Deductive reasoning, also deductive logic, logical deduction is the
process of reasoning from one or more statements (premises) to reach a
logically certain conclusion.[1]
Contents 1 Simple example
2
2.1 Modus ponens 2.2 Modus tollens 2.3 Law of syllogism 3
Simple example[edit] An example of an argument using deductive reasoning: All men are mortal. (First premise) Socrates is a man. (Second premise) Therefore, Socrates is mortal. (Conclusion) 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."
P → Q displaystyle Prightarrow Q ) and as second premise the antecedent ( P displaystyle P ) of the conditional statement. It obtains the consequent ( Q displaystyle Q ) of the conditional statement as its conclusion. The argument form is listed below: P → Q displaystyle Prightarrow Q (First premise is a conditional statement) P displaystyle P (Second premise is the antecedent) Q displaystyle Q (Conclusion deduced is the consequent) In this form of deductive reasoning, the consequent ( Q displaystyle Q ) obtains as the conclusion from the premises of a conditional statement ( P → Q displaystyle Prightarrow Q ) and its antecedent ( P displaystyle P ). However, the antecedent ( P displaystyle P ) cannot be similarly obtained as the conclusion from the premises of the conditional statement ( P → Q displaystyle Prightarrow Q ) and the consequent ( Q displaystyle Q ). Such an argument commits the logical fallacy of affirming the consequent. The following is an example of an argument using modus ponens: If an angle satisfies 90° < A displaystyle A < 180°, then A displaystyle A is an obtuse angle. A displaystyle A = 120°. A displaystyle A is an obtuse angle. Since the measurement of angle A displaystyle A is greater than 90° and less than 180°, we can deduce from the conditional (if-then) statement that A displaystyle A is an obtuse angle. However, if we are given that A displaystyle A is an obtuse angle, we cannot deduce from the conditional statement that 90° < A displaystyle A < 180°. It might be true that other angles outside this range are
also obtuse.
Modus tollens[edit]
Main article: Modus tollens
P → Q displaystyle Prightarrow Q ) and the negation of the consequent ( ¬ Q displaystyle lnot Q ) and as conclusion the negation of the antecedent ( ¬ P displaystyle lnot P ). In contrast to modus ponens, reasoning with modus tollens goes in the opposite direction to that of the conditional. The general expression for modus tollens is the following: P → Q displaystyle Prightarrow Q . (First premise is a conditional statement) ¬ Q displaystyle lnot Q . (Second premise is the negation of the consequent) ¬ P displaystyle lnot P . (Conclusion deduced is the negation of the antecedent) The following is an example of an argument using modus tollens: If it is raining, then there are clouds in the sky. There are no clouds in the sky. Thus, it is not raining. Law of syllogism[edit] Main article: 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. Here is the general form: P → Q displaystyle Prightarrow Q Q → R displaystyle Qrightarrow R Therefore, P → R displaystyle Prightarrow R . The following is an example: If Larry is sick, then he will be absent. If Larry is absent, then he will miss his classwork. Therefore, if Larry is sick, then he will miss his classwork. 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. Another example is the transitive property of equality which can be stated in this form: A = B displaystyle A=B . B = C displaystyle B=C . Therefore, A = C displaystyle A=C .
Deductive arguments are 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 are true. An argument can be “valid” even if one or more of its 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”: Everyone who eats carrots is a quarterback. John eats carrots. Therefore, John is a quarterback. The example’s first premise is false – there are people who eat
carrots who are not quarterbacks – but the conclusion would
necessarily be true, if the premises were true. In other words, it is
impossible for the premises to be true and the conclusion false.
Therefore, the argument is “valid”, but not “sound”. False
generalizations – such as "Everyone who eats carrots is a
quarterback" – are often used to make unsound arguments. The fact
that there are some people who eat carrots but are not quarterbacks
proves the flaw of the argument.
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.
This section needs expansion. You can help by adding to it. (January 2015)
Abductive reasoning
Analogical reasoning
References[edit] ^ Sternberg, R. J. (2009). Cognitive Psychology. Belmont, CA: Wadsworth. p. 578. ISBN 978-0-495-50629-4. ^ Evans, Jonathan St. B. T.; Newstead, Stephen E.; Byrne, Ruth M. J., eds. (1993). Human Reasoning: The Psychology of Deduction (Reprint ed.). Psychology Press. p. 4. ISBN 9780863773136. Retrieved 2015-01-26. In one sense [...] one can see the psychology of deductive reasoning as being as old as the study of logic, which originated in the writings of Aristotle. Further reading[edit] Vincent F. Hendricks, Thought 2 Talk: A Crash Course in Reflection and
Expression, New York: Automatic Press / VIP, 2005,
ISBN 87-991013-7-8
Philip Johnson-Laird, Ruth M. J. Byrne, Deduction, Psychology Press
1991, ISBN 978-0-86377-149-1
Zarefsky, David, Argumentation: The Study of Effective
External links[edit] Wikiquote has quotations related to: Deductive reasoning Look up deductive reasoning in Wiktionary, the free dictionary. Wikiversity has learning resources about Deductive Logic
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