Definition:Universal Negative
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Definition
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:
- $\map {\mathbf E} {S, P}$
Also see
- Results about the universal negative can be found here.
Linguistic Note
The abbreviation $\mathbf E$ for a universal negative originates from the first vowel in the Latin word nEgo, meaning I deny.
Sources
- 1965: E.J. Lemmon: Beginning Logic ... (previous) ... (next): Chapter $4$: The Predicate Calculus $2$: $4$ The Syllogism
- 1973: Irving M. Copi: Symbolic Logic (4th ed.) ... (previous) ... (next): $4$: Propositional Functions and Quantifiers: $4.1$: Singular Propositions and General Propositions
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): categorical proposition
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): syllogism
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): categorical proposition
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): syllogism