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.
1.2 Examples of propositions
Example 1: True propositions
- "$2 + 3 = 5$" → true
- "Tokyo is the capital of Japan" → 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 Examples of non-propositions
The following sentences are not propositions.
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."
→ The criteria for "tall" or "delicious" vary from person to person, so a truth value is not determined.
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, × |
In mathematics, the symbols $\top$ (true) and $\bot$ (false) are also used.
1.5 Sentences containing variables
Consider the following sentence:
"$x$ is even"
Is this a proposition?
Answer: not as it stands
Without knowing the value of $x$, the truth value is not determined.
- 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.
"$x$ is even" is written as $P(x)$.
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 an open sentence (predicate) because its truth depends on the value of $x$.
Q2. What is the difference between an open sentence and a proposition?
A: A proposition has a fixed subject and a determinate truth value. An open sentence (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. By logically determining the truth value of a proposition, one can give a "proof" that holds in every case. The foundation of mathematical rigor lies in clarifying which statements have been proved and which are assumed.