Affirming a disjunct

Affirming a disjunct is a logical fallacy, committed by an invalid argument form “P or Q, P, therefore not Q”.

The affirming the consequent fallacy may be expressed formally as follows:

α ∨ β, α ∴ ¬β

The fallacy occurs when one assumes that when one term within a binary disjunction is true, then the other must be false. This is not the case in the default meaning of a disjunction, ∨, which is an inclusive disjunction—meaning that both connected terms may be true.

Affirming a disjunct example

John is a liar or he is a coward.
John is a liar.
Therefore, John is not a coward.

In this case, “or”, is not meant to be exclusive. The English is a bit ambiguous, but we assume that John may be both a liar and a coward, in which case a fallacy is said.

Consider a more obvious example:

These bones belong to a rat, or a rodent.
These bones belong to a rat.
Therefore, these bones do not belong to a rodent.

This example is clearly in error, since a rat is, in fact, a rodent. The “or” in this case must be inclusive.

Exclusive disjunction

Some logical systems make use of a seperate symbol for an exclusive disjunction, (usually ⊕). In the case of an exclusive disjunction, the form α ⊕ β, α ∴ ¬β is actually valid.

This is equivalent to the English “either/or”. So, the first example could be restated as:

Either John is a liar, or he is a coward.
John is a liar.
Therefore, John is not a coward.

This would not be an example of a fallacy, since the terms “either/or” usually indicate an exclusive disjunction, where only one of the two propositions is posible.