The notion of monotonicity

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.

• 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 downward monotonic?

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.

What is quantification ?

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 banana.

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 "driving"example).

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 downward monotonic.

Negation does reverse the monotonicity, as it exchanges universal and existential quantification.

But I do not think negation has itself an intrinsic role, at least in your examples:

"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".

• Thank you very much for your answer, could you please take a look at addendum, it looks like I have a problem with the concept itself. Thank you. Jun 18 '14 at 17:23
• @user16168 Can you apply my answer to the analysis of the examples in your addendum. The "Des Moines" example is of course wrong ... it could be unsafe to drive slowly (but I never went there:). Jun 18 '14 at 18:03