Inferences are steps in reasoning, moving from to logical consequences; etymologically, the word means to "carry forward". Inference is theoretically traditionally divided into deduction and induction, a distinction that in Europe dates at least to Aristotle (300s BCE). Deduction is inference Formal proof logical conclusions from premises known or assumed to be truth, with the laws of valid inference being studied in logic. Induction is inference from particular premises to a universal conclusion. A third type of inference is sometimes distinguished, notably by Charles Sanders Peirce, 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; 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 variations.
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:
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 Symbology track:
If the premises are true, then the conclusion is necessarily true, too.
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 a false premise may lead to a false conclusion, (this and the following examples do not follow the Greek syllogism):
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:
In this case we have one false premise and one true premise where a true conclusion has been inferred.
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.
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 relevance to its task.
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).)
?- 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:
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.
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.
A relation of inference is monotonic 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.
Psychological investigations about human reasoning: