English translation for duplex negatio and usage examples from the best sources

It is a movement born in a double negation: the rejection of rejection, through the proud, self-conscious union of those who have been defined as belonging to an excluded group.

Both question how a double negation is something other than a positive universal.

Dignāga argues for a way in which language can be construed to signify particulars, through a double negation that excludes that which contradicts the referent object while eliminating the philosophical commitment to universals.

Readers will be familiar with the logical formulation, often encountered in Zen but ubiquitous in Buddhism generally, that A is A; and A is not-A; therefore, A is A. There is a double negation in evidence here: an individual is an individual, and yet an individual is not individual unless one stands opposed to other individuals.

In his helpful remarks on the Eastern intellectual history behind the Kyoto School's idea of absolute nothingness, Whalen Lai writes: "Of the eighteen forms of śūnyatā (emptiness) in Mahāyāna, two have captured the Chinese imagination more than the others: śūnyatā-śūnyatā (the emptiness of emptiness, a double negation that reaffirms the real); and atyanta-śūnyatā … or 'ultimate emptiness'.

It is worth noting that IZF, like many but not all constructive systems, admits a double negation interpretation of the corresponding classical theory ZF; we might think of IZF as a weakened version of ZF from the classical standpoint, but in its own terms it is the theory of a larger, more complex realm in which a copy of the classical universe of set theory is embedded.

Some of Brouwer's later followers even studied intuitionistic type theory (see below), which differs from classical type theory only by the absence of a single axiom (double negation):∀x ∊ Ω(¬¬x ⊃ x), where Ω is the type of truth-values.

As a result, both double negation introduction and double negation elimination are valid in the sense that their truth is supported by every state of any model.

An in-depth investigation of a hierarchy of double negation principles can be found in Kamide 2013.

"There a was period of time where double negation... was the way to negate things, just as it is in French today," said Plotkin.

Not all systems of propositional logic accept a biconditional law of double negation (LDN), ¬ ≡ Φ.