Chapter 1: What is a Proposition?

Introduction (High School to First-Year University Level)

1.1 What is a proposition?

Definition: proposition

A proposition is a sentence whose truth or falsity is definitively determined.

The key point is "definitively determined":

  • If the sentence is correct, it is true (True).
  • If the sentence is incorrect, it is false (False).

A sentence that is neither — vague or ambiguous — is not a proposition.

Note that "determined" does not mean "we can decide it." For example, "every even number greater than or equal to 4 is the sum of two primes" (the Goldbach conjecture) is certainly either true or false, but nobody has yet proved or disproved it. It is still a proposition.

Proposition if correct if incorrect True False
Figure 1.1: A proposition is determined to be either true or false.

1.2 Examples of propositions

Example 1: True propositions

  • "$2 + 3 = 5$" → true
  • "Japan's area is smaller than Russia's" → true
  • "$6$ is divisible by $2$" → true
  • "Every square is a rectangle" → true

Example 2: False propositions

  • "$2 + 3 = 6$" → false
  • "The Earth is flat" → false
  • "$7$ is even" → false
  • "Every rectangle is a square" → false

Key point

A false statement is still a proposition. Even though "$2 + 3 = 6$" is incorrect, the fact that it is incorrect is "definitively determined," so it is a proposition.

1.3 Sentences that are not propositions as they stand

The following sentences are not propositions as they stand.

Example 1: Questions

  • "What day of the week is today?"
  • "What is the value of $x$?"

→ Questions are inquiries, not objects to which a truth value can be assigned.

Example 2: Commands and exclamations

  • "Close the door."
  • "What a beautiful view!"

→ These express actions or emotions; they are not objects for evaluating truth.

Example 3: Vague sentences

  • "He is tall."
  • "This dish is delicious."

→ Without fixing the criteria for "tall" or "delicious" (and what "he" and "this dish" refer to), the truth value cannot be determined uniquely. Once a criterion and a subject are fixed, such as "a person 180 cm or taller counts as tall," it becomes a proposition.

Proposition "2 + 3 = 5" "7 is even" "Mt. Fuji is in Japan" truth value is uniquely determined Not a proposition "What time is it?" "Study!" "It is delicious" truth value is not determined
Figure 1.2: Distinction between propositions and non-propositions.

1.4 Truth values

Definition: truth value

The value "true" or "false" of a proposition is called its truth value.

Truth values are sometimes written as follows:

True False
True, T, 1, ✓ False, F, 0, ✗

Which notation is used depends on the field (1 and 0 are common in computer science; ✓ and ✗ are informal marks). In logic, the symbols $\top$ (true) and $\bot$ (false) are also used.

1.5 Sentences containing variables

Consider the following sentence:

"$x$ is even" ($x$ is an integer)

Is this a proposition?

Answer: not as it stands

Since the value of $x$ is not fixed, this sentence alone has no determined truth value (unlike the vague sentences in 1.3, it becomes determined once a value is substituted for $x$).

  • If $x = 4$, then → true.
  • If $x = 5$, then → false.

Definition: propositional function (predicate)

A sentence that contains a variable and becomes a proposition once a specific value is substituted for the variable is called a propositional function or predicate (also called an open sentence).

Here $x$ denotes an integer. A propositional function makes sense only once the range of the variable (its domain) is fixed. "$x$ is even" is written as $P(x)$.

Note that even a sentence containing a variable becomes a proposition without substitution when it takes the form "for every integer $x$, $P(x)$" or "there is an integer $x$ with $P(x)$." The sentence "every square is a rectangle" in 1.2 has this form.

propositional function P(x): x is even substitute x = 4 substitute x = 5 P(4) = True P(5) = False
Figure 1.3: Substituting a value into a propositional function yields a proposition.

1.6 Chapter summary

Term Meaning
Proposition A sentence whose truth or falsity is definitively determined
Truth value The value "true" or "false"
Propositional function A sentence containing a variable that becomes a proposition upon substitution

Next chapter

The next chapter studies "logical operations" that combine propositions. The precise meanings of "and," "or," and "not" will be examined along with their truth tables.

Frequently Asked Questions

Q1. What is a proposition?

A: A proposition is a sentence that is definitively either true or false. For example, "2 is a prime number" (true) and "every even number is a multiple of 4" (false) are propositions, whereas "$x$ is positive" is a propositional function (predicate) because its truth depends on the value of $x$.

Q2. What is the difference between a propositional function and a proposition?

A: A proposition contains no variable, and its truth value is already determined on its own. A propositional function (predicate) contains a variable, and its truth value is determined only once the variable is given a value. For example, "$n$ is even" is true when $n = 4$ and false when $n = 3$.

Q3. Why is it important to treat propositions rigorously in mathematics?

A: Intuition and the sense that something "seems right" are not sufficient in mathematics. A "proof" is the procedure of establishing the truth value of a proposition logically from definitions and facts already known to be true; it can settle claims that no amount of checking examples could, such as propositions of the form "for every ...". The foundation of mathematical rigor lies in clarifying which statements have been proved and which are assumed.