# Deduction

In a classic presentation of the empirical social sciences, deduction, induction, theory and empiricism are central concepts. In doing so, empirical data is collected and general sentences ( theory ) are obtained from them by induction , and conclusions about individual cases can be obtained from the theory by deduction.

The deduction ( Latin deductio , lead, continue, derivation '), also deductive method or deductive conclusion , is in philosophy and logic a conclusion of given premises on the logically compelling consequences. Deduction was already understood by Aristotle as “inferring from the general to the particular”; H. the inheritance of properties shared by all members of a group on real subsets and individual elements. Aristotle contrasts this with induction as the extraction of general statements from the consideration of several individual cases, and abduction or apagogue , which establishes that certain individual cases fall under a given or yet to be discovered general rule.

## Logic and formal systems

Within modern mathematical logic and all formal systems , a consistent structure with deductive principles is aimed for. The mathematics is largely in place in deductive structure and is mainly taught so; d. H. their results are formally derived from axiom systems . Deductive coherence is an essential feature of formal proof in mathematics . The mathematical methods of complete induction and transfinite induction are, contrary to their names, deductive methods.

The simplest case of using the deductive method is to remove an implication with the help of the separation rule . The logical structure of this rule is the general structure of an argument that draws a conclusion from a set of premises according to an inference rule :

 p (Premise 1) p → q (Premise 2) ——— q (Conclusion)

Are p and pq (read: if p , then q ) true statements, so is q a true statement.

### Decidability

There are logical systems in which expressions occur that can be formulated with the aids of this system, but cannot be decided in it. In their simple structure, deductive and reductive reasoning are rarely used. The actual scientific derivation is a complex system of deductive, reductive and heuristic procedures.

## Philosophy of science

The view that deduction and induction are complementary elements in scientific establishment of truth has also been disputed, most prominently by Karl Popper . According to him, induction is not an evidence process . In Popper's falsificationism, general rules are not derived from empirical evidence according to certain inductive rules; according to him, such rules are at best heuristics for finding general hypotheses . All conclusions that are drawn in science are therefore purely deductive for him, including conclusions from the particular to the general: These are made in the modus tollens , for example when a general hypothesis or theory is falsified based on an observed individual fact.

In the natural sciences , predictions made by deduction must be empirically verifiable in order to be of scientific value. If the observations do not agree with the predictions, the theory must be adjusted or rejected.

The deductive method is generally not considered to be the only method of gaining new scientific knowledge. Such a method must always start from premises which in turn have to be proven to be true , hypothetically assumed to be true , or axiomatically posited as true . Even if such premises can in turn be deductively derived from other premises, this chain of proof must start somewhere (see: Infinite Regress ).

Science has to resort to proof procedures that are non-deductive in nature, i.e. that are based on intentional relationships. So it is about empirical procedures which gain knowledge through observation and experiments. The logical processing of the results of practice into scientific statements or even laws is done with the reductive method .

## psychology

In addition to logic, philosophy and linguistics, thought psychology also deals with human competence and incompetence in reasoning. The most important theories are:

Psychology examines why and how people can infer why they make mistakes and which mistakes are made more often than others. In order to rule out content effects, syllogistic tasks have mostly been used . It turned out that people very often do not think formally and logically, but instead use heuristics , especially under time pressure . Here, let bias seen in the literature as bias (Engl. For prejudice, bias, bias) or Cognitive distortion are called. Illogical conclusions that are credible in terms of content are often held to be true, and logically valid but implausible conclusions are mistakenly wrong (so-called belief bias ). Also widespread tendency to hold conclusions rather true that confirm their own beliefs (so-called. Confirmation bias , English confirmation bias ).