Definition:Universal Negative

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A universal negative is a categorical statement of the form:

No $S$ is $P$

where $S$ and $P$ are predicates.

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

$\forall x: \map S x \implies \neg \map P x$

Its meaning can be amplified in natural language as:

Given any arbitrary object, if it has the property of being $S$, then it does not have the quality of being $P$.

Set Theoretic interpretation of Universal Negative

The universal negative $\forall x: \map S x \implies \neg \map P x$ can be expressed in set language as:

$\set {x: \map S x} \implies \set {x: \map P x} = \O$

or, more compactly:

$S \subseteq \map \complement P$

Also denoted as

Traditional logic abbreviated the universal negative as $\mathbf E$.

Thus, when examining the categorical syllogism, the universal negative $\forall x: \map S x \implies \neg \map P x$ is often abbreviated:

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

Linguistic Note

The abbreviation $\mathbf E$ for a universal negative originates from the first vowel in the Latin word nEgo, meaning I deny.

Also see