Improper Transposition

Taxonomy: Logical Fallacy > Formal Fallacy > Fallacy of Propositional Logic > Improper Transposition

Alias: Negating Antecedent and Consequent

Example:

Tavis Smiley (interviewer): How are you going to respond to folks on the campaign trail when they ask what qualifies you to be the commander-in-chief given that you have not served in the country's military?

Al Sharpton (interviewee): I think that just because one served in the military does not make one a competent commander-in-chief.

Analysis

Forms Similar Validating Forms
(Transposition)
If p then q.
Therefore, if not-p then not-q.		 If p then q.
Therefore, if not-q then not-p.
If not-p then not-q.
Therefore, if p then q.		 If not-p then not-q.
Therefore, if q then p.
Example Counter-Example
If there's a fire, then there's smoke.
Therefore, if there's no fire, then there's no smoke.		 If we guillotine the king, then he will die.
Therefore, if we don't guillotine the king, then he won't die.

Exposition:

"Transposition" is the name of a type of immediate inference of propositional logic involving conditional statements―see the Similar Validating Forms in the table, above. A conditional statement is an "if-then" statement, that is, one that gives a condition―the part after "if"―for the truth of something―the part after "then". The part after "if" is called "the antecedent of the conditional statement", while the part after "then" is called "the consequent". For example, "if this is Tuesday then I have logic class" is a conditional statement, "This is Tuesday" is its antecedent, and "I have logic class" is its consequent.

In a proper transposition, the antecedent and consequent of the conditional premiss are switched and negated in the conclusion. An improper transposition is, of course, an attempted transposition done incorrectly. Specifically, in the fallacy of improper transposition, the antecedent and consequent are negated, but not switched―see the Forms and Example in the table above.

Proper transposition is a validating form of argument, that is, every argument of the form of a proper transposition is valid. However, improper transposition is not validating―see the Counter-Example above, which is an invalid example of the fallacy with a true premiss and false conclusion. In fact, a properly transposed conditional statement is logically equivalent to the transposed sentence, so that either one can be validly inferred from the other.

Exposure:

The forms of improper transposition are similar enough to those of proper transposition to be confused with them, especially since transposition is a complex logical transformation of a conditional statement. This fallacy bears the same type of similarity to Denying the Antecedent as Commutation of Conditionals bears to Affirming the Consequent. As is the case for all of these conditional fallacies, an improper transposition seems most plausible when the converse of the premiss is also true. In such a case, an improper transposition could be a valid enthymeme.

Analysis of the Example:

Smiley says that some people will raise the objection: "If ['given that'] someone has not served in the military then he is not qualified to be Commander-in-Chief". This is equivalent, by proper transposition, to: "If someone is qualified to be Commander-in-Chief then he has served in the military." Since Sharpton had not served in the military, this would imply that he is not qualified for the Presidency, by Modus Tollens.

However, Sharpton says that he will respond to anyone who raises Smiley's objection by denying: "If ['just because'] someone has served in the military then he is qualified to be Commander-in-Chief". This is the improper transposition of Smiley's objection, and is not logically equivalent to it. To see this, notice that Smiley's objection is at least plausible, for it says that military service is a necessary condition for being Commander-in-Chief. However, Sharpton denies an easily refuted claim, namely, that military service is a sufficient condition for being Commander-in-Chief. It isn't a sufficient condition because there are other conditions required to fill the office of President.

Note that the negation in Sharpton's conditional has wide scope, that is, over the entire conditional. Otherwise, the scope would be on the consequent of the conditional, producing the implausible claim: "If someone has served in the military then he is not qualified to be Commander-in-Chief".

So, Sharpton did not answer the objection raised by Smiley, but pulled a logical "bait and switch" by improperly transposing it into an easily refuted claim. The two claims are similar enough that most people will not realize what Sharpton has done, and it is even possible that Sharpton himself did not realize it.

Sources:

Acknowledgments:

Thanks to Michael Koplow for the Sharpton example, to Mark Meyers for criticizing the original analysis of it, and to John R. Owens for a criticism of the wording of the analysis.