Philosophy Index

Alternative Denial

The alternative denial is a truth-functional operator in logic which is equivalent to the English wording “not both“ and”. It is used to deny a conjunction, stating that at least one of the connected propositions is false.

For example, if one of P and Q are false, or if P and Q cannot both be true, we may say formally:

P|Q

This may be read as “not both P and Q” or “P and Q are mutually exclusive”.

In symbolic logic, the vertical line symbol ( | ) is used to indicate an alternative denial. This mark is known as a Sheffer stroke, after the logician Henry M. Sheffer, who defined the operator for use in Boolean algebra. Sometimes, the capitalized term NAND (meaning “not and”), or the upwards arrow symbol ( ↑ ) is used, in contrast to the NOR operation, demonstrated by the Peirce arrow ( ↓ ).

Some, or even most, logical systems do not include the | symbol in their formal language. In these systems, an alternative denial must be formulated using other operators. The following is an alternative expression of an altenrative denial:

¬(P ∧ Q)

Truth values

The following table illustrates the possible truth values of P|Q, given each possible valuation of its terms, P and Q. Note that P|Q does not remove the possibility that both P and Q are false, only that P and Q are both true.

P Q P|Q
T T F
T F T
F T T
F F T

| as the only operator

It is actually the case that a system of logic could be created using only the alternative denial, and it is for this realization that Henry Sheffer is credited. The following table shows equivalent formulas for other operators using only the Sheffer stroke:

Operator Sheffer stroke equivalent
Negation (¬) P|P
Conjunction (∧) (P|Q)|(P|Q)
Disjunction (∨) (P|P)|(Q|Q)
Conditional (→) P|(P|Q)