Definition:Particular Negative
Jump to navigation
Jump to search
Definition
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 $\exists x: \map S x \land \neg \map P x$ can be expressed in set language as:
- $\set {x: \map S x} \cap \set {x: \neg \map P x} \ne \O$
or, more compactly:
- $S \cap \map \complement P \ne \O$
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:
- $\map {\mathbf O} {S, P}$
Also see
- Results about the particular negative can be found here.
Linguistic Note
The abbreviation $\mathbf O$ for a particular negative originates from the second 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