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! == In various fields == === Cognitive psychology === [[Cognitive psychology]] studies the psychological processes responsible for deductive reasoning.<ref name="Schechter"/><ref name="Evans"/> It is concerned, among other things, with how good people are at drawing valid deductive inferences. This includes the study of the factors affecting their performance, their tendency to commit [[Fallacy|fallacies]], and the underlying [[Cognitive bias|biases]] involved.<ref name="Schechter"/><ref name="Evans"/> A notable finding in this field is that the type of deductive inference has a significant impact on whether the correct conclusion is drawn.<ref name="Schechter"/><ref name="Evans"/><ref>{{cite journal |last1=Rips |first1=Lance J. |title=Cognitive processes in propositional reasoning. |journal=Psychological Review |date=1983 |volume=90 |issue=1 |pages=38–71 |doi=10.1037/0033-295X.90.1.38 |url=https://doi.org/10.1037/0033-295X.90.1.38 |access-date=19 March 2022 |language=en |issn=1939-1471}}</ref><ref>{{cite journal |last1=Müller |first1=Ulrich |last2=Overton |first2=Willis F. |last3=Reene |first3=Kelly |title=Development of Conditional Reasoning: A Longitudinal Study |journal=Journal of Cognition and Development |date=February 2001 |volume=2 |issue=1 |pages=27–49 |doi=10.1207/S15327647JCD0201_2|s2cid=143955563 }}</ref> In a meta-analysis of 65 studies, for example, 97% of the subjects evaluated [[modus ponens]] inferences correctly, while the success rate for [[modus tollens]] was only 72%. On the other hand, even some fallacies like [[affirming the consequent]] or [[denying the antecedent]] were regarded as valid arguments by the majority of the subjects.<ref name="Schechter"/> An important factor for these mistakes is whether the conclusion seems initially plausible: the more believable the conclusion is, the higher the chance that a subject will mistake a fallacy for a valid argument.<ref name="Schechter"/><ref name="Evans"/> An important bias is the ''matching bias'', which is often illustrated using the [[Wason selection task]].<ref name="Evans"/><ref name="Schechter"/><ref>{{cite journal |last1=Evans |first1=J. St B. T. |last2=Lynch |first2=J. S. |title=Matching Bias in the Selection Task |journal=British Journal of Psychology |date=August 1973 |volume=64 |issue=3 |pages=391–397 |doi=10.1111/j.2044-8295.1973.tb01365.x}}</ref><ref>{{cite journal |last1=Wagner-Egger |first1=Pascal |title=Conditional reasoning and the Wason selection task: Biconditional interpretation instead of reasoning bias |journal=Thinking & Reasoning |date=1 October 2007 |volume=13 |issue=4 |pages=484–505 |doi=10.1080/13546780701415979 |s2cid=145011175 |url=https://www.tandfonline.com/doi/abs/10.1080/13546780701415979 |issn=1354-6783}}</ref> In an often-cited experiment by [[Peter Wason]], 4 cards are presented to the participant. In one case, the visible sides show the symbols D, K, 3, and 7 on the different cards. The participant is told that every card has a letter on one side and a number on the other side, and that "[e]very card which has a D on one side has a 3 on the other side". Their task is to identify which cards need to be turned around in order to confirm or refute this conditional claim. The correct answer, only given by about 10%, is the cards D and 7. Many select card 3 instead, even though the conditional claim does not involve any requirements on what symbols can be found on the opposite side of card 3.<ref name="Schechter"/><ref name="Evans"/> But this result can be drastically changed if different symbols are used: the visible sides show "drinking a beer", "drinking a coke", "16 years of age", and "22 years of age" and the participants are asked to evaluate the claim "[i]f a person is drinking beer, then the person must be over 19 years of age". In this case, 74% of the participants identified correctly that the cards "drinking a beer" and "16 years of age" have to be turned around.<ref name="Schechter"/><ref name="Evans"/> These findings suggest that the deductive reasoning ability is heavily influenced by the content of the involved claims and not just by the abstract logical form of the task: the more realistic and concrete the cases are, the better the subjects tend to perform.<ref name="Schechter"/><ref name="Evans"/> Another bias is called the "negative conclusion bias", which happens when one of the premises has the form of a negative [[material conditional]],<ref name="Evans"/><ref>{{cite book |last1=Chater |first1=Nick |last2=Oaksford |first2=Mike |last3=Hahn |first3=Ulrike |last4=Heit |first4=Evan |title=Inductive Logic |series=Handbook of the History of Logic |date=1 January 2011 |publisher=North-Holland |pages=553–624 |chapter-url=https://www.sciencedirect.com/science/article/abs/pii/B9780444529367500148 |language=en |chapter=Inductive Logic and Empirical Psychology|volume=10 |doi=10.1016/B978-0-444-52936-7.50014-8 |isbn=9780444529367 }}</ref><ref>{{cite book |last1=Arreckx |first1=Frederique |title=COUNTERFACTUAL THINKING AND THE FALSE BELIEF TASK: A DEVELOPMENTAL STUDY |date=2007 |publisher=University of Plymouth |url=http://hdl.handle.net/10026.1/1758 |language=en |chapter=2. Experiment 1: Affirmative and negative counterfactual questions|doi=10.24382/4506 |hdl=10026.1/1758 |type=Thesis }}</ref> as in "If the card does not have an A on the left, then it has a 3 on the right. The card does not have a 3 on the right. Therefore, the card has an A on the left". The increased tendency to misjudge the validity of this type of argument is not present for positive material conditionals, as in "If the card has an A on the left, then it has a 3 on the right. The card does not have a 3 on the right. Therefore, the card does not have an A on the left".<ref name="Evans"/> ==== Psychological theories of deductive reasoning ==== Various psychological theories of deductive reasoning have been proposed. These theories aim to explain how deductive reasoning works in relation to the underlying psychological processes responsible. They are often used to explain the empirical findings, such as why human reasoners are more susceptible to some types of fallacies than to others.<ref name="Schechter"/><ref name="Johnson-Laird2009"/><ref name="Johnson-Laird1993"/> An important distinction is between ''mental logic theories'', sometimes also referred to as ''rule theories'', and ''mental model theories''. ''Mental logic theories'' see deductive reasoning as a [[language]]-like process that happens through the manipulation of representations.<ref name="Schechter"/><ref name="Johnson-Laird2009"/><ref name="García-Madruga"/><ref name="Johnson-Laird1993">{{cite journal |last1=Johnson-Laird |first1=Philip N. |last2=Byrne |first2=Ruth M. J. |title=Precis of Deduction |journal=Behavioral and Brain Sciences |date=1993 |volume=16 |issue=2 |pages=323–333 |doi=10.1017/s0140525x00030260 |url=https://philpapers.org/rec/JOHPOD-2}}</ref> This is done by applying syntactic rules of inference in a way very similar to how systems of [[#Natural deduction|natural deduction]] transform their premises to arrive at a conclusion.<ref name="Johnson-Laird1993"/> On this view, some deductions are simpler than others since they involve fewer inferential steps.<ref name="Schechter"/> This idea can be used, for example, to explain why humans have more difficulties with some deductions, like the [[modus tollens]], than with others, like the [[modus ponens]]: because the more error-prone forms do not have a native rule of inference but need to be calculated by combining several inferential steps with other rules of inference. In such cases, the additional cognitive labor makes the inferences more open to error.<ref name="Schechter"/> ''Mental model theories'', on the other hand, hold that deductive reasoning involves models or [[mental representation]]s of possible states of the world without the medium of language or rules of inference.<ref name="Schechter"/><ref name="Johnson-Laird2009"/><ref name="Johnson-Laird1993"/> In order to assess whether a deductive inference is valid, the reasoner mentally constructs models that are compatible with the premises of the inference. The conclusion is then tested by looking at these models and trying to find a counterexample in which the conclusion is false. The inference is valid if no such counterexample can be found.<ref name="Schechter"/><ref name="Johnson-Laird2009"/><ref name="Johnson-Laird1993"/> In order to reduce cognitive labor, only such models are represented in which the premises are true. Because of this, the evaluation of some forms of inference only requires the construction of very few models while for others, many different models are necessary. In the latter case, the additional cognitive labor required makes deductive reasoning more error-prone, thereby explaining the increased rate of error observed.<ref name="Schechter"/><ref name="Johnson-Laird2009"/> This theory can also explain why some errors depend on the content rather than the form of the argument. For example, when the conclusion of an argument is very plausible, the subjects may lack the motivation to search for counterexamples among the constructed models.<ref name="Schechter"/> Both mental logic theories and mental model theories assume that there is one general-purpose reasoning mechanism that applies to all forms of deductive reasoning.<ref name="Schechter"/><ref name="García-Madruga">{{cite journal |last1=García-Madruga |first1=Juan A. |last2=Gutiérrez |first2=Francisco |last3=Carriedo |first3=Nuria |last4=Moreno |first4=Sergio |last5=Johnson-Laird |first5=Philip N. |title=Mental Models in Deductive Reasoning |journal=The Spanish Journal of Psychology |date=November 2002 |volume=5 |issue=2 |pages=125–140 |doi=10.1017/s1138741600005904|pmid=12428479 |s2cid=15293848 |url=http://revistas.ucm.es/index.php/SJOP/article/view/SJOP0202220125A }}</ref><ref>{{cite journal |last1=Johnson-Laird |first1=Philip N. |title=Mental models and human reasoning |journal=Proceedings of the National Academy of Sciences |date=18 October 2010 |volume=107 |issue=43 |pages=18243–18250 |doi=10.1073/pnas.1012933107 |pmid=20956326 |pmc=2972923 |issn=0027-8424|doi-access=free }}</ref> But there are also alternative accounts that posit various different special-purpose reasoning mechanisms for different contents and contexts. In this sense, it has been claimed that humans possess a special mechanism for permissions and obligations, specifically for detecting cheating in social exchanges. This can be used to explain why humans are often more successful in drawing valid inferences if the contents involve human behavior in relation to social norms.<ref name="Schechter"/> Another example is the so-called [[Dual process theory|dual-process theory]].<ref name="Evans"/><ref name="Schechter"/> This theory posits that there are two distinct cognitive systems responsible for reasoning. Their interrelation can be used to explain commonly observed biases in deductive reasoning. System 1 is the older system in terms of evolution. It is based on associative learning and happens fast and automatically without demanding many cognitive resources.<ref name="Evans"/><ref name="Schechter"/> System 2, on the other hand, is of more recent evolutionary origin. It is slow and cognitively demanding, but also more flexible and under deliberate control.<ref name="Evans"/><ref name="Schechter"/> The dual-process theory posits that system 1 is the default system guiding most of our everyday reasoning in a pragmatic way. But for particularly difficult problems on the logical level, system 2 is employed. System 2 is mostly responsible for deductive reasoning.<ref name="Evans"/><ref name="Schechter"/> ==== Intelligence ==== The [[ability]] of deductive reasoning is an important aspect of [[intelligence]] and many [[Intelligence test|tests of intelligence]] include problems that call for deductive inferences.<ref name="Johnson-Laird2009"/> Because of this relation to intelligence, deduction is highly relevant to psychology and the cognitive sciences.<ref name="Evans">{{cite book |last1=Evans |first1=Jonathan |editor1-last=Morrison |editor1-first=Robert |title=The Cambridge Handbook of Thinking and Reasoning |date=18 April 2005 |publisher=Cambridge University Press |isbn=978-0-521-82417-0 |url=https://books.google.com/books?id=znbkHaC8QeMC |language=en |chapter=8. Deductive reasoning}}</ref> But the subject of deductive reasoning is also pertinent to the [[computer sciences]], for example, in the creation of [[artificial intelligence]].<ref name="Johnson-Laird2009"/> === Epistemology === Deductive reasoning plays an important role in [[epistemology]]. Epistemology is concerned with the question of [[Justification (epistemology)|justification]], i.e. to point out which beliefs are justified and why.<ref>{{cite web |title=epistemology |url=https://www.britannica.com/topic/epistemology |website=www.britannica.com |access-date=19 March 2022 |language=en}}</ref><ref>{{cite web |last1=Steup |first1=Matthias |last2=Neta |first2=Ram |title=Epistemology |url=https://plato.stanford.edu/entries/epistemology/ |website=The Stanford Encyclopedia of Philosophy |publisher=Metaphysics Research Lab, Stanford University |access-date=19 March 2022 |date=2020}}</ref> Deductive inferences are able to transfer the justification of the premises onto the conclusion.<ref name="Schechter"/> So while logic is interested in the truth-preserving nature of deduction, epistemology is interested in the justification-preserving nature of deduction. There are different theories trying to explain why deductive reasoning is justification-preserving.<ref name="Schechter"/> According to [[reliabilism]], this is the case because deductions are truth-preserving: they are reliable processes that ensure a true conclusion given the premises are true.<ref name="Schechter"/><ref>{{cite web |last1=Becker |first1=Kelly |title=Reliabilism |url=https://iep.utm.edu/reliabil/ |website=Internet Encyclopedia of Philosophy |access-date=19 March 2022}}</ref><ref>{{cite web |last1=Goldman |first1=Alvin |last2=Beddor |first2=Bob |title=Reliabilist Epistemology |url=https://plato.stanford.edu/entries/reliabilism/ |website=The Stanford Encyclopedia of Philosophy |publisher=Metaphysics Research Lab, Stanford University |access-date=19 March 2022 |date=2021}}</ref> Some theorists hold that the thinker has to have explicit awareness of the truth-preserving nature of the inference for the justification to be transferred from the premises to the conclusion. One consequence of such a view is that, for young children, this deductive transference does not take place since they lack this specific awareness.<ref name="Schechter"/> === Probability logic === [[Probability logic]] is interested in how the probability of the premises of an argument affects the probability of its conclusion. It differs from classical logic, which assumes that propositions are either true or false but does not take into consideration the probability or certainty that a proposition is true or false.<ref>{{cite book |last1=Adams |first1=Ernest W. |title=A Primer of Probability Logic |date=13 October 1998 |publisher=Cambridge University Press |isbn=978-1-57586-066-4 |url=https://books.google.com/books?id=YxWMQgAACAAJ |language=en |chapter=1. Deduction and Probability: What Probability Logic Is About}}</ref><ref>{{cite book |last1=Hájek |first1=Alan |title=The Blackwell Guide to Philosophical Logic |date=2001 |publisher=Blackwell |pages=362–384 |url=https://philpapers.org/rec/HJEPLA |chapter=16. Probability, Logic, and Probability Logic}}</ref> The probability of the conclusion of a deductive argument cannot be calculated by figuring out the cumulative probability of the argument's premises. [[Timothy J. McGrew|Dr. Timothy McGrew]], a specialist in the applications of [[probability theory]], and Dr. Ernest W. Adams, a Professor Emeritus at [[UC Berkeley]], pointed out that the theorem on the accumulation of uncertainty designates only a lower limit on the probability of the conclusion. So the probability of the conjunction of the argument's premises sets only a minimum probability of the conclusion. The probability of the argument's conclusion cannot be any lower than the probability of the conjunction of the argument's premises. For example, if the probability of a deductive argument's four premises is ~0.43, then it is assured that the probability of the argument's conclusion is no less than ~0.43. It could be much higher, but it cannot drop under that lower limit.<ref>{{cite book |last= Adams|first= Ernest W.|date= 1998|title= A Primer of Probability Logic|publisher= Cambridge University Press|pages= 31–34|isbn= 157586066X}}</ref><ref name="Argument">{{cite journal |last1= McGrew|first1= Timothy J.|last2= DePoe|first2= John M.|date= 2013|title= Uses of Argument|url= https://philpapers.org/rec/DEPNTA|journal= Philosophia Christi|volume= 15|issue= 2|pages= 299–309|doi= 10.5840/pc201315228|access-date= 13 March 2021}}</ref> There can be examples in which each single premise is more likely true than not and yet it would be unreasonable to accept the conjunction of the premises. [[Henry E. Kyburg Jr.|Professor Henry Kyburg]], who was known for his work in [[probability]] and [[logic]], clarified that the issue here is one of closure – specifically, closure under conjunction. There are examples where it is reasonable to accept P and reasonable to accept Q without its being reasonable to accept the conjunction (P&Q). Lotteries serve as very intuitive examples of this, because in a basic non-discriminatory finite lottery with only a single winner to be drawn, it is sound to think that ticket 1 is a loser, sound to think that ticket 2 is a loser,...all the way up to the final number. However, clearly, it is irrational to accept the conjunction of these statements; the conjunction would deny the very terms of the lottery because (taken with the background knowledge) it would entail that there is no winner.<ref>{{cite journal|last1=Kyburg|first1=Henry|date=1970|title="Conjunctivitis," in M. Swain, ed., Induction, Acceptance, and Rational Belief|url=https://link.springer.com/chapter/10.1007/978-94-010-3390-9_4|journal=SYLI|volume=26|pages=55–82|doi=10.1007/978-94-010-3390-9_4|access-date=13 March 2021}}</ref><ref name="Argument"/> Dr. McGrew further adds that the sole method to ensure that a conclusion deductively drawn from a group of premises is more probable than not is to use premises the conjunction of which is more probable than not. This point is slightly tricky, because it can lead to a possible misunderstanding. What is being searched for is a general principle that specifies factors under which, for any logical consequence C of the group of premises, C is more probable than not. Particular consequences will differ in their probability. However, the goal is to state a condition under which this attribute is ensured, regardless of which consequence one draws, and fulfilment of that condition is required to complete the task. This principle can be demonstrated in a moderately clear way. Suppose, for instance, the following group of premises: {P, Q, R} Suppose that the conjunction ((P & Q) & R) fails to be more probable than not. Then there is at least one logical consequence of the group that fails to be more probable than not – namely, that very conjunction. So it is an essential factor for the argument to “preserve plausibility” (Dr. McGrew coins this phrase to mean “guarantee, from information about the plausibility of the premises alone, that any conclusion drawn from those premises by deductive inference is itself more plausible than not”) that the conjunction of the premises be more probable than not.<ref name="Argument"/> 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