You are addressing two problems:
why is there, apparently, a monotonicity reversal when negating your
first example ("banana"), but not the second ("driving")?
why is your first example upward-monotonic, while the second is
The problem has to do with understanding that general terms like
"fruit" or "driving" may refer to elements of sets that are quantified
existentially or universally, while specific elements, like "banana"
or "fast driving" refer to themselves.
I do not address the examples in your addendum. This answer should tell you how to deal with them
What is quantification ?
Please skip this part if you already know
This is only a sketch. You may find more details on the web or in
textbook. Note that the wikipedia article states that:
The study of quantification in natural languages is much more
difficult than the corresponding problem for formal languages. This
comes in part from the fact that the grammatical structure of
natural language sentences may conceal the logical structure.
This is part of the problem you encounter.
Many natural language statements are actually predicate over some
entities that may be element of sets.
For example you may use the predicate Eat which takes two arguments,
the subject who eats and the object that is eaten.
Then the sentence "I eat banana" can be interpreted semantically as
Eat (I,banana), which establishes a relation between I and
But when you say "I eat fruit", "fruit" stand for Fruit which is the
set of all fruits, that include banana and many others. But, how do
you interpret this sentence: does it mean that you eat all fruits or
that you eat some fruits. This is not obvious though it seems that
most people interpret that as meaning that you eat some
fruits? Furthermore this has to be your interpretation since you
deduce it from the fact that you are eating banana, which cannot imply
that you eat any other fruit. All you know is that you eat one fruit:
banana. So for you, "I eat fruit" must mean that there is a fruit that
you eat, which can be formally written as
∃ f ∈Fruit, Eat (I,f)
or less formally
There is a Fruit f such that Eat (I,f)
In this interpretation, the predicate Eat is exitentially quantified
on its second variable. This mean that we state the existence of at
least one fruit such that the predicate holds.
The predicate could also be universally quantified on its second
variable which one writes logically as
∀ f ∈Fruit, Eat (I,f)
or less formally
For any f that is Fruit, Eat (I,f)
This would probably be stated in English as "I eat all fruits" or "I
eat any fruit".
Now, when you state "I don't eat fruit", it does not mean that ther is
a fruit you do not eat, but rather that you do not eat any fruit,
which one can write formally as:
∀ f ∈Fruit, ¬ Eat (I,f)
or less formally
For any f that is Fruit, not Eat (I,f)
You see that this is a universally quantified statement, using the
negation of the previous predicate.
This is a general rule: if you want to negate a quantified statement,
you change the quantification (from universal to existential, or the
converse) and you use the negation of the predicate.
The rule is simple to use in logic. It is a bit more complex in
natural language in general, and English in particular. What may seem
to be a negation in English is not necessarily a negation of the
logical semantics of the sentence. This can be illustrated by your
second example analyzed in the next section.
Things can be even more complex when there is multiple quantification.
About non-reversal of monotonicity in the second example
Your inference "Driving is dangerous" -> "Fast driving is dangerous"
shows that you understand "Driving is dangerous" as "All driving is dangerous".
Here, "Driving" stands for the set of all forms of driving.
Then, logically, the negation of "Driving is dangerous" is not "Driving is not
dangerous" but "Some driving is not dangerous".
Then this would not imply that "Fast driving is not dangerous".
But if it were true that "Fast driving is not dangerous", that would
imply that "some driving is not dangerous¨.
i.e. you have the same reversal of monotonicity:
"Some driving is not dangerous" <- "Fast driving is not dangerous".
Your problem is with incorrect negation of quantifiers, not with monotonicity.
"Driving is dangerous" means "all forme of driving are
dangerous", which is a universally quantified assertion. When you negate universal quantification, you have to use
existential quantification on the negation of the predicate.
The negation of "for any x, P(x)" is "there is x, not P(x)".
About the fact that the two examples have opposite monotonicity
The second issue still has two do with quantification, and the
respective properties of existential and universal quantification.
To make it more visible I will use two other examples, where I make
quantification explicit. Your examples are harder because they both
have implicit quantification (which I already explicited above for the
The two examples are chosen to be very similar.
"A banana is good" -> "There is a good fruit"
"All fruits are good¨ -> "A banana is good"
You see that you have upward monotony in the first case because the
more general phrase is existentially quantified.
You have downward monotony in the second case because the more general
phrase is universally quantified.
Your first example is upward monotonic for that reason. If you expland
the semantics of "I eat fruit", it is really "There is a fruit I eat".
And we have seen your second example is universally quantified, hence
Negation does reverse the monotonicity, as it exchanges universal and
But I do not think negation has itself an intrinsic role, at least in
"Driving is not dangerous" is negative, but has exactly the same
meanind as "Driving is safe" which is not negative. Recall that the
general/specific opposition here is on "driving".