1
A proposition has exactly one truth value
It is either true or false — not both, not neither, and not "it depends". The value may be unknown to us; what matters is that the sentence has one.
Lesson · Logic and Proof · Propositional logic
Propositions and Truth Values
A proposition is a declarative sentence with exactly one truth value. Establish which sentences qualify, why questions and open sentences do not, and why this restriction is what makes everything later possible.
p : "17 is prime" proposition, true
q : "x + 1 = 5" NOT a proposition (open sentence)
r : "Close the door." NOT a proposition (not declarative)
s : "This sentence is false" NOT a proposition (no consistent value)
Four sentences, one proposition
Core ideas
What this lesson establishes
Everything in this course is built on sentences that have a truth value. The first job is to say precisely which sentences those are.
2
Open sentences are not propositions yet
x > 3 has no truth value until x is given one. It becomes a proposition when the variable is bound, which is what quantifiers do in “Predicates and Quantifiers”.
3
Truth is not the same as provability
"Every even number greater than 2 is a sum of two primes" is a proposition. Nobody knows its value; it still has one.
The material
What a proposition is
Four rules, and the sentences each one excludes.
Definition · Proposition
A proposition is a declarative sentence that is either true or false, but not both. Its truth value is T or F.
Four kinds of sentence fail this definition, and each fails for its own reason.
- Not declarative. "What time is it?" and "Close the door." assert nothing, so there is nothing to be true or false.
- Open. "x + 1 = 5" is true for x = 4 and false otherwise. It has a truth value only once x is fixed. Such a sentence is a predicate, and “Predicates and Quantifiers” gives it the treatment it deserves.
- Self-referentially paradoxical. "This sentence is false" cannot be true (it would then be false) and cannot be false (it would then be true). No truth value can be assigned consistently.
- Vague. "He is tall" has no boundary on which to be true or false. That is a different failure from not knowing the value, and the paragraph after next is about the difference.
The third exclusion is not a technicality. It is the reason logic is built on a restricted class of sentences rather than on English: the moment sentences may talk about their own truth, the two-valued system breaks.
Theorem · Bivalence
Every proposition has exactly one of the two truth values T, F. This is an assumption of classical logic, not a theorem within it, and it is what licenses p ∨ ¬p — the law of the excluded middle — used throughout this path.
Unknown is not a third value. "There are infinitely many twin primes" is a proposition whose value nobody currently knows; the sentence is true or it is false, and our ignorance is a fact about us.
Example · Sorting sentences
"2 + 2 = 5" — proposition, false. "n is even" — open, not a proposition. "Every integer greater than 1 has a prime factor" — proposition, true. "Let x = 3" — an instruction, not a proposition.
Notation: propositions get lower-case letters p, q, r, s. Those letters are propositional variables — placeholders for whole sentences, not for numbers.
Interactive
Truth table
Every assignment, evaluated
Method
Deciding whether a sentence is a proposition
Four questions, in this order. The first "no" settles it.
1
Is it declarative?
Questions, commands and exclamations assert nothing. If the sentence cannot sensibly be preceded by "It is true that…", stop here.
2
Are all its variables fixed?
If the sentence contains a free variable, it is an open sentence. Bind it or substitute a value and it becomes a proposition.
3
Does it refer to its own truth?
Self-reference is where the two-valued assumption breaks. A sentence that asserts something about its own truth value is excluded.
4
Is it vague rather than merely unknown?
"This film is good" has no definite value; "there are infinitely many twin primes" has one we do not know. Only the first fails.
Proposition or not
Score 0 / 0Choose the answer you can justify.
Classifying six sentences
Work down the list and give the reason, not just the verdict.
1. "7 is a prime number." proposition, T
2. "Is 7 prime?" no — interrogative
3. "x is a prime number." no — open in x
4. "7 is prime and 8 is prime." proposition, F
5. "There is a largest prime." proposition, F (Euclid)
6. "P = NP." proposition, value unknown
Line 6 is the one worth dwelling on. It is a perfectly good proposition: a precise mathematical claim that is true or false. That no one has determined which does not put it in a third category.
Common mistakes
1
Treating "unknown" as a truth valueA conjecture is a proposition. Its value exists whether or not anyone has found it, and no rule in this course needs to know which it is.
2
Calling an open sentence false because it is sometimes falsex > 3 is not false; it has no value at all until x is fixed. Confusing "not always true" with "false" is the same error that makes quantifier negation hard in “Negating Quantified Statements”.
3
Assuming vagueness and ignorance are the same problem"Tall" has no sharp boundary, so "he is tall" is not a proposition. "P = NP" has a sharp meaning we cannot currently evaluate, so it is.
Check your understanding
✓
Finish when you can say why a sentence fails, not just that it does.
Given ten sentences you should be able to sort them and name the specific reason for each exclusion: not declarative, open, self-referential, or vague. "It's not a proposition" without the reason is the answer that does not transfer to “Predicates and Quantifiers”.
Note: The truth table lab appears here with a single variable on purpose. Two rows for one proposition, four for two, 2ⁿ for n — the doubling is worth noticing now, because “Truth Tables” depends on it and Combinatorics and Counting counts it.