Deductive reasoning

Deductive reasoning, also called deductive logic, is reasoning which constructs or evaluates deductive arguments. Deductive arguments are attempts to show that a conclusion necessarily follows from a set of premises or hypotheses. A deductive argument is valid if the conclusion does follow necessarily from the premises, i.e., if the conclusion must be true provided that the premises are true. A deductive argument is sound if it is valid and its premises are true. Deductive arguments are valid or invalid, sound or unsound. Deductive reasoning is a method of gaining knowledge. An example of a deductive argument:

  1. All men are mortal
  2. Socrates is a man
  3. Therefore, Socrates is mortal

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 states that "Socrates" must be mortal because he inherits this attribute from his classification as a man.

Contents

Law of detachment

The law of detachment is the first form of deductive reasoning. A single conditional statement is made, and then a hypothesis (P) is stated. The conclusion (Q) is deduced from the hypothesis and the statement. The most basic form is listed below:

  1. P→Q
  2. P (Hypothesis stated)
  3. Q (Conclusion given)

We can conclude Q from P by using the law of detachment from deductive reasoning.[1] However, if the conclusion (Q) is given instead of the hypothesis (P) then there is no valid conclusion.

The following is an example of an argument using the law of detachment in the form of an If-then statement:

  1. If m∠A>90°, then ∠A is an obtuse angle.
  2. m∠A=120°.
  3. ∠A is an obtuse angle.

Since the measurement of angle A is greater than 90°, we can deduce that A is an obtuse angle.

Law of syllogism

The law of syllogism takes two conditional statements and forms a conclusion by combining the hypothesis of one statement with the conclusion of another. The following is an example:

  1. If Larry is sick, then he will be absent from school.
  2. If Larry is absent, then he will miss his classwork.
  3. If Larry is sick, then he will miss his classwork.

We deduced the solution by combining the hypothesis of the first problem with the conclusion of the second statement. We also conclude that this could be a false statement.

Deductive logic

Deductive arguments are generally evaluated in terms of their validity and soundness.

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, whatever they may be, 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.

The following is an example of an argument that is valid, but not sound; a premise is false:

  1. Everyone who eats steak is a quarterback.
  2. John eats steak.
  3. Therefore, John is a quarterback.

The example's first premise is false (there are people who eat steak that 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.

The theory of deductive reasoning known as categorical or 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 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).

Hume's skepticism

Philosopher David Hume presented grounds to doubt deduction by questioning induction. Hume's problem of induction starts by suggesting that the use of even the simplest forms of induction simply cannot be justified by inductive reasoning itself. Moreover, induction cannot be justified by deduction either. Therefore, induction cannot be justified rationally. Consequentially, if induction is not yet justified, then deduction seems to be left to rationally justify itself – an objectionable conclusion to Hume.

Hume did not provide a strictly rational solution per se. He simply explained that we do induce, and that it is useful that we do so, but not necessarily justified. Certainly we must appeal to first principles of some kind, including laws of thought.

See also

References

Further reading


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