"Quantification of the Predicate" question that has me puzzled

Post date: 2022-05-19 08:03:19
Views: 11
Hello, my husband ran across this "thinking about reasoning" question that has me puzzled?

Here's the situation---from the book "Is God a Mathematician"? by Mario Livio:

"One of De Morgan's most important contributions to logic is known as quantification of the predicate.l This is a somewhat bombastic name for what one might view as a surprising oversight on the part of the logicians of the classical period. Aristotelians correctly realized that from premises such as "some Z's are X's" and "some Z's are Y's" no conclusion of necessity can be reached about the relation between the X's and Y's. For instance, "some people eat bread" and "some people eat apples" permit no decisive conclusions about the relation between the apple eaters and the bread eaters. Until the nineteenth century, logicians also assumed that for any relation between the X's and the Y's to follow of necessity, the middle term ("Z" above) must be "universal" in one of the premises. That is, the phrase must include "all Z's". DeMorgan showed this assumption to be wrong. In his book "Formal Logic" (published in 1847) he pointed out that from premises such as "most Z's are X's and most Z's are Y's" it necessarily follows that "some X's are Y's." For example, the phrases "most people eat bread" and "most people eat apples" inevitably imply that "some people eat both bread and apples." DeMorgan went even further and put his new syllogism in precise quantitative form. Imagine that the total number of Z's is z, the number of Z's that are also X's is x, and the number of Z's that are also Y's is y. In the above example, there could be 100 people in total(z=100), of which 57 eat bread (x=57) and 69 eat apples (y=69).Then, DeMorgan noticed, there must be at least (x+y-z) that are also Y's. At least 26 people (obtained from 57+69-100=26) eat both bread and apples."

For myself, a kinesthetic person not familiar with the abstract world of applied logic---use of "could be" implies unknown-unknowns, and so the argument fails to satisfy. My husband explained to me that DeMorgan's formula is "minimal"--in my words, that it would be a kind of boilerplate to apply to situations of precise numbers of x and y.

Calling all hive mind mathematicians and logicians: How would you explain DeMorgan's syllogism in concise and satisfying terms?

Thanks ever so much!
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