Definition:Universal Affirmative

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A universal affirmative is a categorical statement of the form:

Every $S$ is $P$

where $S$ and $P$ are predicates.

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

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

Its meaning can be amplified in natural language as:

Given any arbitrary object, if it has the property of being $S$, then it also has the quality of being $P$.

Set Theoretic interpretation of Universal Affirmative

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

$\left\{{x: S \left({x}\right)}\right\} \subseteq \left\{{x: P \left({x}\right)}\right\}$

or, more compactly:

$S \subseteq P$

Also denoted as

Traditional logic abbreviated the universal affirmative as $\mathbf A$.

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

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

Linguistic Note

The abbreviation $\mathbf A$ for a universal affirmative originates from the first vowel in the Latin word Affirmo, meaning I affirm.

Also see