Variables and Quantifiers: Turning Open Sentences into Real Statements

Why this matters

In the previous post, Simple Statements and Sets, we drew a line between statements (true or false) and sentences that are not definite enough to have a truth value. Here's why that line exists:

A lot of “math-looking sentences” are really templates. They become true or false only after we decide who or what we’re talking about.

That’s where variables, domains, and quantifiers come in.

Variables: placeholders with a job

A variable is a symbol that stands in for an element of some set.

In everyday language, pronouns can act like variables too:

• “He is a Canadian citizen.” This cannot be labeled true or false until we know who “he” refers to.

In mathematics:

• x + 1 > 0 This cannot be labeled true or false until we know what values x is allowed to take.

The domain (also called the universe)

The domain of a variable is the set of allowed replacements for that variable. It answers the question:

What kinds of objects is this variable allowed to represent?

The elements of the domain are sometimes called the values of the variable.

Example: If x is allowed to be any real number, then x + 1 > 0 is sometimes true and sometimes false.

• If x = 2, then 2 + 1 > 0 is true.

• If x = -5, then -5 + 1 > 0 is false.

But if the domain is restricted, the truth can change.

Example: If the domain is the set of positive integers, then x + 1 > 0 becomes true for every allowed value of x.

Constant (in this context)

A constant can be thought of as a “variable” with only one allowed value, meaning its domain has exactly one element.

So the difference is not mystical:

• variables range over many possible values,

• constants are fixed.

Open sentences: statement-templates

A sentence that contains a variable is often called an open sentence (or an open statement). It becomes a true/false statement only after the variable is handled in one of two ways:

1. Substitute a specific value

2. Quantify the variable (use “for all” or “there exists”)

Example open sentence: y + 3 = 4

This is not yet true or false on its own. It is a template.

Open sentences define sets (solution sets / truth sets)

Once you choose a domain, an open sentence splits that domain into two groups:

• values that make the sentence true

• values that make the sentence false

The set of values that make the open sentence true is called the solution set (also commonly: truth set), relative to that domain.

Example

Let the domain be S = {0, 1, 2, 3, 4}

Consider the open sentence y + 3 = 4

Test each value of y in S:

• 0 + 3 = 4 is false

• 1 + 3 = 4 is true

• 2 + 3 = 4 is false

• 3 + 3 = 4 is false

• 4 + 3 = 4 is false

So the solution set in the domain S is:

{1}

When a value makes the open sentence true, we say that value satisfies the sentence. Values in the solution set are also called solutions (and in many contexts, roots).

What “solve” means here

To solve an open sentence over a given domain means:

Find its solution set within that domain.

That wording matters because the solution set can change when the domain changes.

Quantifiers: “how many values make it true?”

Instead of plugging in one value at a time, we often want to make a single statement about many values at once. Words like these do that:

• all, every, each, any

• some, at least one, there exists

These words are called quantifiers because they describe quantity: how many values in the domain satisfy the sentence.

There are two main types.

Universal quantifier: “for all”

A universal statement says the open sentence is true for every value in the domain.

In symbols, you’ll often see: ∀ (read “for all”).

Example: If the domain is the positive integers, then the statement

For every positive integer x, x + 1 > 0.

is true.

Common English equivalents:

• “For all …”

• “For every …”

• “For each …”

• “For any …”

All of these are doing the same logical job.

What it really claims

A universal statement claims there are no exceptions in the domain.

So to prove it false, you only need one counterexample.

Existential quantifier: “there exists”

An existential statement says the open sentence is true for at least one value in the domain.

In symbols, you’ll often see: ∃ (read “there exists”).

Example: If the domain is the positive integers, then the statement There exists a positive integer x such that x + 1 > 0.

is true.

Common English equivalents:

• “There exists …”

• “There is at least one …”

• “For some …”

What it really claims

An existential statement does not claim “all.” It only claims at least one.

To prove it true, you can give a single example that works.

Quantifiers are often hidden in plain sight

A lot of “ordinary” math sentences already contain quantifiers, even when the words are not written.

Geometry example

When someone says: The base angles of an isosceles triangle are equal.

What they mean logically is: For every isosceles triangle T, the base angles of T have equal measure.

That’s a universal statement. It is not about one triangle, it’s about all triangles of that type.

Algebra example

When someone writes:

a² − b² = (a − b)(a + b)

they typically mean something like: For all real numbers a and b, a² − b² = (a − b)(a + b).

Again: universal.

A common ambiguity to fix

If someone writes:

u² = 49

That is an open sentence until you add context. Depending on what the writer intends, it could mean:

• Existential reading: “There exists a number u such that u² = 49.” (true, for example u = 7 or u = -7)

• Universal reading: “For all numbers u, u² = 49.” (false)

In higher-level math, this kind of ambiguity becomes a real problem. The fix is simple: Always say what the domain is, and whether you mean “for all” or “there exists.”

Summary: the “logic upgrade”

• A variable represents an element of a specified domain.

• An open sentence is a template. It becomes a statement only when:

  • you substitute a value, or

  • you apply a quantifier.

• The solution set of an open sentence (over a domain) is the set of all values that make it true.

• Universal statements mean “no exceptions.”

• Existential statements mean “at least one example.”

Mini practice (with answers)

1) Open sentence or statement?

For each, say whether it is an open sentence or a statement.

a) x² = 9 b) “For some integer n, n² = 9.” c) “For every integer n, n² = 9.”

Answers: a) Open sentence. b) Statement (true). c) Statement (false).

2) Same open sentence, different domains

Let the open sentence be: x² < 10

Find the solution set in each domain:

a) Domain D₁ = {0, 1, 2, 3, 4} b) Domain D₂ = {−4, −3, −2, −1, 0, 1, 2, 3, 4}

Answers: a) {0, 1, 2, 3} because 0², 1², 2², 3² are < 10 but 4² = 16 is not. b) {−3, −2, −1, 0, 1, 2, 3} for the same reason.

3) Write the hidden quantifier

Rewrite each as a clear quantified statement.

a) “Squares are nonnegative.” b) “A number whose square is 16 exists.”

Sample answers: a) For every real number x, x² ≥ 0. b) There exists a real number x such that x² = 16.

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