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Deductive reasoning, also deductive logic or logical deduction or, informally, "top-down" logic, is the process of reasoning from one or more statements (premises) to reach a logically certain conclusion.
Deductive reasoning (top-down logic) contrasts with inductive reasoning (bottom-up logic) in the following way: In deductive reasoning, a conclusion is reached reductively by applying general rules that hold over the entirety of a closed domain of discourse, narrowing the range under consideration until only the conclusion is left. In inductive reasoning, the conclusion is reached by generalizing or extrapolating from initial information. As a result, induction can be used even in an open domain, one where there is epistemic uncertainty. Note, however, that the inductive reasoning mentioned here is not the same as induction used in mathematical proofs – mathematical induction is actually a form of deductive reasoning.
An example of a deductive argument:
The first premise states that all objects classified as "men" have the attribute "mortal". The second premise states that "Socrates" is classified as a "man" – a member of the set "men". The conclusion then states that "Socrates" must be "mortal" because he inherits this attribute from his classification as a "man".
The law of detachment (also known as affirming the antecedent and Modus ponens) is the first form of deductive reasoning. A single conditional statement is made, and a hypothesis (P) is stated. The conclusion (Q) is then deduced from the statement and the hypothesis. The most basic form is listed below:
In deductive reasoning, we can conclude Q from P by using the law of detachment. However, if the conclusion (Q) is given instead of the hypothesis (P) then there is no definitive conclusion.
The following is an example of an argument using the law of detachment in the form of an if-then statement:
Since the measurement of angle A is greater than 90° and less than 180°, we can deduce that A is an obtuse angle.
The law of syllogism takes two conditional statements and forms a conclusion by combining the hypothesis of one statement with the conclusion of another. Here is the general form:
The following is an example:
We deduced the final statement by combining the hypothesis of the first statement with the conclusion of the second statement. We also allow that this could be a false statement. This is an example of the Transitive Property in mathematics. The Transitive Property is sometimes phrased in this form:
The following are examples:
An argument is valid if it is impossible for its premises to be true while its conclusion is false. In other words, the conclusion must be true if the premises are true. An argument can be valid even though the premises are false.
An argument is sound if it is valid and the premises are true.
It is possible to have a deductive argument that is logically valid but is not sound. Fallacious arguments often take that form.
The following is an example of an argument that is valid, but not sound:
The example's first premise is false – there are people who eat carrots and are not quarterbacks – but the conclusion must be true, so long as the premises are true (i.e. it is impossible for the premises to be true and the conclusion false). Therefore the argument is valid, but not sound. Generalizations are often used to make invalid arguments, such as "everyone who eats carrots is a quarterback." Not everyone who eats carrots is a quarterback, thus proving the flaw of such arguments.
In this example, the first statement uses categorical reasoning, saying that all carrot-eaters are definitely quarterbacks. This theory of deductive reasoning – also known as term logic – was developed by Aristotle, but was superseded by propositional (sentential) logic and predicate logic.
Deductive reasoning can be contrasted with inductive reasoning, in regards to validity and soundness. In cases of inductive reasoning, even though the premises are true and the argument is "valid", it is possible for the conclusion to be false (determined to be false with a counterexample or other means).
Deductive reasoning is generally thought of as a skill that develops without any formal teaching or training. As a result of this belief, deductive reasoning skills are not taught in secondary schools, where students are expected to use reasoning more often and at a higher level. It is in high school, for example, that students have an abrupt introduction to mathematical proofs – which rely heavily on deductive reasoning.
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