Definition:Particular Negative

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A particular negative is a categorical statement of the form:

Some $S$ is not $P$

where $S$ and $P$ are predicates.

In the language of predicate logic, this can be expressed as:

$\exists x: \map S x \land \neg \map P x$

Its meaning can be amplified in natural language as:

There exists at least one object with the property of being $S$ which does not have the quality of being $P$.

Set Theoretic interpretation of Particular Negative

The particular negative $\exists x: S \left({x}\right) \land \neg P \left({x}\right)$ can be expressed in set language as:

$\left\{{x: S \left({x}\right)}\right\} \cap \left\{{x: \neg P \left({x}\right)}\right\} \ne \varnothing$

or, more compactly:

$S \cap \complement \left({P}\right) \ne \varnothing$

Also denoted as

Traditional logic abbreviated the particular negative as $\mathbf O$.

Thus, when examining the categorical syllogism, the particular negative $\exists x: \map S x \land \neg \map P x$ is often abbreviated:

$\map {\mathbf O} {S, P}$

Linguistic Note

The abbreviation $\mathbf O$ for a particular negative originates from the second vowel in the Latin word negO, meaning I deny.

Also see