Inference
Encyclopedia
Inference is the act or process of deriving logical conclusions from premises known or assumed to be true. The conclusion drawn is also called an idiomatic. The laws of valid inference are studied in the field of logic
.
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
allows for inference from quantitative data.
. 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.
1. A conclusion reached on the basis of evidence and reasoning.
2. The process of reaching such a conclusion: "order, health, and by inference cleanliness".
Greek philosophers
defined a number of syllogism
s, correct three part inferences, that can be used as building blocks for more complex reasoning. We begin with the most famous of them all:
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 the 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:
For the conclusion to be necessarily true, the premises need to be true.
Now we turn to an invalid form.
To show that this form is invalid, we demonstrate how it can lead from true premises to a false conclusion.
A valid argument with false premises may lead to a false conclusion:
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:
In this case we have two false premises that imply a true conclusion.
. 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.
s and later business rule engines.
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.
(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 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.
a new field of application. Being based upon first-order logic
, knowledge expressed using one variant of OWL
can be logically processed, i.e., inferences can be made upon it.
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
, which gave its name to the field.
See Bayesian inference
for examples.
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.
Logic
In philosophy, Logic is the formal systematic study of the principles of valid inference and correct reasoning. Logic is used in most intellectual activities, but is studied primarily in the disciplines of philosophy, mathematics, semantics, and computer science...
.
Human inference (i.e. how humans draw conclusions) is traditionally studied within the field of cognitive psychology
Cognitive psychology
Cognitive psychology is a subdiscipline of psychology exploring internal mental processes.It is the study of how people perceive, remember, think, speak, and solve problems.Cognitive psychology differs from previous psychological approaches in two key ways....
; artificial intelligence
Artificial intelligence
Artificial intelligence is the intelligence of machines and the branch of computer science that aims to create it. AI textbooks define the field as "the study and design of intelligent agents" where an intelligent agent is a system that perceives its environment and takes actions that maximize its...
researchers develop automated inference systems to emulate human inference. Statistical inference
Statistical inference
In statistics, statistical inference is the process of drawing conclusions from data that are subject to random variation, for example, observational errors or sampling variation...
allows for inference from quantitative data.
Definition of inference
The process by which a conclusion is inferred from multiple observations is called inductive reasoningInductive reasoning
Inductive reasoning, also known as induction or inductive logic, is a kind of reasoning that constructs or evaluates propositions that are abstractions of observations. It is commonly construed as a form of reasoning that makes generalizations based on individual instances...
. 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.
1. A conclusion reached on the basis of evidence and reasoning.
2. The process of reaching such a conclusion: "order, health, and by inference cleanliness".
Examples of inference
Incoherence: no definition of deductive inference has been offered. the definition offered was for INDUCTIVE inference.Greek philosophers
Greek philosophy
Ancient Greek philosophy arose in the 6th century BCE and continued through the Hellenistic period, at which point Ancient Greece was incorporated in the Roman Empire...
defined a number of syllogism
Syllogism
A syllogism is a kind of logical argument in which one proposition is inferred from two or more others of a certain form...
s, correct three part inferences, that can be used as building blocks for more complex reasoning. We begin with the most famous of them all:
- All men are mortal
- Socrates is a man
- 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 the 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
Symbology
Symbology concerns the study of symbols.Symbology may also refer to:-Academics:* Semiotics, study of signs and symbols* Iconography, branch of art history which studies images...
track:
- All apples are blue.
- A banana is an apple.
- Therefore, a banana is blue.
For the conclusion to be necessarily true, the premises need to be true.
Now we turn to an invalid form.
- All A are B.
- C is a B.
- 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.
- All apples are fruit. (True)
- Bananas are fruit. (True)
- Therefore, bananas are apples. (False)
A valid argument with false premises may lead to a false conclusion:
- All fat people are Greek.
- John Lennon was fat.
- Therefore, John Lennon was Greek.
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:
- 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.
Incorrect inference
An incorrect inference is known as a fallacyFallacy
In logic and rhetoric, a fallacy is usually an incorrect argumentation in reasoning resulting in a misconception or presumption. By accident or design, fallacies may exploit emotional triggers in the listener or interlocutor , or take advantage of social relationships between people...
. Philosophers who study informal logic
Informal logic
Informal logic, intuitively, refers to the principles of logic and logical thought outside of a formal setting. However, perhaps because of the informal in the title, the precise definition of informal logic is matters of some dispute. Ralph H. Johnson and J...
have compiled large lists of them, and cognitive psychologists have documented many biases in human reasoning
Cognitive bias
A cognitive bias is a pattern of deviation in judgment that occurs in particular situations. Implicit in the concept of a "pattern of deviation" is a standard of comparison; this may be the judgment of people outside those particular situations, or may be a set of independently verifiable...
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 systemExpert system
In artificial intelligence, an expert system is a computer system that emulates the decision-making ability of a human expert. Expert systems are designed to solve complex problems by reasoning about knowledge, like an expert, and not by following the procedure of a developer as is the case in...
s and later business rule engines.
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
Relevance
-Introduction:The concept of relevance is studied in many different fields, including cognitive sciences, logic and library and information science. Most fundamentally, however, it is studied in epistemology...
to its task.
Example using Prolog
PrologProlog
Prolog is a general purpose 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 declarative: the program logic is expressed in terms of...
(for "Programming in Logic") is a programming language
Programming language
A programming language is an artificial language designed to communicate instructions to a machine, particularly a computer. Programming languages can be used to create programs that control the behavior of a machine and/or to express algorithms precisely....
based on a subset
Subset
In mathematics, especially in set theory, a set A is a subset of a set B if A is "contained" inside B. A and B may coincide. The relationship of one set being a subset of another is called inclusion or sometimes containment...
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
Backward chaining
Backward chaining is an inference method that can be described as working backward from the goal...
.
Let us return to our Socrates
Socrates
Socrates was a classical Greek Athenian philosopher. Credited as one of the founders of Western philosophy, he is an enigmatic figure known chiefly through the accounts of later classical writers, especially the writings of his students Plato and Xenophon, and the plays of his contemporary ...
syllogism
Syllogism
A syllogism is a kind of logical argument in which one proposition is inferred from two or more others of a certain form...
. 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
Prolog is a general purpose 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 declarative: the program logic is expressed in terms of...
does not know anything about Plato
Plato
Plato , was a Classical Greek philosopher, mathematician, student of Socrates, writer of philosophical dialogues, and founder of the Academy in Athens, the first institution of higher learning in the Western world. Along with his mentor, Socrates, and his student, Aristotle, Plato helped to lay the...
, and hence defaults to any property about Plato being false (the so-called closed world assumption
Closed world assumption
The closed world assumption is the presumption that what is not currently known to be true, is false. The same name also refers to a logical formalization of this assumption by Raymond Reiter. The opposite of the closed world assumption is the open world assumption , stating that lack of knowledge...
). Finally
?- mortal(X) (Is anything mortal) would result in "Yes" (and in some implementations: "Yes": X=socrates)
Prolog
Prolog
Prolog is a general purpose 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 declarative: the program logic is expressed in terms of...
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 webSemantic Web
The Semantic Web is a collaborative movement led by the World Wide Web Consortium that promotes common formats for data on the World Wide Web. By encouraging the inclusion of semantic content in web pages, the Semantic Web aims at converting the current web of unstructured documents into a "web of...
a new field of application. Being based upon first-order logic
First-order logic
First-order logic is a formal logical system used in mathematics, philosophy, linguistics, and computer science. It goes by many names, including: first-order predicate calculus, the lower predicate calculus, quantification theory, and predicate logic...
, knowledge expressed using one variant of OWL
Web Ontology Language
The Web Ontology Language is a family of knowledge representation languages for authoring ontologies.The languages are characterised by formal semantics and RDF/XML-based serializations for the Semantic Web...
can be logically processed, i.e., inferences can be made upon it.
Bayesian statistics and probability logic
Philosophers and scientists who follow the Bayesian frameworkBayesian inference
In statistics, Bayesian inference is a method of statistical inference. It is often used in science and engineering to determine model parameters, make predictions about unknown variables, and to perform model selection...
for inference use the mathematical rules of probability
Probability
Probability is ordinarily used to describe an attitude of mind towards some proposition of whose truth we arenot certain. The proposition of interest is usually of the form "Will a specific event occur?" The attitude of mind is of the form "How certain are we that the event will occur?" The...
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
Bayes' theorem
In probability theory and applications, Bayes' theorem relates the conditional probabilities P and P. It is commonly used in science and engineering. The theorem is named for Thomas Bayes ....
, which gave its name to the field.
See Bayesian inference
Bayesian inference
In statistics, Bayesian inference is a method of statistical inference. It is often used in science and engineering to determine model parameters, make predictions about unknown variables, and to perform model selection...
for examples.
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, 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
- Reasoning
- Abductive reasoningAbductive reasoningAbduction is a kind of logical inference described by Charles Sanders Peirce as "guessing". The term refers to the process of arriving at an explanatory hypothesis. Peirce said that to abduce a hypothetical explanation a from an observed surprising circumstance b is to surmise that a may be true...
- Deductive reasoningDeductive reasoningDeductive reasoning, also called deductive logic, is reasoning which constructs or evaluates deductive arguments. Deductive arguments are attempts to show that a conclusion necessarily follows from a set of premises or hypothesis...
- Inductive reasoningInductive reasoningInductive reasoning, also known as induction or inductive logic, is a kind of reasoning that constructs or evaluates propositions that are abstractions of observations. It is commonly construed as a form of reasoning that makes generalizations based on individual instances...
- Retroductive reasoning
- Abductive reasoning
- EntailmentEntailmentIn logic, entailment is a relation between a set of sentences and a sentence. Let Γ be a set of one or more sentences; let S1 be the conjunction of the elements of Γ, and let S2 be a sentence: then, Γ entails S2 if and only if S1 and not-S2 are logically inconsistent...
- AnalogyAnalogyAnalogy is a cognitive process of transferring information or meaning from a particular subject to another particular subject , and a linguistic expression corresponding to such a process...
- AxiomAxiomIn traditional logic, an axiom or postulate is a proposition that is not proven or demonstrated but considered either to be self-evident or to define and delimit the realm of analysis. In other words, an axiom is a logical statement that is assumed to be true...
- Bayesian inferenceBayesian inferenceIn statistics, Bayesian inference is a method of statistical inference. It is often used in science and engineering to determine model parameters, make predictions about unknown variables, and to perform model selection...
- Business ruleBusiness ruleA Business rule is a statement that defines or constrains some aspect of the business and always resolves to either true or false. Business rules are intended to assert business structure or to control or influence the behavior of the business. Business rules describe the operations, definitions...
- Business rules engineBusiness rules engineA business rules engine is a software system that executes one or more business rules in a runtime production environment. The rules might come from legal regulation , company policy , or other sources...
- Expert systemExpert systemIn artificial intelligence, an expert system is a computer system that emulates the decision-making ability of a human expert. Expert systems are designed to solve complex problems by reasoning about knowledge, like an expert, and not by following the procedure of a developer as is the case in...
- Fuzzy logicFuzzy logicFuzzy logic is a form of many-valued logic; it deals with reasoning that is approximate rather than fixed and exact. In contrast with traditional logic theory, where binary sets have two-valued logic: true or false, fuzzy logic variables may have a truth value that ranges in degree between 0 and 1...
- Immediate inferenceImmediate inferenceAn immediate inference is an inference which can be made from only one statement or proposition. For instance, from the statement "All toads are green." we can make the immediate inference that "No toads are not green." This new statement is known as the contrapositive of the original statement...
- Inference engineInference engineIn computer science, and specifically the branches of knowledge engineering and artificial intelligence, an inference engine is a computer program that tries to derive answers from a knowledge base. It is the "brain" that expert systems use to reason about the information in the knowledge base for...
- Inferential programmingInferential programmingIn ordinary computer programming, the programmer keeps the program's intended results in mind and painstakingly constructs a computer program to achieve those results. Inferential programming refers to techniques and technologies enabling the inverse...
- InquiryInquiryAn 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.-Deduction:...
- LogicLogicIn philosophy, Logic is the formal systematic study of the principles of valid inference and correct reasoning. Logic is used in most intellectual activities, but is studied primarily in the disciplines of philosophy, mathematics, semantics, and computer science...
- Logic of informationLogic of informationThe 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...
- Logical assertionLogical assertionA logical assertion is a statement that asserts that a certain premise is true, and is useful for statements in proof. It is equivalent to a sequent with an empty antecedent....
- Logical graphLogical graphA logical graph is a special type of diagramatic structure in any one of several systems of graphical syntax that Charles Sanders Peirce developed for logic....
- Nonmonotonic logic
- Rule of inferenceRule of inferenceIn logic, a rule of inference, inference rule, or transformation rule is the act of drawing a conclusion based on the form of premises interpreted as a function which takes premises, analyses their syntax, and returns a conclusion...
- List of rules of inference
- TheoremTheoremIn mathematics, a theorem is a statement that has been proven on the basis of previously established statements, such as other theorems, and previously accepted statements, such as axioms...
- Sherlock HolmesSherlock HolmesSherlock Holmes is a fictional detective created by Scottish author and physician Sir Arthur Conan Doyle. The fantastic London-based "consulting detective", Holmes is famous for his astute logical reasoning, his ability to take almost any disguise, and his use of forensic science skills to solve...