Commutation of Conditionals

Alias:
  • The Fallacy of the Consequent
  • Converting a Conditional
Type: Fallacy of Propositional Logic
Form Similar Validating Form (Contraposition)
If p then q.
Therefore, if q then p.
If p then q.
Therefore, if not-q then not-p.
Example Counter-Example
If James was a bachelor, then he was unmarried.
Therefore, if James was unmarried, then he was a bachelor.
If James was President, then he was over 35.
Therefore, if James was over 35, then he was President.

Exposure:

There is an important linguistic caveat to the application of this fallacy, namely, that it is common mathematical practise to state definitions as conditional propositions, rather than biconditional propositions. For instance:

A vector space V is finite-dimensional if it has a finite basis.

It would not be mistaken to infer from this definition that if a vector space is finite-dimensional then it has a finite basis.

History:

This is one of Aristotle's thirteen fallacies, from the language-independent group, also known as the "Fallacy of the Consequent". The closely-related Fallacy of Affirming the Consequent is often attributed to Aristotle, but his description of the fallacy sounds closer to commuting a conditional than affirming its consequent:

The refutation which depends upon the consequent arises because people suppose that the relation of consequence is convertible. For whenever, suppose A is, B necessarily is, they then suppose also that if B is, A necessarily is. This is also the source of the deceptions that attend opinions based on sense-perception. For people often suppose bile to be honey because honey is attended by a yellow colour: also, since after rain the ground is wet in consequence, we suppose that if the ground is wet, it has been raining; whereas that does not necessarily follow.

Sources:

  • Aristotle, On Sophistical Refutations, 5. 167b1-8.
  • Robert Audi (General Editor), The Cambridge Dictionary of Philosophy (Second Edition) (2001), p. 316.
  • Paul Halmos, Finite-Dimensional Vector Spaces, Springer-Verlag, 1958, p. 10.
  • Thomas P. Kiernan (Editor), Aristotle Dictionary (Philosophical Library, 1962), see "Consequent, Fallacy of".

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