Inferences are steps in reasoning, moving from premises to
Charles Sanders Peirce
1 Definition 2 Examples
2.1 Example for definition #1 2.2 Example for definition #2
3 Incorrect inference 4 Applications
4.1 Inference engines
4.1.1 Prolog engine
4.2 Semantic web 4.3 Bayesian statistics and probability logic 4.4 Fuzzy logic 4.5 Non-monotonic logic
5 See also 6 References 7 Further reading 8 External links
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.
A conclusion reached on the basis of evidence and reasoning. The process of reaching such a conclusion.
Examples Example for definition #1 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.
Example for definition #2
Evidence: It is the early 1950s and you are an American stationed in
the Soviet Union. You read in the
mortal(X) :- man(X). man(socrates).
( Here :- can be read as "if". Generally, if P
(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:
gives the answer "No".
This is because
Prolog does not know anything about Plato, and hence
defaults to any property about
This section needs expansion. You can help by adding to it. (October 2016)
Non-monotonic logic Main article: Non-monotonic logic  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. See also
Abductive reasoning Deductive reasoning Inductive reasoning
^ Fuhrmann, André. Nonmonotonic
Hacking, Ian (2011). An Introduction to
Carnap, Rudolf; Jeffrey, Richard C., eds. (1971). Studies in Inductive
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:
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 Science. 13 (7): 282–287. doi:10.1016/j.tics.2009.04.003. 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. doi:10.1162/089892903321662949. PMID 12803967. 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". Proceedings of the 30th Annual Conference of the Cognitive Science Society. Washington/DC (PDF). pp. 2128–2133.
McCloy, Rachel; Byrne, Ruth M. J.; Johnson-Laird, Philip N. (2009). "Understanding Cumulative Risk" (PDF). The Quarterly Journal of Experimental Psychology. 63: 18. doi:10.1080/17470210903024784. Johnson-Laird, Philip N. (1994). "Mental Models and Probabilistic Thinking" (PDF). Cognition. 50: 189–209. doi:10.1016/0010-0277(94)90028-0. ,
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:10.1037/0278-73220.127.116.112.
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.
Knauff, Markus; Johnson-Laird, P. N. (2002). "Visual imagery can
impede reasoning" (PDF). Memory & Cognition. 30 (3): 363–371.
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
Bucciarelli, Monica; Khemlani, Sangeet; Johnson-Laird, P. N. (Feb 2008). "The Psychology of Moral Reasoning" (PDF). Judgment and Decision Making. 3 (2): 121–139.
Look up inference or infer in Wiktionary, the free dictionary.
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