Unit 2 Lesson 1

What is a Proposition?

 In this lesson, we define precisely what a proposition is and learn to distinguish valid mathematical statements from invalid ones.

Learning Objectives

  • Define a proposition rigorously

  • Distinguish propositions from non-propositions

  • Assign a truth value to a proposition

  • Justify classification decisions using formal criteria

Proposition (Statement)

A proposition or statement is a declarative sentence that has a well-defined truth value. 

Formally:

\[ P \in \{ \text{true}, \text{false} \} \]

Important Clarification

A proposition must satisfy both:

  • It is a declarative statement
  • Its truth value is objectively determined

Valid propositions

  • "\( 2 + 3 = 5 \)" → true
  • "7 is even" → false
  • “There exists a prime number greater than 100” → true

Key Insight

A sentence is not a proposition if:

  • it depends on unknown variables

  • it expresses a command

  • it is ambiguous or subjective

Not propositions

  • "\(x + 2 = 5\)" → depends on x
  • "Close the door!" → command
  • "This number is large." → vague

Truth Value

Every proposition has a truth value.

  • True (T or 1): The statement accurately describes reality or follows from accepted axioms
  • False (F or 0): The statement does not accurately describe reality or contradicts accepted axioms

Truth value \(∈{T,F}\)

This binary nature is fundamental to classical logic and underlies Boolean algebra, which forms the theoretical foundation of computer science.

Critical Distinction

Truth is not:

  • what we believe

  • what we know

  • what we can prove immediately

Truth is an objective property.

“There exists a largest prime number” → false

Even before proof, it is either true or false.

Law of Excluded Middle (Tertium Non Datur)

For any proposition P, either P is true or not P is true. There is no middle ground.

This fundamental principle ensures that every well-formed proposition in classical logic has exactly one truth value.

 Precision vs Ambiguity

Mathematics eliminates ambiguity.

  • “He is tall” → not a proposition

  • “He is taller than 180 cm” → proposition

Exercise 1

For each sentence, decide whether it is a proposition.

a)

\(5 > 2\)

b)

\(x > 2\)

c)

Solve the equation \(𝑥^2 = 4\).

d)

There exists a number divisible by 6.

Exercise 2

Determine the truth value.

a)

10 is a prime number.

b)

Every even number is divisible by 2.

c)

There exists an odd number divisible by 2.

Exercise 3

a)

This number is big.

Is this a proposition?

Why?

b)

There exists a real number \(𝑥\) such that \( 𝑥^2 = 2^n\).

Is this a proposition?

Why?

c)

\(x^2 = 4\)

Is this a proposition?

Why?

Question

Why is

"\(x^2 = 4\)"

not a proposition

but

"There exists \(x\) such that \(x^2 = 4\)."

is a proposition?

Answer

First: depends on \(x\) → no fixed truth value.

Second: quantified → becomes a complete statement → has truth value.

  • A proposition is a statement with a definite truth value
  • Not all sentences are propositions
  • Variables and ambiguity prevent a sentence from being a proposition
  • Precision is essential in mathematics