Philosophy Index

Joint Denial

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 sometimes known as a Peirce arrow, after the logician Charles Sanders Peirce, who defined the operator. It is also known as Nicod‘s dagger after Jean Nicod, who showed that propositional logic could be done with a single rule and a single operator, based either on the joint denial or Sheffer‘s stroke. 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)

Truth values

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

↓ as the only operator

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))