Commutation of Conditionals
- The Fallacy of the Consequent (Aristotle; see the History section, below)
- Converting a Conditional
|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.
|If it's raining then the streets are wet.
Therefore, if the streets are wet then it's raining.
|If Hillary is President then she is over 35.
Therefore, if Hillary is over 35 then she is President.
You may be familiar with commutation from the commutative laws of algebra, namely, that x + y = y + x and xy = yx, for any numbers x and y. The relevant meaning of the word "commutation" is "exchange", as the variables x and y exchange positions from the left side of the equations to the right and vice versa.
There are similar commutative laws in propositional logic that apply to such propositional connectives as conjunction and disjunction, that is, "and" and "or", respectively. For instance, the law for conjunction is that exchanging the conjuncts of a conjunction produces a logically equivalent statement. In other words, "p and q" is logically equivalent to "q and p".
Thus, to commute a compound statement is to switch its propositional components. For example, commuting the compound conjunction: "Today is Sunday and it's raining" is switching its components―"Today is Sunday" and "It's raining"―to get the sentence: "It's raining and today is Sunday."
Commutation is a validating form of immediate inference for some truth-functional connectives, such as conjunction and disjunction. However, the connective "if-then" that occurs in conditional statements is not commutable, that is, the commutation of a conditional statement is not as a general rule logically equivalent to the original statement. For example, "If today is Sunday then it's raining" does not say the same thing as "If it's raining then today is Sunday." Also see the Counter-Example in the table, above.
An argument commits the fallacy of Commutation of Conditionals if its premiss is a conditional statement and its conclusion is the commutation of that conditional statement. However, see the Exposure, below, for an exception.
- As explained in the Exposition, above, commuting a conditional does not produce a logically equivalent statement as a general rule. However, there are exceptions; for instance, if the component statements are themselves logically equivalent, then the commuted conditional statements will be equivalent. Therefore, before concluding that a given argument commits the fallacy of Commuting a Conditional, it is necessary to check whether it is an exception to the general rule that a conditional statement does not imply its commutation.
- Why would someone make the mistake of commuting a conditional proposition? One possible source of such an error is the fact that all other familiar binary connectives, such as "and", "or" and "if and only if", are commutative. Another possible source of confusion is the similar validating form of inference known as "contraposition"―see the table, above. Contraposition is similar to commutation in that it involves switching the antecedent and consequent of a conditional proposition, but in addition each is negated.
- For some reason, it's common mathematical practise to state definitions as conditional propositions, rather than biconditional propositions. This may occur when the unstated direction of the biconditional proposition is thought to be obvious. For this reason, it usually would not be a mistake to commute such a definition.
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.
For this reason, I have included "Fallacy of the Consequent" in the list of aliases for this fallacy.
- Aristotle, On Sophistical Refutations, 5. 167b1-8.
- Robert Audi (General Editor), The Cambridge Dictionary of Philosophy (Second Edition) (2001), p. 316.
- Thomas P. Kiernan (Editor), Aristotle Dictionary (Philosophical Library, 1962), see "Consequent, Fallacy of".
Acknowledgment: Thanks to Martina Rudic for criticism of the previous Example.