In logic, a sequence of steps leading from a set of premisses to a conclusion. Rules of inference are rules for the construction of good and valid arguments.
Inference is the act or process of deriving a conclusion based solely on what one already knows.
Inference is studied within several different fields. Logic studies the laws of valid inference.
The accuracy of inductive and deductive inferences
The conclusion inferred from multiple observations is made by the process of inductive reasoning. In contrast, the conclusion of a valid deductive inference is true if the premises are true. 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.
Valid inferences
Inferences are either valid or invalid, but not both. Philosophical logic has attempted to define the rules of proper inference, i.e.
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. 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:
All A is B All C is A ---------- All C is BSince the syllogism fits this form, then the inference is valid.
In predicate logic (a simple but useful formalization of Aristotelician logic), this syllogism can be stated as follows:
∀ 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".
Consider the following:
All fat people are musicians John Lennon was fat ------------------- Therefore John Lennon was a musicianIn this case we have two false premises that implies a true conclusion. The inference is valid because it follows the form of a correct inference.
An incorrect inference is known as a fallacy.
Automatic logical inference
Although now somewhat past their heyday, AI systems for automated logical inference once were extremely popular research topics, and have known industrial applications under the form of expert systems.
An inference system's job is to extend a knowledge base automatically.
An example: inference using Prolog
Prolog (Programming in Logic) is a programming language based on a subset of predicate calculus.
Inference and uncertainty
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Traditional logic is only concerned with certainty - one progresses from certain premises to certain conclusions.
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).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.
Although that reasoning seems sound, it does not fit in the logical framework described above. 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.
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.
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.
Bayesianists identify probabilities with degrees of beliefs, with certainly true propositions having probability 1, and certainly false propositions having probability 0.
Through the rules of probability, the probability of a conclusion and of alternatives can be calculated. A central rule of Bayesian inference is Bayes' theorem, which gave its name to the field.
See Bayesian inference for examples.
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; 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. 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.).
Three types of logical inference
There are three types of inference:
Deductive reasoning, finding the effect with the cause and the rule.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.
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