Source (from older edition): p 289, A Concise Introduction to Logic (12 Ed, 2014), by Patrick Hurley
Any categorical syllogism that breaks one of the first four rules is invalid from the Aristotelian standpoint. However, if a syllogism breaks only Rule 5, it is valid from the Aristotelian standpoint on condition that the critical term denotes at least one existing thing. (The critical term is the term listed in the farthest right-hand column of the table of conditionally valid syllogistic forms presented in Section 5.1.)
In addition to consulting the table of conditionally valid forms, one way of identifying the critical term is to draw a Venn diagram. The critical term is the one that corresponds to the circle that is all shaded except for one area. In the case of two such circles, it is the one that corresponds to the Venn circle containing the circled X on which the conclusion depends.
What is the intuition behind the bolded assertion? Please do not answer with formal proofs or arguments. My drawing of Venn Diagrams for examples of syllogisms confirms the assertion above, but I still cannot intuit this in general.
For example, the assertion above needs me to choose the circle that is all shaded except for one area in the following Venn Diagram.
Unshaded in 1, 2, and 3, the S circle is shaded the least.
Unshaded in 1 and 2, the R circle is shaded less.
Unshaded only in 1, the C is shaded the most. So C is the critical term. QED.