First, the formulas for reckoning the numerical value of the conclusion from the premises give the strongest conclusions that are entailed, although weaker ones are also entailed as well. For example, if the conclusion drawn by applying the formula is "At least ∅+10 A are C" then the weaker claims that "At least ∅+9, 8, or 7, etc. A are C" certainly follow also. However, "At least ∅+11, 12, 13, etc., A are C" do not follow, for they are "stronger" claims than the premises justify. And the same is the case for the exceptions to the universal conclusions. That is, if "At least ∀-10 A are C" is what the formula concludes, then the weaker claims of "At least ∀-11, 12, 13, etc., A are C" are also entailed while the stronger claims of "At least ∀-9, 8, 7, etc., A are C" are not.
Finally, although there may be an appropriate green arrow path from one extreme term to the other, still no informative conclusion follows if the numerical value of the universal exception(s) is equal to or greater than the numerical value of the particular proposition. For example, in the left argument below both x and y are instantiated by 10 which yields the tautological conclusion that " At least zero A are C" while the right one yields "At least two fewer than zero A are C" which is also a tautology—indeed, an even greater one if there were tautological degrees!
| At least ∅+10 A are B | At least ∅+10 A are B | ||
| At least ∀-10 B are C | At least ∀-12 B are C | ||
| ∴ | At least ∅+0 A are C | ∴ | At least ∅+(-2) A are C |
Still the arguments are valid even though they are useless—useless since the truth of a tautology stands on its own.
