Definition:Square of Opposition/Categorical Statements
Definition
The square of opposition is a diagram whose purpose is to illustrate the relations between the various types of categorical statement.
- $\begin{xy} <-10em,0em>*+{\forall x: \map S x \implies \map P x} = "A", <10em,0em>*+{\forall x: \map S x \implies \neg \map P x} = "E", <-10em,-20em>*+{\exists x: \map S x \land \map P x} = "I", <10em,-20em>*+{\exists x: \map S x \land \neg \map P x} = "O", "A";"E" **@{-} ?<*@{<} ?>*@{>} ?*!/^.8em/{\text{Contraries}}, "A";"I" **@{-} ?>*@{>} ?*!/^3.2em/{\text{Subimplicant}}, "A";"O" **@{-} ?<*@{<} ?>*@{>} ?*!/^4em/{\text{Contradictories}}, "I";"E" **@{-} ?<*@{<} ?>*@{>} ?*!/_4em/{\text{Contradictories}}, "I";"O" **@{-} ?<*@{<} ?>*@{>} ?*!/_.8em/{\text{Subcontraries}}, "E";"O" **@{-} ?>*@{>} ?*!/^-3.2em/{\text{Subimplicant}}, \end{xy}$
This therefore illustrates the relations:
- All $S$ are $P$ is contrary to No $S$ are $P$
- All $S$ are $P$ is contradictory to Some $S$ are not $P$
- Some $S$ are $P$ is contradictory to No $S$ are $P$
- Some $S$ are $P$ is subimplicant to All $S$ are $P$
- Some $S$ are not $P$ is subimplicant to No $S$ are $P$
- Some $S$ are $P$ is subcontrary to Some $S$ are not $P$
where $S$ and $P$ are predicates.
Vacuous Terms
The traditional treatment of the categorical syllogism makes the assumption that no term is vacuous.
However, from the point of view of the full predicate logic, this assumption may not be valid.
Note that if $S$ is empty, then the square of opposition no longer holds.
Although All $S$ are $P$ is vacuously true for such an empty universe, Some $S$ are $P$ is not.
Thus Some $S$ are $P$ is no longer subimplicant to All $S$ are $P$.
Similarly, as Some $S$ are not $P$ is also false, it follows that All $S$ are $P$ and Some $S$ are not $P$ are no longer subcontrary.
Also see
- Universal Affirmative and Particular Negative are Contradictory
- Particular Affirmative and Universal Negative are Contradictory
- Universal Affirmative and Universal Negative are Contrary iff First Predicate is not Vacuous
- Particular Affirmative and Particular Negative are Subcontrary iff First Predicate is not Vacuous
- Universal Affirmative implies Particular Affirmative iff First Predicate is not Vacuous
- Universal Negative implies Particular Negative iff First Predicate is not Vacuous
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