Logic

Quantifications

Quantification means making statements about things.
In logic, this often involves variables that stand for different things in a domain.
Properties are characteristics/attributes that you can talk about. Eg. “even” and “prime” are properties of numbers.

In both these cases, we’re talking about some individual element/variable

Universal Quantifier (∀)
- This symbol (∀) is like saying “for all” or “for every.” It allows us to make statements about every individual element in a certain group or domain. Let’s use an example to illustrate this:
- Example: Consider the domain of natural numbers. If we say ∀x (x is greater than 0), it means that we are making a statement about every natural number. In simpler terms, it’s like saying “Every natural number is greater than 0.” This statement covers all the numbers in the domain of natural numbers.
Existential Quantifier (∃)
- This symbol (∃) is like saying “there exists” or “there is at least one.” It allows us to make statements about the existence of at least one individual element in a certain group or domain. Here’s an example:
- Example: Using the domain of natural numbers again, if we say ∃x (x is even), it means that we are making a statement about the existence of at least one even number in the group of natural numbers. So, it’s like saying “There exists a natural number that is even.” This statement doesn’t specify which number it is; it simply asserts that there is at least one.

In second order, we’re usually quantifying about property in some domain

property as set : Sets of individual things that have that property. For example, the set of all even natural numbers is a set that includes 2, 4, 6, and so on.
property as fn: functions that take in an element and return a Boolean indicating whether the element satisfies the property. (Characteristic function)

In third and other higher orders, we’re usually quantifying about property of properties

Eg. “All properties of sets of cats have the property of being mammal-related.” (Here, you’re not only looking at sets of cats but also at properties of those sets. i.e You’re reasoning about the property)
Even though higher-order logics seem more powerful and expressive, you can still capture their ideas and statements using second-order logic.