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Inference is the act or process of deriving a conclusion based solely on what one already knows. A conclusion can have various specific meanings depending on the context. ...
Inference is studied within several different fields.
- Human inference (i.e. how humans draw conclusions) is traditionally studied within the field of cognitive psychology.
- Logic studies the laws of valid inference.
- Statisticians have developed formal rules for inference from quantitative data.
- Artificial intelligence researchers develop automated inference systems.
Cognitive Psychology is the school of psychology that examines internal mental processes such as problem solving, memory, and language. ...
Logic (from Classical Greek λόγος logos; meaning word, thought, idea, argument, account, reason, or principle) is the study of the principles and criteria of valid inference and demonstration. ...
This article is about the field of statistics. ...
Garry Kasparov playing against Deep Blue, the first machine to win a chess game against a reigning world champion. ...
The accuracy of inductive and deductive inferences
The conclusion inferred from multiple observations is made by the process of inductive reasoning. The conclusion may be correct or incorrect, and may be tested by additional observations. In contrast, the conclusion of a valid deductive inference is true if the premises are true. The conclusion is inferred using the process of deductive reasoning. A valid deductive inference is never false. This is because the validity of a deductive inference is formal. The inferred conclusion of a valid deductive inference is necessarily true if the premises it is based on are true. Aristotle appears first to establish the mental behaviour of induction as a category of reasoning. ...
In theoretical computer science, correctness of an algorithm is asserted when it is said that the algorithm is correct with respect to a specification. ...
In psychology a conclusion is said to be valid, if and only if, it is based on true premises. ...
Deductive reasoning is the process of reaching a conclusion that is guaranteed to follow, if the evidence provided is true and the reasoning used to reach the conclusion is correct. ...
Deductive reasoning is the kind of reasoning where the conclusion is necessitated by previously known premises. ...
The field of half-truths as they relate to the truth of observations, is another area of concern impacting inference based on observations. Half-truths are deceptive statements, that include some element of truth. ...
Valid inferences Inferences are either valid or invalid, but not both. Philosophical logic has attempted to define the rules of proper inference, i.e. the formal rules that, when correctly applied to true premises, lead to true conclusions. Aristotle has given one of the most famous statements of those rules in his Organon. Modern mathematical logic, beginning in the 19th century, has built numerous formal systems that embody Aristotelian logic (or variants thereof). Philosophical logic is the application of formal logical techniques to problems that concern philosophers. ...
Aristotle (Greek: Aristotélēs) (384 BC – 322 BC) was a Greek philosopher, a student of Plato and teacher of Alexander the Great. ...
This article is about Aristotles logical works. ...
Mathematical logic is a major area of mathematics, which grew out of symbolic logic. ...
In logic and mathematics, a formal system consists of two components, a formal language plus a set of inference rules or transformation rules. ...
Aristotelian logic, also known as syllogistic logic, is the particular type of logic created by Aristotle, primarily in his works Prior Analytics and De Interpretatione. ...
An example: the classic syllogism Greek philosophers defined a number of syllogisms, correct three-part inferences, that can be used as building blocks for more complex reasoning. We'll begin with the most famous of them all: Greek philosophy focused on the role of reason and inquiry. ...
A syllogism (Greek: — conclusion, inference), usually the categorical syllogism, is a kind of logical argument in which one proposition (the conclusion) is inferred from two others (the premises) of a certain form. ...
All men are mortal Socrates is a man ------------------ Therefore Socrates is mortal. The reader can check that the premises and conclusion are true. The validity of the inference may not be true. The validity of the inference depends on the form of the inference. That is, a valid inference does not depend on the truth of the premises and conclusion, but on the formal rules of inference being used. In traditional logic, the form of the syllogism is: Traditional logic, also known as term logic, is a loose term for the logical tradition that originated with Aristotle and survived broadly unchanged until the advent of modern predicate logic in the late nineteenth century. ...
All A is B All C is A ---------- All C is B Since the syllogism fits this form, then the inference is valid. And if the premises are true, then the conclusion is necessarily true.
In predicate logic (a simple but useful formalization of Aristotelician logic), this syllogism can be stated as follows: Aristotle (Greek: Aristotélēs) (384 BC – 322 BC) was a Greek philosopher, a student of Plato and teacher of Alexander the Great. ...
∀ X, man(X) → mortal(X) man(Socrates) ------------------------------- ∴mortal(Socrates) Or in its general form: ∀ X, A(X) → B(X) A(x) ------------------------ ∴B(x) ∀, the universal quantifier, is pronounced "for all". It allows us to state a general property. Here it is used to say that "if any X is a man, X is also mortal". In predicate logic, universal quantification is an attempt to formalise the notion that something (a logical predicate) is true for everything, or every relevant thing. ...
∴ is the therefore symbol which denotes the conclusion. In a mathematical proof, the therefore sign () is a symbol that is sometimes placed before a logical consequence, such as the conclusion of a syllogism. ...
Consider the following:
All fat people are musicians John Lennon was fat ------------------- Therefore John Lennon was a musician In this case we have two false premises that imply a true conclusion. The inference is valid because it follows the form of a correct 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. Look up fallacy in Wiktionary, the free dictionary. ...
Informal logic is the study of arguments as presented in ordinary language, as contrasted with the presentations of arguments in an artificial (technical) or formal language (see formal logic). ...
This article or section does not cite its references or sources. ...
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. This article does not cite any references or sources. ...
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. Relevance is a term used to describe how pertinent, connected, or applicable some information is to a given matter. ...
An example: inference using Prolog Prolog (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 the KB using an algorithm called backward chaining. Prolog is a logic programming language. ...
A is a subset of B, and B is a superset of A. In mathematics, especially in set theory, the terms, subset, superset and proper (or strict) subset or superset are used to describe the relation, called inclusion, of one set being contained inside another set. ...
First-order predicate calculus or first-order logic (FOL) permits the formulation of quantified statements such as there exists an x such that. ...
Backward chaining is one of the two main methods of reasoning when using inference rules. ...
Let us return to our Socrates syllogism. We enter into our Knowledge Base the following piece of code: This page is about the ancient Greek philosopher. ...
A syllogism (Greek: — conclusion, inference), usually the categorical syllogism, is a kind of logical argument in which one proposition (the conclusion) is inferred from two others (the premises) of a certain form. ...
mortal(X) :- man(X). man(socrates). This states that all men are mortal and that Socrates is a man. Now we can ask Prolog about Socrates. Prolog is a logic programming language. ...
?- mortal(socrates). Yes On the other hand : ?- mortal(plato). 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). Prolog can be used for vastly more complicated inference tasks. See the corresponding article for further examples. Prolog is a logic programming language. ...
PLATO was one of the first generalized Computer assisted instruction systems, originally built by the University of Illinois (U of I) and later taken over by Control Data Corporation (CDC), who provided the machines it ran on. ...
The closed world assumption is the presumption that what is not currently known to be true is false. ...
Prolog is a logic programming language. ...
Automatic inference and the semantic web Recently automatic reasoners found in semantic web a new field of application. As OWL is based upon first-order logic, knowledge expressed using it can be logically processed, i.e. inference can be made upon it. The semantic web is an evolving extension of the World Wide Web in which web content can be expressed not only in natural language, but also in a form that can be read and used by software agents, thus permitting them to find, share and integrate information more easily. ...
The Web Ontology Language (OWL) is a language for defining and instantiating Web ontologies. ...
First-order logic (FOL) is a universal language in symbolic science, and is in use everyday by mathematicians, philosophers, linguists, computer scientists and practitioners of artificial intelligence. ...
Inference and uncertainty Traditional logic is only concerned with certainty - one progresses from certain premises to certain conclusions. There are several motivations for extending logic to deal with uncertain propositions and weaker modes of reasoning. A related article is titled uncertainty. ...
In discourse, a premise (also premiss in British usage) is a claim which is part of a reason or objection. ...
- Philosophical motivations
- A large part of our everyday reasoning does not follow the strict rules of logic, but is nevertheless effective in many cases
- Science itself is not deductive, but largely inductive, and its process cannot be captured by standard logic (see problem of induction).
- Technical motivations
- Statisticians and scientists wish to be able to infer parameters or test hypothesis on statistical data in a rigorous, quantified way.
- Artificial intelligence systems need to reason efficiently about uncertain quantities.
The problem of induction is the philosophical issue involved in deciding the place of induction in determining empirical truth. ...
Common sense and uncertain reasoning The reason most examples of applying deductive logic, such as the one above, seem artificial is because they are rarely encountered outside fields such as mathematics. Most of our everyday reasoning is of a less "pure" nature. Euclid, Greek mathematician, 3rd century BC, as imagined by by Raphael in this detail from The School of Athens. ...
To take an example: suppose you live in a flat. Late at night, you are awoken by creaking sounds in the ceiling. You infer from these sounds that your neighbour upstairs is having another bout of insomnia and is pacing in his room, sleepless.
Although that reasoning seems sound, it does not fit in the logical framework described above. First, the reasoning is based on uncertain facts: what you heard were creaks, not necessarily footsteps. But even if those facts were certain, the inference is of an inductive nature: perhaps you have often heard your neighbour at night, and the best explanation you have found is that he or she is an insomniac. Hence tonight's footsteps.
It is easy to see that this line of reasoning does not necessarily lead to true conclusions: perhaps your neighbour had a very early plane to catch, which would explain the footsteps just as well. Uncertain reasoning can only find the best explanation among many alternatives.
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). Bayesian inference is statistical inference in which evidence or observations are used to update or to newly infer the probability that a hypothesis may be true. ...
Probability is the likelihood that something is the case or will happen. ...
Edwin Thompson Jaynes (July 5th, 1922 – April 30th, 1998) was Wayman Crow Distinguished Professor of Physics at Washington University in St. ...
Bayesianists 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, which gave its name to the field. Bayesian inference is statistical inference in which evidence or observations are used to update or to newly infer the probability that a hypothesis may be true. ...
Bayess theorem (also known as Bayess rule or Bayess law) is a result in probability theory, which relates the conditional and marginal probability distributions of random variables. ...
See Bayesian inference for examples. Bayesian inference is statistical inference in which evidence or observations are used to update or to newly infer the probability that a hypothesis may be true. ...
Nonmonotonic logic Source: Article of André Fuhrmann about "Nonmonotonic Logic"
A relation of inference is monotonic if the addition of premises does not undermine previously reached conclusions; otherwise the relation is nonmonotonic. Deductive inference, at least according to the canons of classical logic, is monotonic: if a conclusion is reached on the basis of a certain set of premisses, then that conclusion still holds if more premisses 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.
Three types of logical inference There are three types of inference: Deductive reasoning is the kind of reasoning where the conclusion is necessitated by previously known premises. ...
It has been suggested that Abductive validation be merged into this article or section. ...
Aristotle appears first to establish the mental behaviour of induction as a category of reasoning. ...
An example Hooke's law is the rule that gives the elongation of a beam (that's an effect) when a force (that's the cause) is acting on a beam. Hookes law accurately models the physical properties of common mechanical springs for small changes in length. ...
- If the force and Hooke's law are known, the elongation of the beam can be deduced.
- If the elongation and Hooke's law are known, the force acting on the beam can be abduced.
- If the elongation and the force are known, Hooke's law can be induced.
References - Ian Hacking. An Introduction to Probability and Inductive Logic. Cambridge University Press, (2000).
- Edwin Thompson Jaynes. Probability Theory: The Logic of Science. Cambridge University Press, (2003). ISBN 0-521-59271-2.
- David J.C. McKay. Information Theory, Inference, and Learning Algorithms. Cambridge University Press, (2003).
- Stuart Russell, Peter Norvig. Artificial Intelligence: A Modern Approach. Prentice Hall, (2002).
- Henk Tijms. Understanding Probability. Cambridge University Press, (2004).
- André Fuhrmann: Nonmonotonic Logic.
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