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*Deductive reasoning*

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Deductive reasoning

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**Deductive reasoning**, also

**deductive logic**,

**logical deduction**is the process of reasoning from one or more statements (premises) to reach a logically certain conclusion.BOOK, Sternberg, R. J., Cognitive Psychology, 2009, Wadsworth, Belmont, CA, 978-0-495-50629-4, 578, Deductive reasoning goes in the same direction as that of the conditionals, and links premises with conclusions. If all premises are true, the terms are clear, and the rules of deductive logic are followed, then the conclusion reached is necessarily true.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 which hold over the entirety of a closed domain of discourse, narrowing the range under consideration until

*only*the conclusion(s) is left. In inductive reasoning, the conclusion is reached by generalizing or extrapolating from specific cases to general rules, i.e., there is epistemic uncertainty. However, the inductive reasoning mentioned here is not the same as induction used in mathematical proofs â€“ mathematical induction is actually a form of deductive reasoning.Deductive reasoning differs from abductive reasoning by the direction of the reasoning relative to the conditionals. Deductive reasoning goes in the ''same direction as that of the conditionals, whereas abductive reasoning goes in the opposite direction to that of the conditionals.

## Simple example

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)

## Reasoning with modus ponens, modus tollens, and the law of syllogism

### Modus ponens

Modus ponens (also known as "affirming the antecedent" or "the law of detachment") is the primary deductive rule of inference. It applies to arguments that have as first premise a conditional statement (P rightarrow Q) and as second premise the antecedent (P) of the conditional statement. It obtains the consequent (Q) of the conditional statement as its conclusion. The argument form is listed below:- P rightarrow Q (First premise is a conditional statement)
- P (Second premise is the antecedent)
- Q (Conclusion deduced is the consequent)

- If an angle satisfies 90Â° < A < 180Â°, then A is an obtuse angle.
- A = 120Â°.
- A is an obtuse angle.

### Modus tollens

Modus tollens (also known as "the law of contrapositive") is a deductive rule of inference. It validates an argument that has as premises a conditional statement (P rightarrow Q) and the negation of the consequent (lnot Q) and as conclusion the negation of the antecedent (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 rightarrow Q. (First premise is a conditional statement)
- lnot Q. (Second premise is the negation of the consequent)
- lnot P. (Conclusion deduced is the negation of the antecedent)

- 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

In proposition logic 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 rightarrow Q
- Q rightarrow R
- Therefore, P rightarrow R.

- 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.

- A = B.
- B = C.
- Therefore, A = C.

## Validity and soundness

(File:Argument terminology used in logic.png|thumb|400px|Argument terminology)Deductive arguments are evaluated in terms of their*validity*and

*soundness*.An argument is â€œ

**valid**â€ 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 â€œvalidâ€ even if one or more of its premises are false.An argument is â€œ

**sound**â€ 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 â€œvalidâ€, but not â€œsoundâ€:

- Everyone who eats carrots is a quarterback.
- John eats carrots.
- Therefore, John is a quarterback.

## History

{{expand section|date=January 2015}}Aristotle started documenting deductive reasoning in the 4th century BC.BOOK, Evans, Jonathan St. B. T., Newstead, Stephen E., Byrne, Ruth M. J., Ruth M. J. Byrne, Human Reasoning: The Psychology of Deduction,weblink Reprint, Psychology Press, 1993, 4, 9780863773136, 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.,## See also

{{div col|colwidth=30em}}- Abductive reasoning
- Analogical reasoning
- Argument (logic)
- Correspondence theory of truth
- Decision making
- Decision theory
- Defeasible reasoning
- Fallacy
- Fault Tree Analysis
- Geometry
- Hypothetico-deductive method
- Inference
- Inquiry
- Legal syllogism
- Logic and rationality
- Logical consequence
- Mathematical logic
- Natural deduction
- Peirce's theory of deductive reasoning
- Propositional calculus
- Retroductive reasoning
- Scientific method
- Subjective logic
- Theory of justification

## References

{{reflist}}## Further reading

- 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 Reasoning Parts I and II*, The Teaching Company 2002 - Bullemore, Thomas, The Pragmatic Problem of Induction.

## External links

{{wiktionary|deductive reasoning}}{{Wikiversity|Deductive Logic}}- {{PhilPapers|category|deductive-reasoning}}
- {{InPho|idea|636}}
- IEP, ded-ind, Deductive reasoning,

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