Mr. Nance,

On the Square of Opposition, the particular affirmative is “Some S is P” and the particular negative is “Some S is not P.” Why does the universal “All S is P” not have the contrary universal “All S is not P”? Why instead do you use “No S is P”?

Just curious!

Great question! The reason I prefer to use “No S is P” is that in ordinary English usage, “All S is not P” is ambiguous. Without context, it is hard to discern if the statement is a universal negative or a particular negative.

For example, were I to claim that “All my students are not here,” most people would take that to mean the same thing as “Not all of my students are here.” This, however, is equivalent to “Some of my students are not here.” In short, this “All S is not P” sounds like “Some S is not P.” 

But were I to state, “All the plays from that theater are not worth watching,” most people would take this to mean “No plays from that theater are worth watching.” In this case, this “All S is not P” sounds like “No S is P.”

This ambiguity appears in different verses in the Bible. Joshua 5:5 tells us that “all the people born in the wilderness had not been circumcised.” This clearly means that none of the people born in the wilderness had been circumcised. Here, the All S is not P = No S is P.

But in 1 Corinthians 6:12 Paul tells us that, though all things are lawful, “All things are not profitable.” This just as clearly means “Some things are not profitable.” Here then, the All S is not P = Some S is not P.  

Interestingly, in the Greek, 1 Corinthians 6:12 and 1 Corinthians 10:23 use exactly the same phrase, but the NKJV translates them differently. The Greek is

οὐ πάντα συμφέρει 

Or “not all things are profitable.” In the New King James, the two verses say this:
 6:12  “all things are not helpful”
10:23 “not all things are helpful”

Things that make you go “Huh.”