Denying a Conjunct

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

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

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.
Therefore, not q.
Not both p and q.
Therefore, not p.


A conjunctive statement, or conjunction for short, is a statement of the "both-and" form. For example, "It's both rainy and sunny" is a conjunction. The conjuncts of a conjunction are its component statements, so the conjuncts of the example are "It's rainy" and "It's sunny".

To deny or negate a conjunction is to claim that at least one of the conjuncts is false, but it leaves open the possibility that both may be false. To return to the example, denying 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 negated conjunction 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―however, see the Exception section.

Given that there are two conjuncts in any binary conjunction, 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.1


As mentioned in the Exposition section, above, the form of Denying a Conjunct is non-validating, which means that not every argument of that form is valid. This doesn't mean that every argument that denies a conjunct is invalid; rather, it means that some arguments of that form are invalid. There are arguments of that form that are formally valid, but all of them are such that the second premiss alone implies the conclusion, that is, the immediate inference from the second premiss to the conclusion is valid. Therefore, before pronouncing an instance of denying the conjunct invalid, check to see whether the second premiss implies the conclusion.



  1. See, also, Howard Pospesel, Introduction to Logic: Propositional Logic (Third Edition) (Prentice Hall, 1998), p. 67.
  2. William L. Reese, Dictionary of Philosophy and Religion (Humanities, 1980), see under "Fallacies".