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)

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:

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

Exception:

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.

Exposure:

• Presumably, it is the similarity between the two argument forms―Conjunctive Argument and Denying a Conjunct―that is the psychological source of the fallacy. In other words, people confuse the fallacious form―Denying a Conjunct―with the validating form―Conjunctive Argument.

Moreover, Denying a Conjunct is likely to seem more plausible when we have independent reasons for thinking that at least one of the two conjuncts is true. Suppose that we add to Denying a Conjunct the further disjunctive premiss:

Either p or q.

The resulting argument form is validating. So, when it is reasonable to suppose that the disjunctive premiss has been suppressed, the argument will be a valid enthymeme, rather than fallacious.

• The confusing alias "The Fallacy of the Disjunctive Syllogism" comes from W. L. Reese's Dictionary of Philosophy and Religion2. Here is the complete entry:
The Fallacy of the Disjunctive Syllogism: affirming and denying. The disjunctive syllogism utilizes a stronger sense of either-or, that of mutual exclusion. When the intention of the premises is to assert: Not both A and B, it is clear that one cannot argue "Not A. Therefore, B." … For example, on the dinner menu one may have a choice of green beans or cauliflower. It would not do to argue:

"Either green beans or cauliflower.
I do not care for green beans.
Therefore, I must take cauliflower."

Disjunctive Syllogism (D.S.) is the name usually given to a validating form of argument, so to call it a "fallacy" is puzzling, though perhaps it might be used as the name of a fallacious form of argument similar to D.S. The example at the end of the entry is actually an instance of D.S. and, therefore, valid. Of course, there does seem to be something wrong with the argument, but this is because Reese has misrepresented its form. Specifically, the first premiss is not fully expressed, since neither "green beans" nor "cauliflower" are propositions. The argument is more accurately analyzed as follows:

"I may take either green beans or cauliflower (but not both).
I do not care for green beans.
Therefore, I may take cauliflower."

If the premisses are true, then the conclusion must be true, so the argument is valid. However, the fact that you may select the cauliflower does not obligate you to do so, and you may choose neither.

Logicians usually distinguish two senses of "either-or":

1. Inclusive: The logically weaker sense in which at least one disjunct is true, and both may be true. Reese calls this sense: "alternation".
2. Exclusive: The logically stronger sense in which exactly one disjunct is true, and both cannot be true. Reese calls this sense: "disjunction".

However, D.S. is a validating form of argument for both senses.

The exclusive sense of "either p or q" is equivalent to "either p or q but not both p and q", where "either…or" is inclusive. Reese seems to have mistaken the exclusive form of "either…or" for "not both…and", by somehow dropping the "either…or". For these reasons, the alias should be avoided.

Notes:

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".