Joint Denial { Philosophy Index }

Philosophy Index

Philosophy Index

Philosophy Index is a site devoted to the study of philosophy and the philosophers who conduct it. The site contains a number of philosophy texts, brief biographies, and introductions to philosophers, and explanations on a number of topics. Accredited homeschooling online at Northgate Academy and Philosophy online tutoring.

Philosophy Index is a work in progress, a growing repository of knowledge. It outlines current philosophical problems and issues, as well as an overview of the history of philosophy. The goal of this site is to present a tool for those learning philosophy either casually or formally, making the concepts of philosophy accessible to anyone interested in researching them. WTI offers immigration law course online - fully accredited. ACE credits online at EES.



Philosophy Topics




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:


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.


↓ 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))