{{Short description|Steps in reasoning}} {{For multi|the music album|Inference (album){{!}}''Inference'' (album)}} {{No footnotes|date=July 2023}} {{Use dmy dates|date=November 2024}} {{peirce_deduction_induction_abduction_venn.svg}} '''Inferences''' are steps in [[logical reasoning]], moving from [[premise]]s to [[logical consequence]]s. Inference is traditionally divided into [[deductive reasoning|deduction]] and [[inductive reasoning|induction]], a distinction that dates at least to [[Aristotle]] (300s BC). A third type of inference, [[Abductive reasoning|abduction]], has been proposed, notably by [[Charles Sanders Peirce]].Francesco Bellucci, “Eco and Peirce on Abduction”, European Journal of Pragmatism and American Philosophy [Online], X-1 | 2018, Online since 20 July 2018, connection on 16 March 2026. URL: http://journals.openedition.org/ejpap/1122; DOI: https://doi.org/10.4000/ejpap.1122 Deduction is inference [[Formal proof|deriving]] [[Logical consequence|logical conclusions]] from premises known or assumed to be [[truth|true]], with the [[Rule of inference|laws of valid inference]] being studied in [[logic]]. Induction is inference from [[particular]] evidence to a [[Universal (metaphysics)|universal]] conclusion.Vlasáková, M. (2023). "Aristotle’s Notion of Deduction." ''Disputatio'', vol. 15, no. 68, University of Lisbon, 2023, pp. 90-114. https://doi.org/10.2478/disp-2023-0004. Abduction seeks neither logical certainty nor a universal conclusion but a "best explanation" based on likelihood given the evidence. Various fields study how inference is done in practice. Human inference (i.e. how humans draw conclusions) is traditionally studied within the fields of logic, argumentation studies, and [[cognitive psychology]]; [[artificial intelligence]] researchers develop automated inference systems to emulate human inference. [[Statistical inference]] uses mathematics to draw conclusions in the presence of uncertainty. This generalizes deterministic reasoning, with the absence of uncertainty as a special case. Statistical inference uses quantitative or qualitative ([[categorical data|categorical]]) data which may be subject to random variations."Foundations of Inference" ''Statistics & Data Science Dietrich College of Humanities and Social Sciences Carnegie Mellon University''. Retrieved 03.17.2026 https://www.cmu.edu/dietrich/statistics-datascience/research/foundations-of-inference.html ==Definition== The process by which a general conclusion is inferred from multiple [[observations]] is called [[inductive reasoning]]. The conclusion may be correct or incorrect, or correct to within a certain degree of accuracy, or correct in certain situations. Conclusions inferred from multiple observations may be tested by additional observations. This definition is disputable (due to its lack of clarity. Ref: Oxford English dictionary: "induction ... 3. Logic the inference of a general law from particular instances."{{clarify|reason=Some information, probably encoded by different type fonts in the dictionary, seems to have got lost. I assume, under the keyword 'induction ([[logic]])', the dictionary says 'the inference of a general law from particular instances'. Please check with the source and clarify.|date=August 2013}}) The definition given thus applies only when the "conclusion" is general. Two possible definitions of "inference" are: # A conclusion reached on the basis of evidence and reasoning. # The process of reaching such a conclusion. ==Examples== ===Example for definition #1=== [[Ancient Greek philosophy|Ancient Greek philosophers]] defined a number of [[syllogism]]s, correct three part inferences, that can be used as building blocks for more complex reasoning. We begin with a famous example: # All humans are mortal. # All Greeks are humans. # All Greeks are mortal. The reader can check that the premises and conclusion are true, but logic is concerned with inference: does the truth of the conclusion follow from that of the premises? The validity of an inference depends on the form of the inference. That is, the word "valid" does not refer to the truth of the premises or the conclusion, but rather to the form of the inference. An inference can be valid even if the parts are false, and can be invalid even if some parts are true. But a valid form with true premises will always have a true conclusion. For example, consider the form of the following [[Symbology|symbological]] track: #All meat comes from animals. #All beef is meat. #Therefore, all beef comes from animals. If the premises are true, then the conclusion is necessarily true, too. Now we turn to an invalid form. #All A are B. #All C are B. #Therefore, all C are A. To show that this form is invalid, we demonstrate how it can lead from true premises to a false conclusion. #All apples are fruit. (True) #All bananas are fruit. (True) #Therefore, all bananas are apples. (False) A valid argument with a false premise may lead to a false conclusion, (this and the following examples do not follow the Greek syllogism): #All tall people are French. (False) #John Lennon was tall. (True) #Therefore, John Lennon was French. (False) When a valid argument is used to derive a false conclusion from a false premise, the inference is valid because it follows the form of a correct inference. A valid argument can also be used to derive a true conclusion from a false premise: #All tall people are musicians. (Valid, False) #John Lennon was tall. (Valid, True) #Therefore, John Lennon was a musician. (Valid, True) In this case we have one false premise and one true premise where a true conclusion has been inferred. ===Example for definition #2=== Evidence: It is the early 1950s and you are an American stationed in the [[Soviet Union]]. You read in the [[Moscow]] newspaper that a [[soccer]] team from a small city in [[Siberia]] starts winning game after game. The team even defeats the Moscow team. Inference: The small city in Siberia is not a small city anymore. The Soviets are working on their own nuclear or high-value secret weapons program. Knowns: The Soviet Union is a [[command economy]]: people and material are told where to go and what to do. The small city was remote and historically had never distinguished itself; its soccer season was typically short because of the weather. Explanation: In a [[command economy]], people and material are moved where they are needed. Large cities might field good teams due to the greater availability of high quality players; and teams that can practice longer (possibly due to sunnier weather and better facilities) can reasonably be expected to be better. In addition, you put your best and brightest in places where they can do the most good—such as on high-value weapons programs. It is an anomaly for a small city to field such a good team. The anomaly indirectly described a condition by which the observer inferred a new meaningful pattern—that the small city was no longer small. Why would you put a large city of your best and brightest in the middle of nowhere? To hide them, of course. ==Incorrect inference== An incorrect inference is known as a [[fallacy]]. Philosophers who study [[informal logic]] have compiled large [[List of fallacies|lists of them]]. Cognitive psychologists have documented many [[cognitive bias|biases in human reasoning]] that favor incorrect reasoning, and explain them with the use of [[Heuristic (psychology)|heuristics]] in human reasoning.{{Cite book |last=Goldstein |first=E. Bruce |title=Cognitive psychology: connecting mind, research, and everyday experience |last2=Hale |first2=Ralph G. |date=2026 |publisher=Cengage |isbn=979-8-214-14338-5 |edition=6th |location=Australia Brazil Canada Mexico Singapore United Kingdom United States}} One example of human reasoning bias is the [[confirmation bias]], where people tend to seek information that confirms their beliefs rather than information that may contradict it, even though the latter (falsifications) is more informative for deductive reasoning. This is demonstrated by the [[Wason selection task]].{{Cite journal |last=Wason |first=P. C. |last2=Shapiro |first2=Diana |date=1971 |title=Natural and contrived experience in a reasoning problem |url=https://journals.sagepub.com/doi/10.1080/00335557143000068 |journal=Quarterly Journal of Experimental Psychology |language=en |volume=23 |issue=1 |pages=63–71 |doi=10.1080/00335557143000068 |issn=0033-555X|url-access=subscription }}{{Cite journal |last=Wason |first=P. C. |date=1968 |title=Reasoning about a Rule |url=https://journals.sagepub.com/doi/10.1080/14640746808400161 |journal=Quarterly Journal of Experimental Psychology |language=en |volume=20 |issue=3 |pages=273–281 |doi=10.1080/14640746808400161 |issn=0033-555X|url-access=subscription }} Another example, involving probabilistic reasoning, is the [[conjunction fallacy]], where people judge a conjunction A \wedge B to be more probable than a single conjunct A, because B contains more "representative" content. This is demonstrated by the "[[Conjunction fallacy|Linda problem]]" and explained with the use of a [[representativeness heuristic]].{{Cite journal |last=Tversky |first=Amos |last2=Kahneman |first2=Daniel |date=1983 |title=Extensional versus intuitive reasoning: The conjunction fallacy in probability judgment. |url=https://doi.apa.org/doi/10.1037/0033-295X.90.4.293 |journal=Psychological Review |language=en |volume=90 |issue=4 |pages=293–315 |doi=10.1037/0033-295X.90.4.293 |issn=1939-1471|url-access=subscription }} ==Applications== ===Inference engines=== {{main|Reasoning system|Inference engine|expert system|business rule engine}} AI systems first provided automated logical inference and these were once extremely popular research topics, leading to industrial applications under the form of [[expert system]]s and later [[business rule engine]]s. More recent work on [[automated theorem proving]] has had a stronger basis in formal logic. An inference system's job is to extend a knowledge base automatically. The [[knowledge base]] (KB) is a set of propositions that represent what the system knows about the world. Several techniques can be used by that system to extend KB by means of valid inferences. An additional requirement is that the conclusions the system arrives at are [[relevance|relevant]] to its task. Additionally, the term 'inference' has also been applied to the process of generating predictions from trained [[Artificial neural network|neural networks]]. In this context, an 'inference engine' refers to the system or hardware performing these operations. This type of inference is widely used in applications ranging from [[image recognition]] to [[natural language processing]]. ====Prolog engine==== [[Prolog]] (for "Programming in Logic") is a [[programming language]] based on a [[subset]] of [[predicate calculus]]. Its main job is to check whether a certain proposition can be inferred from a KB (knowledge base) using an algorithm called [[backward chaining]]. Let us return to our [[Socrates]] [[syllogism]]. We enter into our Knowledge Base the following piece of code: mortal(X) :- man(X). man(socrates). ( Here ''''':-''''' can be read as "if". Generally, if ''P {{imp}} Q'' (if P then Q) then in Prolog we would code ''Q''':-'''P'' (Q if P).)
This states that all men are mortal and that Socrates is a man. Now we can ask the Prolog system about Socrates: ?- mortal(socrates). (where ''?-'' signifies a query: Can ''mortal(socrates).'' be deduced from the KB using the rules) gives the answer "Yes". On the other hand, asking the Prolog system the following: ?- mortal(plato). gives the answer "No". This is because [[Prolog]] does not know anything about [[Plato]], and hence defaults to any property about Plato being false (the so-called [[closed world assumption]]). Finally ?- mortal(X) (Is anything mortal) would result in "Yes" (and in some implementations: "Yes": X=socrates)
[[Prolog]] can be used for vastly more complicated inference tasks. See the corresponding article for further examples. ===Semantic web=== Recently automatic reasoners found in [[semantic web]] a new field of application. Being based upon [[description logic]], knowledge expressed using one variant of [[Web Ontology Language|OWL]] can be logically processed, i.e., inferences can be made upon it. ===Bayesian statistics and probability logic=== {{main|Bayesian inference}} Philosophers and scientists who follow the [[Bayesian inference|Bayesian framework]] for inference use the mathematical rules of [[probability]] to find this best explanation. The Bayesian view has a number of desirable features—one of them is that it embeds deductive (certain) logic as a subset (this prompts some writers to call Bayesian probability "probability logic", following [[E. T. Jaynes]]). Bayesians identify probabilities with degrees of beliefs, with certainly true propositions having probability 1, and certainly false propositions having probability 0. To say that "it's going to rain tomorrow" has a 0.9 probability is to say that you consider the possibility of rain tomorrow as extremely likely. Through the rules of probability, the probability of a conclusion and of alternatives can be calculated. The best explanation is most often identified with the most probable (see [[Bayesian decision theory]]). A central rule of Bayesian inference is [[Bayes' theorem]]. ===Fuzzy logic=== {{main||Fuzzy logic}} {{expand section|date=October 2016}} === Non-monotonic logic === {{main|Non-monotonic logic}} For example, logicians have worked to develop a formal logic of reasons, using variants of non-monotonic logic.{{Cite book |last=Horty |first=John |title=Reasons as defaults |date=2014 |publisher=Oxford Univ. Press |isbn=978-0-19-939644-3 |edition=1. issued as paperback |location=Oxford}} ==See also== {{Portal|Philosophy|Psychology}} * {{Annotated link|A priori and a posteriori|''A priori'' and ''a posteriori''}} * {{Annotated link|Abductive reasoning}} * {{Annotated link|Deductive reasoning}} * {{Annotated link|Inductive reasoning}} * {{Annotated link|Entailment}} * {{Annotated link|Epilogism}} * {{Annotated link|Analogy}} * {{Annotated link|Axiom system}} ** {{Annotated link|Axiom}} * {{Annotated link|Immediate inference}} * {{Annotated link|Inferential programming}} * {{Annotated link|Inquiry}} * {{Annotated link|Logic}} * {{Annotated link|Logic of information}} * {{Annotated link|Logical assertion}} * {{Annotated link|Logical graph}} * {{Annotated link|Rule of inference}} * {{Annotated link|List of rules of inference}} * {{Annotated link|Theorem}} * {{Annotated link|Transduction (machine learning)}} ==References== {{Reflist}} ==Further reading== {{refbegin}} * {{cite book |first=Ian |last=Hacking |title=An Introduction to Probability and Inductive Logic |publisher=Cambridge University Press |year=2001 |ref=Hacking, 2001 |isbn=978-0-521-77501-4 }} * {{cite book |first=Edwin Thompson |last=Jaynes |url=http://titles.cambridge.org/catalogue.asp?isbn=0521592712 |title=Probability Theory: The Logic of Science |publisher=Cambridge University Press |year=2003 |isbn=978-0-521-59271-0 |ref=Jaynes, 2003 |access-date=29 November 2004 |archive-url=https://web.archive.org/web/20041011085524/http://titles.cambridge.org/catalogue.asp?isbn=0521592712 |archive-date=11 October 2004 |url-status=dead }} * {{cite book |first=David J.C. |last=McKay |author-link=David J. C. MacKay |title=Information Theory, Inference, and Learning Algorithms |publisher=Cambridge University Press |year=2003 |isbn=978-0-521-64298-9 |url=http://www.inference.phy.cam.ac.uk/mackay/itila/book.html |ref=McKay, 2003 }} * {{Russell Norvig 2003}} * {{cite book |last=Tijms |first=Henk |author-link=Henk Tijms |title=Understanding Probability |url=https://archive.org/details/understandingpro0000tijm |url-access=registration |publisher=Cambridge University Press |year=2004 |ref=Tijms, 2004 |isbn=978-0-521-70172-3 }} {{refend}} '''Inductive inference:''' * {{cite book| title=Studies in Inductive Logic and Probability| year=1971| volume=1| publisher=The University of California Press| editor1-first=Rudolf |editor1-last=Carnap |editor2-first=Richard C. |editor2-last=Jeffrey}} * {{cite book| title=Studies in Inductive Logic and Probability| year=1980| volume=2| publisher=The University of California Press| editor-first=Richard C. |editor-last=Jeffrey|url=https://books.google.com/books?id=Qfe0SEazn3oC| isbn=9780520038264}} * {{cite thesis| type=Ph.D.| first=Dana |last=Angluin| title=An Application of the Theory of Computational Complexity to the Study of Inductive Inference| year=1976| publisher=University of California at Berkeley}} * {{cite journal| first=Dana| last=Angluin| title=Inductive Inference of Formal Languages from Positive Data| journal=Information and Control| year=1980| volume=45| issue=2| pages=117–135| doi=10.1016/s0019-9958(80)90285-5| doi-access=free}} * {{cite journal| first1=Dana |last1=Angluin|first2= Carl H. |last2=Smith| title=Inductive Inference: Theory and Methods| journal=Computing Surveys|date=Sep 1983| volume=15| number=3| pages=237–269| url=http://users.dsic.upv.es/asignaturas/facultad/apr/AngluinSmith83.pdf| doi=10.1145/356914.356918|s2cid=3209224}} * {{cite book| title=Inductive Logic| year=2009| volume=10| publisher=Elsevier| editor1-first=Dov M. |editor1-last=Gabbay |editor2-first=Stephan |editor2-last=Hartmann |editor3-first=John |editor3-last=Woods| series=Handbook of the History of Logic|isbn=978-0-444-52936-7}} * {{cite book| first=Nelson |last=Goodman| title=Fact, Fiction, and Forecast| year=1983| publisher=Harvard University Press|url=https://books.google.com/books?id=i97_LdPXwrAC|isbn=9780674290716}} '''Abductive inference:''' * {{cite book| title=Automated abduction: Inference to the best explanation| year=1997| publisher=AAAI Press| editor1-first=P. |editor1-last=O'Rourke |editor2-first=J. |editor2-last=Josephson}} * {{cite book| first=Stathis |last=Psillos| title=An Explorer upon Untrodden Ground: Peirce on Abduction|chapter=An Explorer upon Untrodden Ground | year=2009| volume=10| pages=117–152| publisher=Elsevier| editor1-first=Dov M. |editor1-last=Gabbay |editor2-first=Stephan |editor2-last=Hartmann |editor3-first=John |editor3-last=Woods| series=Handbook of the History of Logic| url=http://users.uoa.gr/~psillos/PapersI/11-Peirce-Abduction.pdf|doi=10.1016/B978-0-444-52936-7.50004-5|isbn=978-0-444-52936-7 }} * {{cite thesis| type=Ph.D.| first=Oliver |last=Ray| title=Hybrid Abductive Inductive Learning|date=Dec 2005| publisher=University of London, Imperial College| citeseerx = 10.1.1.66.1877}} '''Psychological investigations about human reasoning:''' * '''deductive:''' **{{cite book| first1 = Philip Nicholas |last1 = Johnson-Laird| author-link = Philip Johnson-Laird | first2 = Ruth M. J. | last2 = Byrne| title=Deduction| year=1992| publisher=Erlbaum}} **{{cite journal| first1=Ruth M. J.| last1=Byrne| first2=P. N.| last2=Johnson-Laird| authorlink2=Philip Johnson-Laird| title="If" and the Problems of Conditional Reasoning| journal=Trends in Cognitive Sciences| year=2009| volume=13| number=7| pages=282–287| url=https://psych.princeton.edu/psychology/research/johnson_laird/pdfs/2009ifandtheproblemof.pdf| doi=10.1016/j.tics.2009.04.003| pmid=19540792| s2cid=657803| access-date=9 August 2013| archive-url=https://web.archive.org/web/20140407061602/https://psych.princeton.edu/psychology/research/johnson_laird/pdfs/2009ifandtheproblemof.pdf| archive-date=7 April 2014| url-status=dead}} **{{cite journal | first1 = Markus | last1 = Knauff | first2 = Thomas | last2 = Fangmeier | first3 = Christian C. | last3 = Ruff | first4 = P. N. | last4 = Johnson-Laird | authorlink4 = Philip Johnson-Laird | title = Reasoning, Models, and Images: Behavioral Measures and Cortical Activity | journal = Journal of Cognitive Neuroscience | year = 2003 | volume = 15 | number = 4 | pages = 559–573 | url = http://www.uni-giessen.de/cms/fbz/fb06/psychologie/abt/kognition/dateien/kfrjl_JOCN.pdf | doi = 10.1162/089892903321662949 | pmid = 12803967 | hdl = 11858/00-001M-0000-0013-DC8B-C | citeseerx = 10.1.1.318.6615 | s2cid = 782228 | access-date = 9 August 2013 | archive-url = https://web.archive.org/web/20150518094338/http://www.uni-giessen.de/cms/fbz/fb06/psychologie/abt/kognition/dateien/kfrjl_JOCN.pdf | archive-date = 18 May 2015 | url-status = dead }} **{{cite book| first = Philip N. | last = Johnson-Laird| author-link = Philip Johnson-Laird | title=Mental Models, Deductive Reasoning, and the Brain| year=1995| pages=999–1008| publisher=MIT Press| editor-first = M. S. | editor-last = Gazzaniga| url=http://nbu.bg/cogs/events/2002/materials/Markus/mental_models.pdf}} **{{cite book| first1 = Sangeet | last1 = Khemlani | first2 = P. N. | last2 = Johnson-Laird| authorlink2 = Philip Johnson-Laird | chapter=Illusory Inferences about Embedded Disjunctions| title=Proceedings of the 30th Annual Conference of the Cognitive Science Society. Washington/DC| year=2008| pages=2128–2133| chapter-url=http://mentalmodels.princeton.edu/papers/2008disjillusions.pdf}} * '''statistical:''' **{{cite journal | first1 = Rachel | last1 = McCloy | first2 = Ruth M. J. | last2 = Byrne | first3 = Philip N. | last3 = Johnson-Laird | authorlink3 = Philip Johnson-Laird | title = Understanding Cumulative Risk | journal = The Quarterly Journal of Experimental Psychology | year = 2009 | pages = 499–515 | url = http://psych.princeton.edu/psychology/research/johnson_laird/pdfs/2009%20Understanding%20cumulative%20risk.pdf | doi = 10.1080/17470210903024784 | pmid = 19591080 | volume = 63 | issue = 3 | s2cid = 7741180 | access-date = 9 August 2013 | archive-url = https://web.archive.org/web/20150518073242/http://psych.princeton.edu/psychology/research/johnson_laird/pdfs/2009%20Understanding%20cumulative%20risk.pdf | archive-date = 18 May 2015 | url-status = dead }} **{{cite journal| first = Philip N. | last = Johnson-Laird | author-link = Philip Johnson-Laird | title=Mental Models and Probabilistic Thinking| journal=Cognition| year=1994| volume=50| issue = 1–3 | pages=189–209| url=http://mentalmodels.princeton.edu/papers/1994probabilistic.pdf| doi=10.1016/0010-0277(94)90028-0| pmid = 8039361 | s2cid = 9439284 }}, * '''analogical:''' **{{cite journal| first = B. D. | last = Burns| title=Meta-Analogical Transfer: Transfer Between Episodes of Analogical Reasoning| journal=Journal of Experimental Psychology: Learning, Memory, and Cognition| year=1996| volume=22| number=4| pages=1032–1048| doi=10.1037/0278-7393.22.4.1032}} * '''spatial:''' **{{cite journal| first1 = Georg | last1 = Jahn | first2 = Markus | last2 = Knauff | first3 = P. N. | last3 = Johnson-Laird| authorlink3 = Philip Johnson-Laird | title=Preferred mental models in reasoning about spatial relations| journal=Memory & Cognition| year=2007| volume=35| number=8| pages=2075–2087| url=http://mentalmodels.princeton.edu/papers/2007preferredmodels.pdf| doi=10.3758/bf03192939| pmid = 18265622 | s2cid = 25356700 | doi-access = free}} **{{cite journal| first1 = Markus | last1 = Knauff | first2 = P. N. | last2 = Johnson-Laird | authorlink2 = Philip Johnson-Laird | title=Visual imagery can impede reasoning| journal=Memory & Cognition| year=2002| volume=30| number=3| pages=363–371| url=http://mentalmodels.princeton.edu/papers/2002imagery.pdf| doi=10.3758/bf03194937| pmid = 12061757 | s2cid = 7330724 | doi-access=free}} **{{cite journal| first1 = James A. | last1 = Waltz | first2 = Barbara J. | last2 = Knowlton | first3 = Keith J. | last3 = Holyoak | first4 = Kyle B. | last4 = Boone | first5 = Fred S. | last5 = Mishkin | first6 = Marcia | last6 = de Menezes Santos | first7 = Carmen R. | last7 = Thomas | first8 = Bruce L. | last8 = Miller| title=A System for Relational Reasoning in Human Prefrontal Cortex| journal=Psychological Science|date=Mar 1999| volume=10| number=2| pages=119–125| url=https://www.researchgate.net/publication/228906574| doi=10.1111/1467-9280.00118| s2cid = 44019775 }} * '''moral:''' **{{cite journal| first1 = Monica | last1 = Bucciarelli | first2 = Sangeet | last2 = Khemlani | first3 = P. N. | last3 = Johnson-Laird | authorlink3 = Philip Johnson-Laird | title=The Psychology of Moral Reasoning| journal=Judgment and Decision Making|date=Feb 2008| volume=3| number=2| pages=121–139| doi = 10.1017/S1930297500001479 | s2cid = 327124 | url=http://journal.sjdm.org/jdm8105.pdf}} == External links == {{Wiktionary|inference|infer}} * {{PhilPapers|category|inference}} * [http://philosophyterms.com/inference/ Inference example and definition] * {{InPho|taxonomy|2397}} {{Logic}} {{Philosophy topics}} {{Authority control}} [[Category:Inference| ]] [[Category:Concepts in epistemology]] [[Category:Concepts in logic]] [[Category:Concepts in metaphilosophy]] [[Category:Concepts in metaphysics]] [[Category:Concepts in the philosophy of mind]] [[Category:History of logic]] [[Category:Intellectual history]] [[Category:Logic]] [[Category:Logic and statistics]] [[Category:Logical consequence]] [[Category:Metaphysics of mind]] [[Category:Reasoning]] [[Category:Semantics]] [[Category:Sources of knowledge]] [[Category:Thought]]