Denying a Conjunct

Taxonomy: Logical Fallacy > Formal Fallacy > Propositional Fallacy > Denying a Conjunct

Alias: The Fallacy of the Disjunctive Syllogism (see Exposure)

Forms
Not both p and q.
Not p.
Therefore, q.
Not both p and q.
Not q.
Therefore, p.
Example Counter-Example
It isn't both sunny and overcast.
It isn't sunny.
Therefore, it's overcast.
It isn't both raining and snowing.
It isn't raining.
Therefore, it's snowing.
Similar Validating Forms
(Conjunctive Argument)
Not both p and q.
p.
Therefore, not q.
Not both p and q.
q.
Therefore, not p.

Exposition:

To negate a conjunctive proposition is to claim that at least one of the conjuncts is false, but it leaves open the possibility that both are false. For example, "It's rainy and it's sunny" is a conjunctive proposition; negating it produces: "It's not both rainy and sunny." For this to be true it has to be the case that either it's not rainy, it's not sunny, or both.

So, if we know that one of the conjuncts of a negative conjunctive proposition is true, we may validly infer that the other is false by Conjunctive Argument―see Similar Validating Forms, above. In contrast, if we know that one of the conjuncts is false, we cannot validly infer from that information alone that the other is true, since it may be false as well. To do so anyway would be to commit the fallacy of denying a conjunct.

Given that there are two conjuncts in any binary conjunctive proposition, there are two forms of denying a conjunct, depending upon which conjunct is denied―see the table, above. The Example given in the table is an example of the first form of denying a conjunct; an example of the second form would simply deny the second conjunct and conclude the first. The Counter-Example in the table is also an instance of the first form, but its point is to show that denying a conjunct is not a validating form of argument, since the Counter-Example itself is obviously invalid, as it might be neither raining nor snowing.

Exposure:

Sources: