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. 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. 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 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 data which may be subject to random variation.
Examples 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 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 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.
Beef is a type of meat. 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. 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.
Bananas are fruit. Therefore, bananas are apples.
A valid argument with false premises may lead to a false conclusion:
All tall people are Greek. John Lennon was tall.
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 tall people are musicians
John Lennon was tall Therefore, John Lennon was a musician
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 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
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 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 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 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 then in Prolog we would code Q:-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. Finally ?- mortal(X)
would result in "Yes"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
logic as a subset. 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. A central rule of Bayesian inference is Bayes' theorem.
See Bayesian inference 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 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. 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
References
Further reading
Inductive inference: Rudolf Carnap and Richard C. Jeffrey, ed..
Studies in Inductive Logic and Probability 1. The University of California Press.
Richard C. Jeffrey, ed.. Studies in Inductive Logic and Probability 2. The University of
California Press. Dana Angluin. An Application of the Theory
of Computational Complexity to the Study of Inductive Inference. University of California
at Berkeley. Dana Angluin. "Inductive Inference of Formal
Languages from Positive Data". Information and Control 45: 117–135. doi:10.1016/s0019-9958(80)90285-5.
Dana Angluin, Carl H. Smith. "Inductive Inference: Theory and Methods". Computing Surveys 15:
237–269. doi:10.1145/356914.356918. Dov M. Gabbay and Stephan Hartmann and John
Woods, ed.. Inductive Logic. Handbook of the History of Logic 10. Elsevier.
Nelson Goodman. Fact, Fiction, and Forecast. Bobbs-Merrill Co. Inc.
Abductive inference: P. O'Rourke and J. Josephson, ed.. Automated
abduction: Inference to the best explanation. AAAI Press.
Stathis Psillos. Dov M. Gabbay and Stephan Hartmann and John Woods, ed. An Explorer upon
Untrodden Ground: Peirce on Abduction. Handbook of the History of Logic 10. Elsevier. pp. 117–152.
Oliver Ray. Hybrid Abductive Inductive Learning. University of London, Imperial College.
Psychological investigations about human reasoning: deductive:
Philip Nicholas Johnson-Laird, Ruth M.J. Byrne. Deduction. Erlbaum.
Ruth M.J. Byrne, P.N. Johnson-Laird. ""If" and the Problems of Conditional Reasoning".
Trends in Cognitive Science 13: 282–287. doi:10.1016/j.tics.2009.04.003.
Markus Knauff, Thomas Fangmeier, Christian C. Ruff, P.N. Johnson-Laird. "Reasoning, Models,
and Images: Behavioral Measures and Cortical Activity". Journal of Cognitive Neuroscience
15: 559–573. doi:10.1162/089892903321662949. Philip N. Johnson-Laird. M.S. Gazzaniga, ed.
Mental Models, Deductive Reasoning, and the Brain. MIT Press. pp. 999–1008.
Sangeet Khemlani, P.N. Johnson-Laird. "Illusory Inferences about Embedded Disjunctions". Proceedings
of the 30th Annual Conference of the Cognitive Science Society. Washington/DC. pp. 2128–2133.
statictical: Rachel McCloy, Ruth M.J. Byrne, Philip N.
Johnson-Laird. "Understanding Cumulative Risk". The Quarterly Journal of Experimental Psychology:
18. Philip N. Johnson-Laird. "Mental Models and
Probabilistic Thinking". Cognition 50: 189–209. doi:10.1016/0010-0277(94)90028-0. ,
analogical: B.D. Burns. "Meta-Analogical Transfer: Transfer
Between Episodes of Analogical Reasoning". Journal of Experimental Psychology: Learning,
Memory, and Cognition 22: 1032–1048. doi:10.1037/0278-7393.22.4.1032.
spatial: Georg Jahn, Markus Knauff, P.N. Johnson-Laird.
"Preferred mental models in reasoning about spatial relations". Memory & Cognition 35:
2075–2087. doi:10.3758/bf03192939. Markus Knauff, P.N. Johnson-Laird. "Visual
imagery can impede reasoning". Memory \& Cognition 30: 363–371. doi:10.3758/bf03194937.
James A. Waltz, Barbara J. Knowlton, Keith J. Holyoak, Kyle B. Boone, Fred S. Mishkin,
Marcia de Menezes Santos, Carmen R. Thomas, and Bruce L. Miller. "A System for Relational
Reasoning in Human Prefrontal Cortex". Psychological Science 10: 119–125. doi:10.1111/1467-9280.00118.
moral: Monica Bucciarelli, Sangeet Khemlani, P.N.
Johnson-Laird. "The Psychology of Moral Reasoning". Judgment and Decision Making 3: 121–139.
External links Inference at PhilPapers
Inference at the Indiana Philosophy Ontology Project