Alias: The Fallacy of the Disjunctive Syllogism (see Exposure)
Negating a conjunction"not both", which is sometimes abbreviated as "nand"means that at least one of the conjuncts is false, but it leaves open the possibility that both conjuncts are false. So, if we know that one of the conjuncts 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 (Denying a Conjunct).
Presumably, it is the similarity between these 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 confusing alias "The Fallacy of the Disjunctive Syllogism" comes from W. L. Reese's Dictionary of Philosophy and Religion. 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:
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).
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":
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.