The joint denial is a truth-functional operator in logic which is equivalent to the English wording “neither… nor”. It is used to state that both conjoined propositions are false.
For example, if both P and Q are false, or ¬P and ¬Q are both true:
P↓Q
This may be read as “neither P nor Q”.
In symbolic logic, the downwards arrow symbol ( ↓ ) is used to indicate a joint denial. This mark is known as a Peirce arrow, after the logician Charles Sanders Peirce, who defined the operator. Sometimes, the capitalized term NOR (meaning “not or”), or the ⊥ symbol, is used.
Often, logical systems do not include the ↓ symbol in their formal language. In these systems, a joint denial must be formulated using other operators. The following is an alternative expression of a joint denial:
(¬P ∧ ¬Q)
The following table illustrates the possible truth values of P↓Q, given each possible valuation of its terms, P and Q.
| P | Q | P|Q |
|---|---|---|
| T | T | F |
| T | F | F |
| F | T | F |
| F | F | T |
It is actually the case that a system of logic could be created using only the joint denial, as was demonstrated by Peirce.
| Operator | Peirce arrow equivalent |
|---|---|
| Negation (¬) | P↓P |
| Conjunction (∧) | (P↓P)↓(Q↓Q) |
| Disjunction (∨) | (P↓Q)↓(P↓Q) |
| Conditional (→) | ((P↓P)↓Q)↓((P↓P)↓Q)) |
