Definition:Standard Instance of Categorical Syllogism
Definition
For each categorical syllogism, there are four figures:
- $\begin{array}{r|rl}
\text I & & \\ \hline \\ \text{Major Premise}: & \mathbf \Phi_1 & \tuple {M, P} \\ \text{Minor Premise}: & \mathbf \Phi_2 & \tuple {S, M} \\ \hline \\ \text{Conclusion}: & \mathbf \Phi_3 & \tuple {S, P} \\ \end{array} \qquad \begin{array}{r|rl} \text {II} & & \\ \hline \\ \text{Major Premise}: & \mathbf \Phi_1 & \tuple {P, M} \\ \text{Minor Premise}: & \mathbf \Phi_2 & \tuple {S, M} \\ \hline \\ \text{Conclusion}: & \mathbf \Phi_3 & \tuple {S, P} \\ \end{array}$
- $\begin{array}{r|rl}
\text {III} & & \\ \hline \\ \text{Major Premise}: & \mathbf \Phi_1 & \tuple {M, P} \\ \text{Minor Premise}: & \mathbf \Phi_2 & \tuple {M, S} \\ \hline \\ \text{Conclusion}: & \mathbf \Phi_3 & \tuple {S, P} \\ \end{array} \qquad \begin{array}{r|rl} \text {IV} & & \\ \hline \\ \text{Major Premise}: & \mathbf \Phi_1 & \tuple {P, M} \\ \text{Minor Premise}: & \mathbf \Phi_2 & \tuple {M, S} \\ \hline \\ \text{Conclusion}: & \mathbf \Phi_3 & \tuple {S, P} \\ \end{array}$
where $\mathbf \Phi_1$, $\mathbf \Phi_2$ and $\mathbf \Phi_3$ each denote one of the categorical statements $\mathbf A$, $\mathbf E$, $\mathbf I$ or $\mathbf O$.
A standard instance of a categorical syllogism is obtained by substituting $\mathbf A$, $\mathbf E$, $\mathbf I$ or $\mathbf O$ for each of $\mathbf \Phi_1$, $\mathbf \Phi_2$ and $\mathbf \Phi_3$ in one of the above figures.
Not all of these standard instances are valid.
Also known as
A standard instance of a categorical syllogism is also known as a syllogistic form.
Also see
Sources
- 1965: E.J. Lemmon: Beginning Logic ... (previous) ... (next): Chapter $4$: The Predicate Calculus $2$: $4$ The Syllogism
- 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): syllogism