I am slightly confused bu the notion of upward-monotonicity and downward-monotonicity.
I cannot understand what exactly can be defined as upward-monoty and down-ward-monotony, is this definition of relation between two phrases, or definition of some keywords that change monotonicity.
For example, "I eat banana"->"I eat fruit" this is example of upward-monotonicity (more specific phrase entail more common phrase), so the whole sentence is called upward-monotony?
Then apply negation, "I don't eat banana"<-"I don't eat fruit"
What happens, can I call this relation downward-monotony, or applying negation is downward-monotonicity? or "don't" itself is downward-monotony.
Negation is known for it's property to reverse the monotonicity.
"Driving is dangerous" -> "Fast driving is dangerous". Here, no negation, and still is downward-monotonicity.
Apply negation,
"Driving is not dangerous" -> "Fast driving is not dangeroius".
So here, negation didn't reverse the monotonicity, so not always negation reverse monotonicity, maybe negation can only change from upward to downward, here what can be called downward-monotonicity, it looks the relation itself is whether [downward-monoty|upward-monotony|no monotony], and some "keyword" could change monotonicity only in one direction.
Please, help me to understand the notion of monotonicity.
Addendum: I am competely confused, I found the following examples in Monotonicity. page 7.
It's dangerous to drive in Rome ➝ It's dangerous to drive fast in Rome.
According to inference relation, reading the first part must imply the truth of the second part. If my understanding is wrong, please write a correct definition.
Let's check the following example.
It's safe to drive fast in Des Moines ➝ It's safe to drive in Des Moines.
I am not sure that it's a correct instance of inference relation, however it's in the manual, as a counterexample I might assume highway in Des Moines, where driving slow is not safe.
Where my understanding is wrong or example is not successful.