9.2. Order of Quantification

Now let us try to understand Nested Quantifiers, so there is very often we encounter statements where we need to have a nested form of quantification and this is similar to nested loops in programming languages.

Nested quantifiers allow us to express complex statements involving multiple variables. Just like nested loops in programming, where one loop runs inside another, nested quantifiers mean that one quantification depends on another. For example, if we want to assert that every person has a mother, we represent it with a nested quantifier. The outer quantifier ('for all x') specifies every person, while the inner quantifier ('there exists a y') indicates that for each person, there is a mother.

Think of nested quantifiers like a classroom where each student (the outer quantifier x) must present their project (the inner quantifier y). For each student, there exists a specific project that they worked on, illustrating the relationship.

And, I want to represent a statement that every person in this world has a mother. So my claim is that this can be represented by this expression for all x there exist y such that M(x,y) is true.

The expression 'for all x there exists y such that M(x,y) is true' captures the idea that for each individual (x), there is at least one person (y) that satisfies the predicate M (mother). This means that if you pick any person in the world, you can find their mother, which is a universal claim.

Imagine you have a big box of chocolates (people), and for every chocolate (each person), there is a specific wrapping paper (mother) it comes in. No matter which chocolate you pick from the box, you will always find that it has wrapping paper around it.

Now when you are dealing with nested quantification the order of the quantification matters a lot because if you change the order of the quantification then the logical interpretation of the statement changes completely.

The sequence in which quantifications are presented is crucial for interpreting statements accurately. Changing the order can alter the meaning of the entire logical statement. For example, 'there exists y for all x, M(x,y)' claims that there is one person (y) that is the mother of every person (x), which is fundamentally different from the earlier statement where each person can have a different mother.

Think of it like saying 'every student can choose their favorite subject' versus 'there exists a favorite subject that all students choose.' The first allows for individual choices, while the second suggests a single favorite subject for everyone.

Swapping of quantifications are not always possible, it is possible only when you have the quantifications of the same type occurring throughout the expression.

In expressions where the same type of quantifier appears multiple times, such as 'for all x for all y', you can switch their order without changing the meaning. However, if they are different types, like 'for all x' and 'there exists y', swapping them may completely change the statement's interpretation.

This can be likened to organizing a tournament. If you say 'every team plays every other team', swapping the order doesn’t change the outcome. However, saying 'some team wins against every other team' conveys a drastically different scenario.

So suppose I want to represent a statement that if a person is female and is a parent then this person is someone's mother.

To translate this statement into logic, we define predicates: F(x) for 'x is female', P(x) for 'x is a parent', and M(x, y) for 'x is the mother of y'. The logical form would be: for all x (if F(x) and P(x), then there exists a y such that M(x, y)). This expresses that if all conditions are met (being female and a parent), then the existence of at least one child follows contextually.

Imagine you are in a bakery, and you find that every baker (x) who specializes in wedding cakes (is female and is a parent) has at least one cake design (y) she is known for, representing her 'signature cake' in every wedding she caters.

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