consequent (<math>Q</math>) obtains as the conclusion from the premises of a conditional statement (<math>P \rightarrow Q</math>) and its antecedent (<math>P</math>). However, the antecedent (<math>P</math>) cannot be similarly obtained as the conclusion from the premises of the conditional statement (<math>P \rightarrow Q</math>) and the consequent (<math>Q</math>). Such an argument commits the logical fallacy of [[affirming the consequent]]. The following is an example of an argument using modus ponens: # If it is raining, then there are clouds in the sky. # It is raining. # Thus, there are clouds in the sky. ==== Modus tollens ==== {{Main|Modus tollens}} Modus tollens (also known as "the law of contrapositive") is a deductive rule of inference. It validates an argument that has as premises a conditional statement (formula) and the negation of the consequent (<math>\lnot Q</math>) and as conclusion the negation of the antecedent (<math>\lnot P</math>). In contrast to [[modus ponens]], reasoning with modus tollens goes in the opposite direction to that of the conditional. The general expression for modus tollens is the following: # <math>P \rightarrow Q</math>. (First premise is a conditional statement) # <math>\lnot Q</math>. (Second premise is the negation of the consequent) # <math>\lnot P</math>. (Conclusion deduced is the negation of the antecedent) The following is an example of an argument using modus tollens: # If it is raining, then there are clouds in the sky. # There are no clouds in the sky. # Thus, it is not raining. ==== Hypothetical syllogism ==== {{main|hypothetical syllogism}} A ''hypothetical [[syllogism]]'' is an inference that takes two conditional statements and forms a conclusion by combining the hypothesis of one statement with the conclusion of another. Here is the general form: # <math>P \rightarrow Q</math> # <math>Q \rightarrow R</math> # Therefore, <math>P \rightarrow R</math>. In there being a subformula in common between the two premises that does not occur in the consequence, this resembles syllogisms in [[term logic]], although it differs in that this subformula is a proposition whereas in Aristotelian logic, this common element is a term and not a proposition. The following is an example of an argument using a hypothetical syllogism: # If there had been a thunderstorm, it would have rained. # If it had rained, things would have gotten wet. # Thus, if there had been a thunderstorm, things would have gotten wet.<ref>{{cite journal |last1=Morreau |first1=Michael |title=The Hypothetical Syllogism |journal=Journal of Philosophical Logic |date=2009 |volume=38 |issue=4 |pages=447β464 |doi=10.1007/s10992-008-9098-y |jstor=40344073 |s2cid=34804481 |url=https://www.jstor.org/stable/40344073 |issn=0022-3611}}</ref> === Fallacies === Various formal fallacies have been described. They are invalid forms of deductive reasoning.<ref name="IEPFallacies">{{cite web |last1=Dowden |first1=Bradley |title=Fallacies |url=https://iep.utm.edu/fallacy/ |website=Internet Encyclopedia of Philosophy |access-date=12 March 2022}}</ref><ref name="Stump">{{cite book |last1=Stump |first1=David J. |title=New Dictionary of the History of Ideas |url=https://www.encyclopedia.com/history/dictionaries-thesauruses-pictures-and-press-releases/fallacy-logical |chapter=Fallacy, Logical}}</ref> An additional aspect of them is that they appear to be valid on some occasions or on the first impression. They may thereby seduce people into accepting and committing them.<ref>{{cite web |last1=Hansen |first1=Hans |title=Fallacies |url=https://plato.stanford.edu/entries/fallacies/ |website=The Stanford Encyclopedia of Philosophy |publisher=Metaphysics Research Lab, Stanford University |access-date=12 March 2022 |date=2020}}</ref> One type of formal fallacy is [[affirming the consequent]], as in "if John is a bachelor, then he is male; John is male; therefore, John is a bachelor".<ref>{{cite web |title=Expert thinking and novice thinking: Deduction |url=https://www.britannica.com/topic/thought/Expert-thinking-and-novice-thinking |website=Encyclopedia Britannica |access-date=12 March 2022}}</ref> This is similar to the valid rule of inference named [[modus ponens]], but the second premise and the conclusion are switched around, which is why it is invalid. A similar formal fallacy is [[denying the antecedent]], as in "if Othello is a bachelor, then he is male; Othello is not a bachelor; therefore, Othello is not male".<ref name="BritannicaThought">{{cite web |title=Thought |url=https://www.britannica.com/topic/thought |website=Encyclopedia Britannica |access-date=14 October 2021 |language=en}}</ref><ref>{{cite journal |last1=Stone |first1=Mark A. |title=Denying the Antecedent: Its Effective Use in Argumentation |journal=Informal Logic |date=2012 |volume=32 |issue=3 |pages=327β356 |doi=10.22329/il.v32i3.3681 |url=https://philpapers.org/rec/STODTA|doi-access=free }}</ref> This is similar to the valid rule of inference called [[modus tollens]], the difference being that the second premise and the conclusion are switched around. Other formal fallacies include [[affirming a disjunct]], [[denying a conjunct]], and the [[fallacy of the undistributed middle]]. All of them have in common that the truth of their premises does not ensure the truth of their conclusion. But it may still happen by coincidence that both the premises and the conclusion of formal fallacies are true.<ref name="IEPFallacies"/><ref name="Stump"/> === Definitory and strategic rules === Rules of inferences are definitory rules: they determine whether an argument is deductively valid or not. But reasoners are usually not just interested in making any kind of valid argument. Instead, they often have a specific point or conclusion that they wish to prove or refute. So given a set of premises, they are faced with the problem of choosing the relevant rules of inference for their deduction to arrive at their intended conclusion.<ref name="Hintikka"/><ref name="BritannicaSystems">{{cite web |title=Logical systems |url=https://www.britannica.com/topic/logic/Logical-systems |website=www.britannica.com |access-date=4 December 2021 |language=en}}</ref><ref name="Pedemonte">{{cite journal |last1=Pedemonte |first1=Bettina |title=Strategic vs Definitory Rules: Their Role in Abductive Argumentation and their Relationship with Deductive Proof |journal=Eurasia Journal of Mathematics, Science and Technology Education |date=25 June 2018 |volume=14 |issue=9 |pages=em1589 |doi=10.29333/ejmste/92562 |s2cid=126245285 |url=https://www.ejmste.com/article/strategic-vs-definitory-rules-their-role-in-abductive-argumentation-and-their-relationship-with-5539 |language=english |issn=1305-8215|doi-access=free }}</ref> This issue belongs to the field of strategic rules: the question of which inferences need to be drawn to support one's conclusion. The distinction between definitory and strategic rules is not exclusive to logic: it is also found in various games.<ref name="Hintikka"/><ref name="BritannicaSystems"/><ref name="Pedemonte"/> In [[chess]], for example, the definitory rules state that [[Bishop (chess)|bishops]] may only move diagonally while the strategic rules recommend that one should control the center and protect one's [[King (chess)|king]] if one intends to win. In this sense, definitory rules determine whether one plays chess or something else whereas strategic rules determine whether one is a good or a bad chess player.<ref name="Hintikka"/><ref name="BritannicaSystems"/> The same applies to deductive reasoning: to be an effective reasoner involves mastering both definitory and strategic rules.<ref name="Hintikka"/> Summary: Please note that all contributions to Christianpedia may be edited, altered, or removed by other contributors. If you do not want your writing to be edited mercilessly, then do not submit it here. You are also promising us that you wrote this yourself, or copied it from a public domain or similar free resource (see Christianpedia:Copyrights for details). Do not submit copyrighted work without permission! Cancel Editing help (opens in new window) Discuss this page