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Title: Inference  
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Subject: Logic, Logical consequence, Outline of logic, Mathematical logic, History of logic
Collection: Concepts in Logic, Logic and Statistics, Logical Consequence, Reasoning, Sources of Knowledge
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Inference is the act or process of deriving logical conclusions from premises known or assumed to be true.[1] The conclusion drawn is also called an idiomatic. The laws of valid inference are studied in the field of logic.

Alternatively, inference may be defined as the non-logical, but rational means, through observation of patterns of facts, to indirectly see new meanings and contexts for understanding. Of particular use to this application of inference are anomalies and symbols. Inference, in this sense, does not draw conclusions but opens new paths for inquiry. (See second set of Examples.) In this definition of inference, there are two types of inference: inductive inference and deductive inference. Unlike the definition of inference in the first paragraph above, meaning of word meanings are not tested but meaningful relationships are articulated.

Human inference (i.e. how humans draw conclusions) is traditionally studied within the field of 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 variation.


  • Examples 1
    • Example for definition #2 1.1
  • Incorrect inference 2
  • Automatic logical inference 3
    • Example using Prolog 3.1
    • Use with the semantic web 3.2
    • Bayesian statistics and probability logic 3.3
    • Nonmonotonic logic 3.4
  • See also 4
  • References 5
  • Further reading 6
  • External links 7


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:

  1. All men are mortal
  2. Socrates is a man
  3. Therefore, Socrates is 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:

  1. All meat comes from animals.
  2. Beef is a type of meat.
  3. Therefore, beef comes from an animal.

If the premises are true, then the conclusion is necessarily true, too.

Now we turn to an invalid form.

  1. All A are B.
  2. C is a B.
  3. Therefore, C is an A.

To show that this form is invalid, we demonstrate how it can lead from true premises to a false conclusion.

  1. All apples are fruit. (Correct)
  2. Bananas are fruit. (Correct)
  3. Therefore, bananas are apples. (Wrong)

A valid argument with false premises may lead to a false conclusion:

  1. All tall people are Greek.
  2. John Lennon was tall.
  3. Therefore, John Lennon was Greek. (wrong)

When a valid argument is used to derive a false conclusion from false premises, 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 false premises:

  1. All tall people are musicians (although wrong)
  2. John Lennon was tall (right, Valid)
  3. Therefore, John Lennon was a musician (Right)

In this case we have two false premises that imply a true conclusion.

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 (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.

Incorrect inference

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.

Automatic logical inference

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.

Example using Prolog

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).

( Here :- can be read as "if". Generally, if P → 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.

Use with the 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 OWL can be logically processed, i.e., inferences can be made upon it.

Bayesian statistics and probability logic

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.

See Bayesian inference for examples.

Nonmonotonic logic[2]

A relation of inference is monotonic if the addition of premises does not undermine previously reached conclusions; otherwise the relation is nonmonotonic. 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 nonmonotonic 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


  1. ^
  2. ^ Fuhrmann, André. Nonmonotonic Logic (PDF). Archived from the original (PDF) on 9 December 2003. 

Further reading

  • Hacking, Ian (2011). An Introduction to Probability and Inductive Logic. Cambridge University Press.  
  • Jaynes, Edwin Thompson (2003). Probability Theory: The Logic of Science. Cambridge University Press.  

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. (1979). Studies in Inductive Logic and Probability 2. The University of California Press. 
  • 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: 117–135.  
  • Angluin, Dana; Smith, Carl H. (Sep 1983). "Inductive Inference: Theory and Methods" (PDF). Computing Surveys 15 (3): 237–269.  
  • Gabbay, Dov M.; Hartmann, Stephan; Woods, John, eds. (2009). Inductive Logic. Handbook of the History of Logic 10. Elsevier. 
  • Goodman, Nelson (1973). Fact, Fiction, and Forecast. Bobbs-Merrill Co. Inc. 

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.  

Psychological investigations about human reasoning:

  • deductive:
    • Byrne, Ruth M. J.;  
    • Knauff, Markus; Fangmeier, Thomas; Ruff, Christian C.;  
    • Khemlani, Sangeet;  
  • statistical:
    • McCloy, Rachel; Byrne, Ruth M. J.;  
    •  ,
  • 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.  
  • spatial:
    • Jahn, Georg; Knauff, Markus;  
    • Knauff, Markus;  
    • 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" (PDF). Psychological Science 10 (2): 119–125.  
  • moral:
    • Bucciarelli, Monica; Khemlani, Sangeet;  

External links

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