# Are these generalized quantifiers correct?

According to Kearns (2011), I know that "the ten apples are bruised" can be interpreted as "‘The ten apples are bruised’ is true if and only if |A ∩ B| = 10."

But how about this sentence "Neither artist is Bulgarian."? Should it be interpreted as "'Neither artist is Bulgarian' is true if and only if |A ∩ B| = 0"? Also, should "Both avenues are broad." be "‘Both avenues are broad’ is true if and only if |A ∩ B| = 2"?

• My instinct says "yes", though English is not my first language. I have, however, taken logical semantics in university (in English) and I think I would interpret "neither" as "must be none" and "both" as "bust be all". Oct 30 '20 at 10:00

Correct; "neither" does essentially assert "0" and "both" asserts "2".
What makes them more complicated is that "neither" and "both" bear an additional presupposition, namely that |A| = 2. This presupposition can be expressed as ∂(|A| = 2). If the presupposition is not met, the sentence will have no truth value, otherwise it evaluates to whether the remaining content is true. You could then formulate the meanings more precisely as

neither A B ⇔
∂(|A| = 2) and |A ∩ B| = 0

both A B ⇔
∂(|A| = 2) and |A ∩ B| = 2

Though note that the key feature of presuppositions is that they survive under negation, so if you negate the quantifiers, what is negated is the cardinality of the intersection while the cardinality of the subject terms reimains, and you end up with

not neither A B ⇔
∂(|A| = 2) and |A ∩ B| ≠ 0

not both A B ⇔
∂(|A| = 2) and |A ∩ B| ≠ 2

• ♦ Thank you for your detailed reply! Now I get it! And I really appreciate that you always answer my questions! :) Oct 30 '20 at 12:25