Negative conclusion from affirmative premises

{{Short description|In logic, a type of syllogistic fallacy}}

Negative conclusion from affirmative premises is a syllogistic fallacy committed when a categorical syllogism has a negative conclusion yet both premises are affirmative. The inability of affirmative premises to reach a negative conclusion is usually cited as one of the basic rules of constructing a valid categorical syllogism.

Statements in syllogisms can be identified as the following forms:

• a: All A is B. (affirmative)
• e: No A is B. (negative)
• i: Some A is B. (affirmative)
• o: Some A is not B. (negative)

The rule states that a syllogism in which both premises are of form a or i (affirmative) cannot reach a conclusion of form e or o (negative). Exactly one of the premises must be negative to construct a valid syllogism with a negative conclusion. (A syllogism with two negative premises commits the related fallacy of exclusive premises.)

Example (invalid aae form):

:Premise: All colonels are officers.

:Premise: All officers are soldiers.

:Conclusion: Therefore, no colonels are soldiers.

The aao-4 form is perhaps more subtle as it follows many of the rules governing valid syllogisms, except it reaches a negative conclusion from affirmative premises.

Invalid aao-4 form:

:All A is B.

:All B is C.

:Therefore, some C is not A.

This is valid only if A is a proper subset of B and/or B is a proper subset of C. However, this argument reaches a faulty conclusion if A, B, and C are equivalent.{{cite book |title=The use of words in reasoning |author=Alfred Sidgwick |year=1901 |publisher=A. & C. Black |url=https://archive.org/details/useofwordsinreas00sidgiala |pages=[https://archive.org/details/useofwordsinreas00sidgiala/page/297 297]–300}}{{cite web |page=16 |title=Equivalence of syllogisms |author=Fred Richman |date=July 26, 2003 |publisher=Florida Atlantic University |url=http://www.math.fau.edu/richman/docs/syllog-4.pdf |url-status=dead |archive-url=https://web.archive.org/web/20100619151753/http://math.fau.edu/Richman/Docs/syllog-4.pdf |archive-date=June 19, 2010 }} In the case that A = B = C, the conclusion of the following simple aaa-1 syllogism would contradict the aao-4 argument above:

:All B is A.

:All C is B.

:Therefore, all C is A.