Deductive reasoning
The deduction (Latin deductio, discharge, continuation, derivative '), also deductive method or deductive conclusion is in philosophy and logic of a conclusion from given premises to logically compelling consequences. Deduction has been understood already by Aristotle as " circuit from the general to the particular ," that is, the inheritance of properties shared by all members of a group on real subsets and individual items. The Aristotle is the induction as extraction of general statements from the consideration of several individual cases, and the abduction or Apagoge opposite, stating that certain individual cases under a given or yet to be discovered general rule fall.
Logic and formal systems
Within the modern mathematical logic and all formal systems, a continuous structure as possible, the aim of deductive principles. The mathematics is largely present in deductive structure and is taught mainly as; that is, their results are derived from formal axiomatic systems. Deductive coherence is an essential feature of formal proofs in mathematics (see a proof (mathematics) ), the mathematical method of induction and the transfinite induction are contrary to its name, deductive method.
The simplest case of the application of the method is the elimination of a deductive implication using the separating rule. The logical structure of this rule is the general structure of an argument, which closes a set of premises following a rule of inference to a conclusion:
P → q --- q
( Premise 2 ) ( Conclusion )
If p and p → q (read: if p, then q) is true statements, as well as q is a true statement.
Decidability
There are logical systems in which expressions occur, although with the tools of this system can be formulated in it but they are not decidable. Deductive and reductive of inference are rarely applied in its simple structure. The actual scientific derivation is a complex system of deductive, reductive and heuristic methods.
Philosophy of Science
The view that deduction and induction are complementary elements of scientific truth, has also been disputed, most prominently by Karl Popper. According to him, it is in the induction is not a proof procedure. General rules are not derived in Popper's falsificationism, according to certain rules from the inductive empiricism, such rules are according to him, at best heuristics for finding general hypotheses. All conclusions that are drawn in science, for him, therefore, purely deductive, and conclusions from the particular to the general: These are made in the modus tollens, for example, when a general hypothesis or theory, based on an observed single fact is falsified (see also indirect evidence ).
In the natural sciences determined by deduction predictions must be empirically verifiable to have a scientific value. If the observations do not match the predictions, the theory must be modified or discarded.
The deductive method is generally not thought to be the only method of gaining new scientific knowledge. Such a method must always start from premises that are, in turn prove to be true, assuming hypothetically be true or are set as axiomatically true. Even if those assumptions turn can be derived deductively from different premises, this chain of evidence must be somewhere to start (see: Infiniter recourse).
Science has to resort to methods of proof, the non- deductive in nature, which therefore intensional relations are based. It involves empirical method, which gain knowledge through observation and experimentation. The logical processing of the results of the practice to scientific statements or even laws happens to the reductive method.
Psychology
In addition to logic, philosophy and linguistics and the psychology of thinking for human competence and incompetence employed by the deductive thinking. The main theories are:
- The theory of mental models by Philip Johnson -Laird (1983 ),
- The theory of mental logic of Jean Piaget (1958), Rips (1994 ), Ford (1995 ) and others, and
- The information gain theory of Oaksford and Chater (1994).