Definition:Universal Negative

From ProofWiki
Jump to navigation Jump to search


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 $\forall x: S \left({x}\right) \implies \neg P \left({x}\right)$ can be expressed in set language as:

$\left\{{x: S \left({x}\right)}\right\} \implies \left\{{x: \neg P \left({x}\right)}\right\} = \varnothing$

or, more compactly:

$S \subseteq \complement \left({P}\right)$

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