Inference

In conclusion ( inference from Latin infero " into bear "; " infer ", " close ", english inference ) is called in the logic of one of three closely related issues:

In computer science and statistics, the conclusion is also sometimes referred to by the other in the German unusual foreign word inference, probably as a translation of the English inference (, conclusion inference '); but usually the word inference in computer science is more specifically used for such conclusions, automated, ie computer- supported by an inference engine were drawn.

The logical conclusion is quoted at the follow arrow.

2.1 A first approximation
2.2 clarification
2.3 Correctness in formal logic

Types of logical inference

Three types of logical inference can be distinguished: deduction, induction and abduction. Wherein in each case the condition (also premise or "cause" ), the consistency (also result or "effect" ) and the rule (including law ) is referred to. Each of these references is in practice usually occurs more than once.

Deduction is the closing of the condition and usually the consequence.

Induction is the closing of the condition and the consequence of the rule.

Abduction is the conclusion of the rule and the consequence of the condition.

Regardless of experience and compellingly is only the deduction. The induction and abduction represent only possible logical conclusions

Examples

The following cases demonstrate the fact with the braking of a vehicle.

Case of deduction

Upon actuation of the brake, the vehicle is decelerating. (the Act)
The brake is operated. ( the observed cause)
The vehicle is slowing down. ( deductive conclusion to the effect )

Case of the induction

The brake is operated. ( the observed cause)
The vehicle is decelerating. ( the observed effect )
When you press the brake the vehicle ( each time) will slow down. ( However, inductive conclusion on the law., other conceivable laws that require, for example, other conditions)

Case of abduction

Upon actuation of the brake, the vehicle is decelerating. (the Act)
The vehicle is decelerating. ( the observed effect )
The brake is applied. (but abductive conclusion on the cause., other causes conceivable, for example an increase in the roadway )

Correctness of a conclusion

A first approximation

In a first approximation, one can say that a conclusion is correct or valid if it is impossible that the premises are true but the conclusion is false - succinctly formulated: From truths only truth follows. An example:

Premises: "All men are Bavaria ", " Socrates is a man "
Conclusion: " Socrates is Bayer "

Apparently, here one of the premises is false, as well as the conclusion. However, for the validity of an inference is not relevant to the actual truth of the premises, the above conclusion is valid because if the premises were true, the conclusion would be true. ( Had namely actually all people Bavaria, so would also be a Socrates because he is a man. ) So, are at a valid conclusion the premises are true, then so is the conclusion. However, if at least one premise is false, the conclusion may be true but also false. An example of a conclusion with a false premise and a true conclusion would be:

Premises: "All men are Greeks ", " Socrates is a man "
Conclusion: " Socrates is Greek".

Despite its catchiness makes the final concept shown here room for different interpretations " from truths only truth follows ". Thus, there is both intuitive and quite philosophical disagreement about the validity of different arguments or different types of arguments. As examples, the double negation are ( also the conclusion of "It does not rain not " to " It's raining " ) and the closing of an all- for an existence statement ( the conclusion of " All pigs are pink " to " There are pink pigs " ) called, and the " all " may " not " be considered valid or invalid, among others, depending on the specific understanding of the terms.

Clarification

More precisely, one can grasp the concept of correctness, if a distinction between logical and non- logical expressions. Logical expressions are testimony links ( connectives ) such as " and", " or " and " not," with which one or more statements to be linked to a new, more complex statement, and quantifiers such as "for all", "all", "any / r " (so-called universal quantifiers ), and" for some "," some " ," there " (so-called existential ); other expressions are called non- logical. An argument is valid if each substitution of one or more non- logical expressions in it, in which the premises are true, makes the conclusion true ( " met "). If we replace in the above example the non-logical term " Bayer " for example, by " mortal" and " man " with " Greek", we get:

Premises: " All Greeks are mortal ", " Socrates is a Greek"
Conclusion: " Socrates is mortal ".

Here are true both premises, as well as the conclusion. In fact, there may be in this case, no replacement of non- logical expressions in which both premises are true, the conclusion, however, is wrong. From this it also follows a test to prove the invalidity of a Conclusion: It is to provide a replacement of the non-logical terms, which makes the premises are true, the conclusion is, however, wrong. Consider this example, the following invalid argument:

Premises: "Some Bayern Munich are ", " Some Bavaria are Schwabinger "
Conclusion: "Some of Munich's Schwabing are "

Here both premises and the conclusion are true. However, it is not a valid argument, because we replace " Schwabing " with " Nuremberg ", the premises remain true, the conclusion will be wrong.

Correctness in formal logic

To an even more accurate and more general characterization of the correctness of a conclusion sought formal logic. Because of the greater complexity and ambiguity of natural language natural language statements are translated into statements of a precisely defined formal language. A Ableitbarkeitsbegriff is defined on these formal objects then, which is usually symbolized by the sign. The motivation here is often that just then a Ableitbarkeitsbeziehung exists between the formal objects when the natural language structures whose translations they represent follow apart.

At the latest at the stage of formalization can be the philosophical and intuitive differences in the understanding of inference and thus as to which arguments are valid, no longer overlap. Accordingly, there are different, mutually non-equivalent Ableitbarkeitsbegriffe that reflect the different varieties of intuitive and scientific circuit concept. The most frequently used of the classical and the intuitionistic Ableitbarkeitsbegriff whose distinction goes back to both a very different understanding of the logical expressions (eg, the connectives " and", " or " and " not " ) and on a different concept of truth.

The definition of Ableitbarkeitsbegriffs is done by rules of inference and, where appropriate axioms. A formal system that defines rules of inference and axioms, is called Calculus. See also the general article Proof ( logic). An introductory presentation of a specific logical system with a detailed formulation of the Ableitbarkeitsbegriffs can be found in the article propositional logic. For automatic inference is in computer science, the inference engine available.

Deadline Procedure

Closing procedures are in different areas and methods used, such as judicial syllogism, probabilistic reasoning, nonmonotonic reasoning, etc.