Existential Quantifier

When we say:

There are men.

We mean that the propositional function:

x is a man.

is sometimes true, i.e. the above propositional function is true for some values of x.

Similarly, when we say:

Some men are Greeks.

This means that the propositional function:

x is a man and a Greek

is sometimes true, i.e. is true for some values of x.

To say:

There are at least n indivduals in the world.

is to say that the propositional function:

'a' is a class of individuals and a member of the cardinal number n.

is sometimes true, or, is true for certain values of 'a.'

This form of expression is more convenient since it is telling us about the variable that we are taking as the argument to our propositional function. For example, the propositional function:

'a' is a class of individuals and a member of the cardinal number 'n'

can be written as:

'a' is a class of 'n' individuals.

And this form clearly indicates that we have two variables in the propositional function namely, 'a' and 'n.' In the language of propositional functions, the axiom of infinity is:

If n is an inductive number, is true for all some values of "a" that "a" is a class of 'n" individuals.

is true for all possible values of n. Here, we are saying two things:

1. There is a secondary function, 'a' is a class of n individuals," which is sometimes true with respect of variable "a."

2. The assertion that this happens if n is an inductive number is always true with respect to the variable n.

Now that we have a little bit idea of the philosophy of this concept, let us work on this concept from computational view point. As usual, let us start with an example. Consider the following propositional function:

x > y + 1

It is not difficult to see that this propositional function (formula) fails to be satisfied by every pair of numbers. For instance, when x = 3 and y = 4. The operative words here are "every pair of numbers" i.e. "every pair" (see universal propositions). But, there are pairs of numbers which satisfy the above  propositional function. For example, when x = 4 and y = 2. This situation can be expressed by the following phrase:

For some numbers x and y, x > y + 1

or 

There are numbers x and y such that x > y + 1

Clearly, the above expressions (or any linguistic equivalent) are true propositions. These are the examples of existential propositions, which state the existence of things with certain property.

Conditionally Existential Proposition

Consider the following proposition:

For any number x and y there exist a number z such that x = y + z.

Propositions of this type are called conditionally existential proposition because they state the existence of numbers (numbers in this particular example- but propositions in general) having a certain property, but on condition that certain other number exist. As an example, consider the following:

All men are mortal.

We can formulate the same with the help of quantifiers as follows:

For any x, if x is a man, then x is mortal.

Returning to our original example:

For any number x and y there exist a number z such that x = y + z.

And this conditionally existential propositional function assumes the following form:

∀ x, y, ∃z such that x = y + z.

Now its time to formalize the above concept. Consider a sentence:

There is a student in my class.

Another way to write the above sentence is:

∃ a person s such that s is a student in my class.

or more formally,

∃ s ∈ S such that s is a student in my class.

where S denotes the set of all people the symbol ∃ denotes "there exists" and is called the existential quantifier.

Note that the domain of the predicate variable is generally indicated:

Existential Proposition

Example: Truth of Existential Proposition.

Consider the existential proposition:

∃ m ∈ Z such that m² = m.

The problem is to show that this proposition is true.

Observe that 1² = 1. Thus, "m² = m" is true for at least one integer m. Hence, according to the definition, the given existential proposition "∃ m ∈ Z such that m² = m" is true.

Example: Falsity of Existential Proposition.

Consider the existential proposition:

∃ m ∈ Z such that m² = m.

But this time, the domain of a predicate variable is the set E = {5, 6, 7, 8, 9}.

Now our problem is to show that the given existential proposition is false.

Here, we apply the method of exhaustion to show the falsity of the given existential proposition. That is, check that "m² = m" is not true for each element m in E.

Since, m² = m is not not true for any integer m in E. Therefore, the given existential proposition "∃ m ∈ Z such that m² = m" is false.