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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. enter image description here

  • Is there any chance you could share a little more of the original context, and possibly the diagrams you drew? – Joseph Weissman Dec 28 '15 at 22:17
  • @JosephWeissman I have quoted some more; does my change suffice? I fear lengthening my post too much. Per your request, I also included a Venn Diagram. – Greek - Area 51 Proposal Dec 29 '15 at 4:57
  • Much appreciated! :) – Joseph Weissman Dec 29 '15 at 22:32
  • @JosephWeissman My pleasure. It is I who should thank all of you for educating me with all my questions. – Greek - Area 51 Proposal Dec 29 '15 at 22:33
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You have to recall the Existential fallacy.

If there are no Ss, we have that the "set of Ss" is empty. Then we have "all R are S", which means that the "set of Rs" in included into the "set of Ss". Thus : no Ss implies no Rs.

But "all C are R"; thus, again : no Cs.

If we have no Cs, the sentence "some C are S" is false : this is the Existential fallacy.

Thus, from the "modern" (Boolean) standpoint, the form is invalid.

From the Aristotelian standpoint, the syllogism is valid provided that the condition stated in Table Sect.5.1 [Figure 1 - schema AAI] is satisfied :

the subject of the conclusion (the minor term) exists [in your example : the "set of Cs" is not empty].

This is why C is the critical term : in order to avoid the Existential fallacy from the Aristotelian standpoint, you have to apply the schema to cases when the term C has "existential import" [see examples : "Some tigers are animals" versus "Some unicorns are animals".]

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