Pronunciation: /dɪˈdʌk ʃən/ ?

Deduction is a form of reasoning that uses premises that are agreed upon to support a conclusion.[1] In mathematics, the premises are called axioms and theorems.

If the premises on which deduction is based are true, and if a deductive argument is valid, then the conclusion must be true. If the premises are all true and the deduction is valid, then the argument is sound. If any of the premises are false, or if the deduction is invalid, then the argument is unsound.

If any of the premises are false, then the conclusion may or may not be true. Look at table 2. The first premise, 'All humans are mammals,' is correct. The second premise, 'All mammals have toes,' is incorrect. Hooved animals such as horses do not have toes. Yet the conclusion, 'All humans have toes,' is correct. The conclusion is true, but the argument is unsound.


All men are mortal.Premise
Socrates is a man.Premise
Therefore Socrates is mortal.Conclusion
Table 1: Aristotle's example of deduction.

All humans are mammals.Correct premise
All mammals have toes.Incorrect premise
Therefore all humans have toes.Conclusion
Table 2: Example of unsound deduction.

Related Words

  • Assumption: A premise that is assumed to be true without proof.
  • Axiom: A mathematical statement that is taken to be true without proof.
  • Deductive: Based on deduction - deductive reasoning.
  • Hypothesis: A claim that is proposed to be true, but has not been proved.
  • Conjecture: A claim that is consistent with known data and is believed to be true, but has not been proved.
  • Postulate: A premise that is assumed to be true without proof.
  • Premise: An axiom or previously proved theorem used to support a conclusion.
  • Theorem: A claim that has already been proved.


  1. deduction. WordNet. Princeton University. (Accessed: 2011-01-08).
  2. Jevons, W. Stanley. Logic, pp 12-14. American Book Company, 189?. (Accessed: 2010-01-22).
  3. Bain, Alexander. Logic, Part First, Deduction. Longmans, Green, & Co., 1879. (Accessed: 2010-01-22).

More Information

  • O'Connor, J J and Robertson, E F. Aristotle. School of Mathematics and Statistics, University of St Andrews, Scotland. 2009-03-12.

Printed Resources

Cite this article as:

Deduction. 2010-01-22. All Math Words Encyclopedia. Life is a Story Problem LLC.


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Revision History

2010-01-22: Added "References" (McAdams, David.)
2009-01-27: Added second paragraph to description. Added example 2 (McAdams, David.)
2008-07-10: Added conjecture to the list of related words (McAdams, David.)
2008-04-14: Initial version (McAdams, David.)

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