# Fallacy of: the Undistributed Middle vs Denying the Antecedent

Primary Source: Levi, Edward | An Introduction to Legal Reasoning (2013 ed, not 1949), p. 3 Footnote 5.
2ndary Source: Weinreb. Legal Reason: The Use of Analogy in Legal Argument (1 ed 2005, but ∃ 2 ed 2016), p. 9 Footnote.

5The logical fallacy is the fallacy of the undistributed middle[,] or the fallacy of assuming the antecedent is true because the consequent has been affirmed.

[ Wikipedia : ] The fallacy is similar to affirming the consequent and denying the antecedent.

See my own tabulation below: I understand that 4 logically equals 5.
But is Wikipedia right: how does 2 logically equal 3? 2 lacks, but 3 contains, negation.

To see the similarity let's formulate the arguments as follows.

Affirming the consequent

``````1. if p then q
q
------------
p
``````

Denying the antecedent

``````2. if ~q then ~p
q
------------
p
``````

These are equivalent since 2 is logically equivalent to 1 (they are contrapositives).

Affirming the consequent is indeed equivalent to denying the antecedent.

However, undistributed middle is a different problem. When the middle term of a syllogism is undistributed, there is no link created between the major and minor premises. Accordingly, nothing follows beyond the plain statements of the premises themselves.

One example is AAA in the second figure, which is invalid. All P are M. All S are M. Thus: All S are P. The middle term M is not distributed in either premise, and the argument fails.