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**Inferences** are steps in reasoning, moving from premises to logical consequences; etymologically, the word * infer * means to "carry forward". Inference is theoretically traditionally divided into deduction and induction, a distinction that in Europe dates at least to Aristotle (300s BCE). Deduction is inference deriving logical conclusions from premises known or assumed to be true, with the laws of valid inference being studied in logic. Induction is inference from particular premises to a universal conclusion. A third type of inference is sometimes distinguished, notably by Charles Sanders Peirce, contradistinguishing abduction from induction.

- Definition
- Examples
- Example for definition #1
- Example for definition #2
- Incorrect inference
- Applications
- Inference engines
- Semantic web
- Bayesian statistics and probability logic
- Fuzzy logic
- Non-monotonic logic
- See also
- References
- Further reading
- External links

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 which may be subject to random variations.

The process by which a 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." ^{[ clarification needed ]}) 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.

Ancient Greek philosophers defined a number of syllogisms, 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 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.

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 (weather, 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 (i.e. the soccer scores and great soccer team) 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.

An incorrect inference is known as a fallacy. Philosophers who study informal logic have compiled large lists of them, and cognitive psychologists have documented many biases in human reasoning that favor incorrect reasoning.

AI systems first provided automated logical inference and these were once extremely popular research topics, leading to industrial applications under the form of expert systems and later business rule engines. 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 relevant to its task.

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

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.

Recently automatic reasoners found in semantic web a new field of application. Being based upon description logic, knowledge expressed using one variant of OWL can be logically processed, i.e., inferences can be made upon it.

Philosophers and scientists who follow the 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.

^{ [1] }

A relation of inference is monotonic if the addition of premises does not undermine previously reached conclusions; otherwise the relation is non-monotonic. Deductive inference is monotonic: if a conclusion is reached on the basis of a certain set of premises, then that conclusion still holds if more premises are added.

By contrast, everyday reasoning is mostly non-monotonic because it involves risk: we jump to conclusions from deductively insufficient premises. We know when it is worth or even necessary (e.g. in medical diagnosis) to take the risk. Yet we are also aware that such inference is defeasible—that new information may undermine old conclusions. Various kinds of defeasible but remarkably successful inference have traditionally captured the attention of philosophers (theories of induction, Peirce's theory of abduction, inference to the best explanation, etc.). More recently logicians have begun to approach the phenomenon from a formal point of view. The result is a large body of theories at the interface of philosophy, logic and artificial intelligence.

*A priori and a posteriori*- Abductive reasoning
- Deductive reasoning
- Inductive reasoning
- Entailment
- Epilogism
- Analogy
- Axiom system
- Immediate inference
- Inferential programming
- Inquiry
- Logic
- Logic of information
- Logical assertion
- Logical graph
- Rule of inference
- List of rules of inference
- Theorem
- Transduction (machine learning)

In logic, more precisely in deductive reasoning, an argument is **sound** if it is both valid in form and its premises are true. Soundness also has a related meaning in mathematical logic, wherein logical systems are sound if and only if every formula that can be proved in the system is logically valid with respect to the semantics of the system.

A **syllogism** is a kind of logical argument that applies deductive reasoning to arrive at a conclusion based on two or more propositions that are asserted or assumed to be true.

A **fallacy** is the use of invalid or otherwise faulty reasoning, or "wrong moves" in the construction of an argument. A fallacious argument may be deceptive by appearing to be better than it really is. Some fallacies are committed intentionally to manipulate or persuade by deception, while others are committed unintentionally due to carelessness or ignorance. The soundness of legal arguments depends on the context in which the arguments are made.

**Abductive reasoning** is a form of logical inference formulated and advanced by American philosopher Charles Sanders Peirce beginning in the last third of the 19th century. It starts with an observation or set of observations and then seeks to find the simplest and most likely conclusion from the observations. This process, unlike deductive reasoning, yields a plausible conclusion but does not positively verify it. Abductive conclusions are thus qualified as having a remnant of uncertainty or doubt, which is expressed in retreat terms such as "best available" or "most likely". One can understand abductive reasoning as **inference to the best explanation**, although not all usages of the terms *abduction* and *inference to the best explanation* are exactly equivalent.

**Deductive reasoning**, also **deductive logic**, is the process of reasoning from one or more statements (premises) to reach a logical conclusion.

The **problem of induction** is the philosophical question of what are the justifications, if any, for any growth of knowledge understood in the classic philosophical sense—knowledge that goes beyond a mere collection of observations—highlighting the apparent lack of justification in particular for:

- Generalizing about the properties of a class of objects based on some number of observations of particular instances of that class or
- Presupposing that a sequence of events in the future will occur as it always has in the past. Hume called this the principle of uniformity of nature.

**Inductive reasoning** is a method of reasoning in which the premises are viewed as supplying *some* evidence, but not full assurance, of the truth of the conclusion. It is also described as a method where one's experiences and observations, including what are learned from others, are synthesized to come up with a general truth. Many dictionaries define inductive reasoning as the derivation of general principles from specific observations, although there are many inductive arguments that do not have that form.

Two kinds of **logical reasoning** are often distinguished in addition to formal deduction: induction and abduction. Given a precondition or *premise*, a conclusion or *logical consequence* and a rule or *material conditional* that implies the *conclusion* given the *precondition*, one can explain the following.

- Deductive reasoning determines whether the truth of a
*conclusion*can be determined for that*rule*, based solely on the truth of the premises. Example: "When it rains, things outside get wet. The grass is outside, therefore: when it rains, the grass gets wet." Mathematical logic and philosophical logic are commonly associated with this type of reasoning. - Inductive reasoning attempts to support a determination of the
*rule*. It hypothesizes a*rule*after numerous examples are taken to be a*conclusion*that follows from a*precondition*in terms of such a*rule*. Example: "The grass got wet numerous times when it rained, therefore: the grass always gets wet when it rains." This type of reasoning is commonly associated with generalization from empirical evidence. While they may be persuasive, these arguments are not deductively valid: see the problem of induction. - Abductive reasoning, sometimes called
*inference to the best explanation*, selects a cogent set of*preconditions*. Given a true*conclusion*and a*rule*, it attempts to select some possible*premises*that, if true also, can support the*conclusion*, though not uniquely. Example: "When it rains, the grass gets wet. The grass is wet. Therefore, it might have rained." This kind of reasoning can be used to develop a hypothesis, which in turn can be tested by additional reasoning or data. Diagnosticians, detectives, and scientists often use this type of reasoning.

A **mental model** is an explanation of someone's thought process about how something works in the real world. It is a representation of the surrounding world, the relationships between its various parts and a person's intuitive perception about his or her own acts and their consequences. Mental models can help shape behaviour and set an approach to solving problems and doing tasks.

In philosophy, a **formal fallacy**, **deductive fallacy**, **logical fallacy** or **non sequitur** is a pattern of reasoning rendered invalid by a flaw in its logical structure that can neatly be expressed in a standard logic system, for example propositional logic. It is defined as a deductive argument that is invalid. The argument itself could have true premises, but still have a false conclusion. Thus, a formal fallacy is a fallacy where deduction goes wrong, and is no longer a logical process. This may not affect the truth of the conclusion, since validity and truth are separate in formal logic.

An **inquiry** is any process that has the aim of augmenting knowledge, resolving doubt, or solving a problem. A theory of inquiry is an account of the various types of inquiry and a treatment of the ways that each type of inquiry achieves its aim.

Appeal to the stone, also known as ‘*argumentum ad lapidem’*, is a logical fallacy that dismisses an argument as untrue or absurd and proves this assertion by stating that the argument is absurd. This theory is closely tied to proof by assertion due to the lack of evidence behind the statement and its attempt to persuade without providing any evidence.

In logic and philosophy, an **argument** is a series of statements, called the premises or premisses, intended to determine the degree of truth of another statement, the conclusion. The logical form of an argument in a natural language can be represented in a symbolic formal language, and independently of natural language formally defined "arguments" can be made in math and computer science.

**Analytical skill** is the ability to deconstruct information into smaller categories in order to draw conclusions. Analytical skill consists of categories that include logical reasoning, critical thinking, communication, research, data analysis and creativity. Analytical skill is taught in contemporary education with the intention of fostering the appropriate practises for future professions. The professions that adopt analytical skill include educational institutions, public institutions, community organisations and industry.

The **psychology of reasoning** is the study of how people reason, often broadly defined as the process of drawing conclusions to inform how people solve problems and make decisions. It overlaps with psychology, philosophy, linguistics, cognitive science, artificial intelligence, logic, and probability theory.

In logic, more precisely in deductive reasoning, an argument is **valid** if and only if it takes a form that makes it impossible for the premises to be true and the conclusion nevertheless to be false. It is not required for a valid argument to have premises that are actually true, but to have premises that, if they were true, would guarantee the truth of the argument's conclusion. Valid arguments must be clearly expressed by means of sentences called well-formed formulas. The **validity** of an argument—its being valid—can be tested, proved or disproved, and depends on its logical form.

**Plausible reasoning** is a method of deriving new conclusions from given known premises, a method different from the classical syllogistic argumentation methods of Aristotelian two-valued logic. The syllogistic style of argumentation is illustrated by the oft-quoted argument "All men are mortal, Socrates is a man, and therefore, Socrates is mortal." In contrast, consider the statement "if it is raining then it is cloudy." The only logical inference that one can draw from this is that "if it is not cloudy then it is not raining." But ordinary people in their everyday lives would conclude that "if it is not raining then being cloudy is less plausible," or "if it is cloudy then rain is more plausible." The unstated and unconsciously applied reasoning, arguably incorrect, that made people come to their conclusions is typical of plausible reasoning.

**Logic** is the systematic study of valid rules of inference, i.e. the relations that lead to the acceptance of one proposition on the basis of a set of other propositions (premises). More broadly, logic is the analysis and appraisal of arguments.

As the study of argument is of clear importance to the reasons that we hold things to be true, logic is of essential importance to rationality. Arguments may be logical if they are "conducted or assessed according to strict principles of validity", while they are rational according to the broader requirement that they are based on reason and knowledge.

In argumentation theory, an **argumentation scheme** or **argument scheme** is a template that represents a common type of argument used in ordinary conversation. Many different argumentation schemes have been identified. Each one has a name and presents a type of connection between *premises* and a *conclusion* in an argument, and this connection is expressed as a rule of inference. Argumentation schemes can include inferences based on different types of reasoning—deductive, inductive, abductive, probabilistic, etc.

- ↑ Fuhrmann, André.
*Nonmonotonic Logic*(PDF). Archived from the original (PDF) on 9 December 2003.

- Hacking, Ian (2001).
*An Introduction to Probability and Inductive Logic*. Cambridge University Press. ISBN 978-0-521-77501-4. - Jaynes, Edwin Thompson (2003).
*Probability Theory: The Logic of Science*. Cambridge University Press. ISBN 978-0-521-59271-0. Archived from the original on 2004-10-11. Retrieved 2004-11-29. - McKay, David J.C. (2003).
*Information Theory, Inference, and Learning Algorithms*. Cambridge University Press. ISBN 978-0-521-64298-9. - Russell, Stuart J.; Norvig, Peter (2003),
*Artificial Intelligence: A Modern Approach*(2nd ed.), Upper Saddle River, New Jersey: Prentice Hall, ISBN 0-13-790395-2 - Tijms, Henk (2004).
*Understanding Probability*. Cambridge University Press. ISBN 978-0-521-70172-3.

**Inductive inference:**

- Carnap, Rudolf; Jeffrey, Richard C., eds. (1971).
*Studies in Inductive Logic and Probability*.**1**. The University of California Press. - Jeffrey, Richard C., ed. (1980).
*Studies in Inductive Logic and Probability*.**2**. The University of California Press. ISBN 9780520038264. - Angluin, Dana (1976).
*An Application of the Theory of Computational Complexity to the Study of Inductive Inference*(Ph.D.). University of California at Berkeley. - Angluin, Dana (1980). "Inductive Inference of Formal Languages from Positive Data" (PDF).
*Information and Control*.**45**(2): 117–135. doi:10.1016/s0019-9958(80)90285-5. - Angluin, Dana; Smith, Carl H. (Sep 1983). "Inductive Inference: Theory and Methods" (PDF).
*Computing Surveys*.**15**(3): 237–269. doi:10.1145/356914.356918. S2CID 3209224. - Gabbay, Dov M.; Hartmann, Stephan; Woods, John, eds. (2009).
*Inductive Logic*. Handbook of the History of Logic.**10**. Elsevier. - Goodman, Nelson (1983).
*Fact, Fiction, and Forecast*. Harvard University Press. ISBN 9780674290716.

**Abductive inference:**

- O'Rourke, P.; Josephson, J., eds. (1997).
*Automated abduction: Inference to the best explanation*. AAAI Press. - Psillos, Stathis (2009). Gabbay, Dov M.; Hartmann, Stephan; Woods, John (eds.).
*An Explorer upon Untrodden Ground: Peirce on Abduction*(PDF). Handbook of the History of Logic.**10**. Elsevier. pp. 117–152. - Ray, Oliver (Dec 2005).
*Hybrid Abductive Inductive Learning*(Ph.D.). University of London, Imperial College. CiteSeerX 10.1.1.66.1877 .

**Psychological investigations about human reasoning:**

**deductive:**- Johnson-Laird, Philip Nicholas; Byrne, Ruth M. J. (1992).
*Deduction*. Erlbaum. - Byrne, Ruth M. J.; Johnson-Laird, P. N. (2009). ""If" and the Problems of Conditional Reasoning" (PDF).
*Trends in Cognitive Sciences*.**13**(7): 282–287. doi:10.1016/j.tics.2009.04.003. PMID 19540792. S2CID 657803. Archived from the original (PDF) on 2014-04-07. Retrieved 2013-08-09. - Knauff, Markus; Fangmeier, Thomas; Ruff, Christian C.; Johnson-Laird, P. N. (2003). "Reasoning, Models, and Images: Behavioral Measures and Cortical Activity" (PDF).
*Journal of Cognitive Neuroscience*.**15**(4): 559–573. CiteSeerX 10.1.1.318.6615 . doi:10.1162/089892903321662949. PMID 12803967. S2CID 782228. Archived from the original (PDF) on 2015-05-18. Retrieved 2013-08-09. - Johnson-Laird, Philip N. (1995). Gazzaniga, M. S. (ed.).
*Mental Models, Deductive Reasoning, and the Brain*(PDF). MIT Press. pp. 999–1008. - Khemlani, Sangeet; Johnson-Laird, P. N. (2008). "Illusory Inferences about Embedded Disjunctions" (PDF).
*Proceedings of the 30th Annual Conference of the Cognitive Science Society. Washington/DC*. pp. 2128–2133.

- Johnson-Laird, Philip Nicholas; Byrne, Ruth M. J. (1992).
**statistical:**- McCloy, Rachel; Byrne, Ruth M. J.; Johnson-Laird, Philip N. (2009). "Understanding Cumulative Risk" (PDF).
*The Quarterly Journal of Experimental Psychology*.**63**(3): 499–515. doi:10.1080/17470210903024784. PMID 19591080. S2CID 7741180. Archived from the original (PDF) on 2015-05-18. Retrieved 2013-08-09. - Johnson-Laird, Philip N. (1994). "Mental Models and Probabilistic Thinking" (PDF).
*Cognition*.**50**(1–3): 189–209. doi:10.1016/0010-0277(94)90028-0. PMID 8039361. S2CID 9439284.,

- McCloy, Rachel; Byrne, Ruth M. J.; Johnson-Laird, Philip N. (2009). "Understanding Cumulative Risk" (PDF).
**analogical:**- Burns, B. D. (1996). "Meta-Analogical Transfer: Transfer Between Episodes of Analogical Reasoning".
*Journal of Experimental Psychology: Learning, Memory, and Cognition*.**22**(4): 1032–1048. doi:10.1037/0278-7393.22.4.1032.

- Burns, B. D. (1996). "Meta-Analogical Transfer: Transfer Between Episodes of Analogical Reasoning".
**spatial:**- Jahn, Georg; Knauff, Markus; Johnson-Laird, P. N. (2007). "Preferred mental models in reasoning about spatial relations" (PDF).
*Memory & Cognition*.**35**(8): 2075–2087. doi:10.3758/bf03192939. PMID 18265622. S2CID 25356700. - Knauff, Markus; Johnson-Laird, P. N. (2002). "Visual imagery can impede reasoning" (PDF).
*Memory & Cognition*.**30**(3): 363–371. doi:10.3758/bf03194937. PMID 12061757. S2CID 7330724. - Waltz, James A.; Knowlton, Barbara J.; Holyoak, Keith J.; Boone, Kyle B.; Mishkin, Fred S.; de Menezes Santos, Marcia; Thomas, Carmen R.; Miller, Bruce L. (Mar 1999). "A System for Relational Reasoning in Human Prefrontal Cortex".
*Psychological Science*.**10**(2): 119–125. doi:10.1111/1467-9280.00118. S2CID 44019775.

- Jahn, Georg; Knauff, Markus; Johnson-Laird, P. N. (2007). "Preferred mental models in reasoning about spatial relations" (PDF).
**moral:**- Bucciarelli, Monica; Khemlani, Sangeet; Johnson-Laird, P. N. (Feb 2008). "The Psychology of Moral Reasoning" (PDF).
*Judgment and Decision Making*.**3**(2): 121–139.

- Bucciarelli, Monica; Khemlani, Sangeet; Johnson-Laird, P. N. (Feb 2008). "The Psychology of Moral Reasoning" (PDF).

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