Inferences are steps in

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

Inference example and definition

* {{Authority control Concepts in epistemology Concepts in logic Concepts in metaphilosophy Concepts in metaphysics Concepts in the philosophy of mind History of logic Intellectual history Logic Logic and statistics Logical consequence Metaphysics of mind Reasoning Sources of knowledge Thought

reason
Reason is the capacity of consciously applying logic by drawing conclusions from new or existing information, with the aim of seeking the truth. It is closely associated with such characteristically human activities as philosophy, science, l ...

ing, moving from premise
A premise or premiss is a true or false statement that helps form the body of an argument, which logically leads to a true or false conclusion. A premise makes a declarative statement about its subject matter which enables a reader to either agr ...

s to logical consequence
Logical consequence (also entailment) is a fundamental concept in logic, which describes the relationship between statements that hold true when one statement logically ''follows from'' one or more statements. A valid logical argument is on ...

s; etymologically, the word ''infer
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 ...

'' means to "carry forward". Inference is theoretically traditionally divided into deduction and induction, a distinction that in Europe dates at least to Aristotle
Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Greek philosopher and polymath during the Classical period in Ancient Greece. Taught by Plato, he was the founder of the Peripatetic school of ...

(300s BCE). Deduction is inference deriving logical conclusions from premises known or assumed to be true
True most commonly refers to truth, the state of being in congruence with fact or reality.
True may also refer to:
Places
* True, West Virginia, an unincorporated community in the United States
* True, Wisconsin, a town in the United States
* ...

, with the laws of valid inference being studied in logic
Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from prem ...

. Induction is inference from particular
In metaphysics, particulars or individuals are usually contrasted with universals. Universals concern features that can be exemplified by various different particulars. Particulars are often seen as concrete, spatiotemporal entities as opposed to ...

evidence to a universal conclusion. A third type of inference is sometimes distinguished, notably by Charles Sanders Peirce
Charles Sanders Peirce ( ; September 10, 1839 – April 19, 1914) was an American philosopher, logician, mathematician and scientist who is sometimes known as "the father of pragmatism".
Educated as a chemist and employed as a scientist for ...

, contradistinguishing abduction from induction.
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
Cognitive psychology is the scientific study of mental processes such as attention, language use, memory, perception, problem solving, creativity, and reasoning.
Cognitive psychology originated in the 1960s in a break from behaviorism, which h ...

; artificial intelligence
Artificial intelligence (AI) is intelligence—perceiving, synthesizing, and inferring information—demonstrated by machines, as opposed to intelligence displayed by animals and humans. Example tasks in which this is done include speec ...

researchers develop automated inference systems to emulate human inference. Statistical inference
Statistical inference is the process of using data analysis to infer properties of an underlying distribution of probability.Upton, G., Cook, I. (2008) ''Oxford Dictionary of Statistics'', OUP. . Inferential statistical analysis infers properti ...

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.
Definition

The process by which a conclusion is inferred from multipleobservations
Observation is the active acquisition of information from a primary source. In living beings, observation employs the senses. In science, observation can also involve the perception and recording of data via the use of scientific instruments. The ...

is called inductive reasoning
Inductive reasoning is a method of reasoning in which a general principle is derived from a body of observations. It consists of making broad generalizations based on specific observations. Inductive reasoning is distinct from ''deductive'' r ...

. 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." ) 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 philosophers defined a number ofsyllogism
A syllogism ( grc-gre, συλλογισμός, ''syllogismos'', 'conclusion, inference') is a kind of logical argument that applies deductive reasoning to arrive at a conclusion based on two propositions that are asserted or assumed to be true ...

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 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 theSoviet Union
The Soviet Union,. officially the Union of Soviet Socialist Republics. (USSR),. was a transcontinental country that spanned much of Eurasia from 1922 to 1991. A flagship communist state, it was nominally a federal union of fifteen nati ...

. You read in the Moscow
Moscow ( , US chiefly ; rus, links=no, Москва, r=Moskva, p=mɐskˈva, a=Москва.ogg) is the capital and largest city of Russia. The city stands on the Moskva River in Central Russia, with a population estimated at 13.0 million ...

newspaper that a soccer
Association football, more commonly known as football or soccer, is a team sport played between two teams of 11 players who primarily use their feet to propel the ball around a rectangular field called a pitch. The objective of the game is ...

team from a small city in Siberia
Siberia ( ; rus, Сибирь, r=Sibir', p=sʲɪˈbʲirʲ, a=Ru-Сибирь.ogg) is an extensive geographical region, constituting all of North Asia, from the Ural Mountains in the west to the Pacific Ocean in the east. It has been a part ...

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
A planned economy is a type of economic system where investment, production and the allocation of capital goods takes place according to economy-wide economic plans and production plans. A planned economy may use centralized, decentralized, p ...

: 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
A planned economy is a type of economic system where investment, production and the allocation of capital goods takes place according to economy-wide economic plans and production plans. A planned economy may use centralized, decentralized, p ...

, 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 afallacy
A fallacy is the use of invalid or otherwise faulty reasoning, or "wrong moves," in the construction of an argument which may appear stronger than it really is if the fallacy is not spotted. The term in the Western intellectual tradition was i ...

. Philosophers who study informal logic
Informal logic encompasses the principles of logic and logical thought outside of a formal setting (characterized by the usage of particular statements). However, the precise definition of "informal logic" is a matter of some dispute. Ralph H. ...

have compiled large lists of them, and cognitive psychologists have documented many biases in human reasoning that favor incorrect reasoning.
Applications

Inference engines

AI systems first provided automated logical inference and these were once extremely popular research topics, leading to industrial applications under the form ofexpert system
In artificial intelligence, an expert system is a computer system emulating the decision-making ability of a human expert.
Expert systems are designed to solve complex problems by reasoning through bodies of knowledge, represented mainly as if ...

s and later business rule engines. More recent work on automated theorem proving
Automated theorem proving (also known as ATP or automated deduction) is a subfield of automated reasoning and mathematical logic dealing with proving mathematical theorems by computer programs. Automated reasoning over mathematical proof was a ma ...

has had a stronger
basis in formal logic.
An inference system's job is to extend a knowledge base automatically. The knowledge base
A knowledge base (KB) is a technology used to store complex structured and unstructured information used by a computer system. The initial use of the term was in connection with expert systems, which were the first knowledge-based systems. ...

(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
Relevant is something directly related, connected or pertinent to a topic; it may also mean something that is current.
Relevant may also refer to:
* Relevant operator, a concept in physics, see renormalization group
* Relevant, Ain, a commune ...

to its task.
Prolog engine

Prolog
Prolog is a logic programming language associated with artificial intelligence and computational linguistics.
Prolog has its roots in first-order logic, a formal logic, and unlike many other programming languages, Prolog is intended primaril ...

(for "Programming in Logic") is a programming language
A programming language is a system of notation for writing computer programs. Most programming languages are text-based formal languages, but they may also be graphical. They are a kind of computer language.
The description of a programmin ...

based on a subset
In mathematics, set ''A'' is a subset of a set ''B'' if all elements of ''A'' are also elements of ''B''; ''B'' is then a superset of ''A''. It is possible for ''A'' and ''B'' to be equal; if they are unequal, then ''A'' is a proper subset o ...

of predicate calculus
Predicate or predication may refer to:
* Predicate (grammar), in linguistics
* Predication (philosophy)
* several closely related uses in mathematics and formal logic:
**Predicate (mathematical logic)
**Propositional function
**Finitary relation, o ...

. 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
Socrates (; ; –399 BC) was a Greek philosopher from Athens who is credited as the founder of Western philosophy and among the first moral philosophers of the ethical tradition of thought. An enigmatic figure, Socrates authored no ...

syllogism
A syllogism ( grc-gre, συλλογισμός, ''syllogismos'', 'conclusion, inference') is a kind of logical argument that applies deductive reasoning to arrive at a conclusion based on two propositions that are asserted or assumed to be true ...

. 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 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
Prolog is a logic programming language associated with artificial intelligence and computational linguistics.
Prolog has its roots in first-order logic, a formal logic, and unlike many other programming languages, Prolog is intended primaril ...

does not know anything about Plato
Plato ( ; grc-gre, Πλάτων ; 428/427 or 424/423 – 348/347 BC) was a Greek philosopher born in Athens during the Classical period in Ancient Greece. He founded the Platonist school of thought and the Academy, the first instit ...

, 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
Prolog is a logic programming language associated with artificial intelligence and computational linguistics.
Prolog has its roots in first-order logic, a formal logic, and unlike many other programming languages, Prolog is intended primaril ...

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 ofOWL
Owls are birds from the order Strigiformes (), which includes over 200 species of mostly solitary and nocturnal birds of prey typified by an upright stance, a large, broad head, binocular vision, binaural hearing, sharp talons, and feathers ...

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 ofprobability
Probability is the branch of mathematics concerning numerical descriptions of how likely an event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and 1, where, roughly speaking ...

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
In probability theory and statistics, Bayes' theorem (alternatively Bayes' law or Bayes' rule), named after Thomas Bayes, describes the probability of an event, based on prior knowledge of conditions that might be related to the event. For exam ...

.
Fuzzy logic

Non-monotonic logic

A relation of inference ismonotonic
In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of orde ...

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

* ''A priori and a posteriori
("from the earlier") and ("from the later") are Latin phrases used in philosophy to distinguish types of knowledge, justification, or argument by their reliance on empirical evidence or experience. knowledge is independent from current ...

''
* Abductive reasoning
Abductive reasoning (also called abduction,For example: abductive inference, or retroduction) is a form of logical inference formulated and advanced by American philosopher Charles Sanders Peirce beginning in the last third of the 19th centu ...

* Deductive reasoning
Deductive reasoning is the mental process of drawing deductive inferences. An inference is deductively valid if its conclusion follows logically from its premises, i.e. if it is impossible for the premises to be true and the conclusion to be fal ...

* Inductive reasoning
Inductive reasoning is a method of reasoning in which a general principle is derived from a body of observations. It consists of making broad generalizations based on specific observations. Inductive reasoning is distinct from ''deductive'' r ...

* Entailment
Logical consequence (also entailment) is a fundamental concept in logic, which describes the relationship between statements that hold true when one statement logically ''follows from'' one or more statements. A valid logical argument is on ...

* Epilogism
* Analogy
Analogy (from Greek ''analogia'', "proportion", from ''ana-'' "upon, according to" lso "against", "anew"+ ''logos'' "ratio" lso "word, speech, reckoning" is a cognitive process of transferring information or meaning from a particular subject ...

* Axiom system
** Axiom
An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that which is thought worthy or ...

* Immediate inference An immediate inference is an inference
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 deduc ...

* Inferential programming
* Inquiry
An inquiry (also spelled as enquiry in British English) 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 ...

* Logic
Logic is the study of correct reasoning. It includes both formal and informal logic. Formal logic is the science of deductively valid inferences or of logical truths. It is a formal science investigating how conclusions follow from prem ...

* Logic of information The logic of information, or the logical theory of information, considers the information content of logical signs and expressions along the lines initially developed by Charles Sanders Peirce. In this line of work, the concept of information serve ...

* Logical assertion
In mathematical logic, a judgment (or judgement) or assertion is a statement or enunciation in a metalanguage. For example, typical judgments in first-order logic would be ''that a string is a well-formed formula'', or ''that a proposition is tru ...

* Logical graph
* Rule of inference
In the philosophy of logic, a rule of inference, inference rule or transformation rule is a logical form consisting of a function which takes premises, analyzes their syntax, and returns a conclusion (or conclusions). For example, the rule of ...

* List of rules of inference
* Theorem
In mathematics, a theorem is a statement that has been proved, or can be proved. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of ...

* Transduction (machine learning)
References

Further reading

* * * * * Inductive inference: * * * * * * * Abductive inference: * * * Psychological investigations about human reasoning: * deductive: ** ** ** ** ** * statistical: ** **, * analogical: ** * spatial: ** ** ** * moral: **External links

*Inference example and definition

* {{Authority control Concepts in epistemology Concepts in logic Concepts in metaphilosophy Concepts in metaphysics Concepts in the philosophy of mind History of logic Intellectual history Logic Logic and statistics Logical consequence Metaphysics of mind Reasoning Sources of knowledge Thought