# conclusion

Conclusion , inference , inference , inference (from Latin inferre " carry in "; "infer", "close"; English inference ) or conclusion ( Latin conclusio "conclusion") and implication are in logic terms for several closely related facts:

• A conclusion is firstly a linguistic structure that consists of a series of truthful statements on the one hand, the premises or assumptions (for example axioms or scientific hypotheses ), and on the other hand, the conclusion. Such a structure is also called a (logical) conclusion or an argument . In German, the transition between premises and conclusion is often introduced with therefore , therefore , thus , consequently or on the basis of it . A distinction is made between correct and incorrect conclusions (see also below). This distinction is central to logic; logic can almost be called the science of correct inference.
• In a second sense, a conclusion is a part of the linguistic structure just mentioned, namely the conclusion. The synonyms Conclusion or Final Clause also exist for these . In rhetoric , the word conclusion is also used generally for the conclusion of a speech.
• Thirdly, the conclusion is the result of reflection , i.e. the (mostly step-by-step) recognition of conclusions or the implementation of a proof . These conclusions can also be drawn from unconscious cultural, social or religious background assumptions.

In computer science and statistics , the conclusion is sometimes referred to using the foreign word inference , which is otherwise unusual in German , probably as a translation of the English inference ('conclusion, consequence'); but mostly the word inference in computer science is used more specifically for conclusions that are automated, i.e. H. computer-aided by an inference machine .

The logical conclusion is noted with the following arrow .

## Types of logical inference Simplified overview: `A=>B`is the rule, `A`the condition and `B`the consequence.

There are three types of logical reasoning: deduction , induction and abduction . Reference is made to the condition (also premise or “cause”), the consequence (also result or “effect”) and the rule (also law ); each of these three references usually occurs several times in practical application.

• Deduction is the conclusion from the condition and the rule to the consequence.
Short form: cause `&`law → effect
• Induction is the conclusion of the condition and the consequence of the rule.
Short form: cause `&`effect → law
• Abduction is the conclusion of the rule and the consequence on the condition.
Short form: law `&`effect → cause

Only deduction can be used independently of experience. Induction and abduction are possible logical conclusions.

### Examples

The following cases demonstrate the situation using the braking of a vehicle:

• Case of deduction
• When the brakes are applied, the vehicle slows down. (the law)
• The brake is applied. (the observed cause)
• The vehicle will slow down. (deductive conclusion on the effect)
• Case of induction
• The brake is applied. (the observed cause)
• The vehicle slows down. (the observed effect)
• When the brake is applied, the vehicle slows down (every time). (Inductive conclusion on the law. However, other laws are also conceivable, which, for example, require further conditions)
• Case of abduction
• When the brakes are applied, the vehicle slows down. (the law)
• The vehicle slows down. (the observed effect)
• The brake has been applied. (Abductive conclusion on the cause. However, other causes are also conceivable, e.g. an increase in the roadway)

## Correctness of a conclusion

### A first approximation

As 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 - to put it succinctly: from truth only follows truth. An example:

• Premises: "All people are Bavarians", "Socrates is a person"
• Conclusion: "Socrates is Bavarian"

Apparently one of the premises is wrong here, as is the conclusion. However, for the validity of a conclusion is not relevant to the actual truth of the premises, the above conclusion is valid because if the premises are true would be , the conclusion true would . (If all people were actually Bavaria, then Socrates would be one too, since he is a person.) If the premises are true for a valid conclusion, then the conclusion is also true. However, if at least one premise is false, the conclusion can be true or false. An example of a conclusion with a wrong premise and a true conclusion would be:

• Premises: "All people are Greeks", "Socrates is a person"
• Conclusion: "Socrates is Greek".

Despite its catchiness, the final term presented here, “From truth only follows truth” leaves room for different interpretations. Both intuitively and philosophically, there is definitely disagreement regarding the validity of different arguments or different types of arguments. Examples are the double negation (the conclusion from "It does not rain " to "It rains") and the conclusion from an all to one existence statement (the conclusion from "All pigs are pink" to "There are pink pigs") which, among other things, can be regarded as valid or invalid depending on the specific understanding of the terms “not” and “all”.

### Clarification

The concept of correctness can be grasped a little more precisely if one distinguishes between logical and non-logical expressions. Logical expressions are statement links ( joiners ) such as "and", "or" and "not", with which one or more statements are linked to a new, more complex statement, as well as quantifiers such as "for all", "all", "each / r ”(so-called universal quantifiers) and“ for some ”,“ some ”,“ there are ”(so-called existential quantifiers); other expressions are called non-logical. An argument is valid if every substitution in it of one or more non-logical expressions for which the premises are true also makes the conclusion true (“fulfilled”). If, in the above example, we replace the non-logical expression “Bayer” with “mortal” and “human” with “Greek”, we get:

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

Both premises are true here, as is the conclusion. In fact, in this case there can be no substitution of nologiological expressions where both premises are true but the conclusion is false. This also results in a test to prove the invalidity of a conclusion: A replacement of the non-logical terms is to be given, which makes the premises true, but the conclusion false. For example, consider the following invalid argument:

• Premises: "Some Bavarians are Munich", "Some Bavarians are Schwabing"
• Conclusion: "Some Munich residents are Schwabing"

Here both premises as well as the conclusion are true. Nevertheless, it is not a valid argument, because if we replace “Schwabinger” with “Nürnberger”, the premises remain true, but the conclusion becomes false.

### Correctness in formal logic

Formal logic endeavors to characterize the correctness of a conclusion more precisely and more generally. Because of the greater complexity and ambiguity of natural languages, natural language statements are translated into statements of a precisely defined formal language . A concept of derivability is then defined on these formal objects , which is usually symbolized by the sign . The motivation here is often that there is a deducibility relationship between the formal objects precisely when the natural language structures whose translations they represent follow one another. ${\ displaystyle \ vdash}$ At the latest in the formalization stage , the philosophical and intuitive differences in the understanding of “conclusion” - and thus with regard to which arguments are valid - can no longer be covered. Accordingly, there are different, non-equivalent deducibility terms that reflect the different varieties of the intuitive and scientific final term. The classical and the intuitionist concept of derivability are most frequently used, the distinction of which is based on a very different understanding of the logical expressions (e.g. the connectives “and”, “or” and “not”) and a different concept of truth.

The definition of the concept of derivability is done by rules of inference and possibly by axioms . A formal system that establishes inference rules and axioms is called a calculus . See also the general article Proof (logic) . An introductory presentation of a concrete logical system with a detailed formulation of the concept of derivability can be found in the article propositional logic . Is in computer science which for the automatic inference inference engine available.

## Final procedure

Closing procedures are used in different methods in different areas, e.g. B. Justice syllogism , probabilistic reasoning , non-monotonous reasoning, etc.