Predicates and Open Sentences
A free Logic lesson from the “Quantifiers and Predicates” unit, with a worked example and practice problems including step-by-step solutions.
A predicate contains a free variable. Its truth is evaluated using an assignment in a specified domain. An existential claim needs one allowed witness; a universal claim must survive every allowed value. Changing the domain can change whether witnesses and counterexamples exist.
What you'll learn
- Recognize claims that become statements when a variable is given a value
- Explain the idea in plain English before using symbols
- Use examples, non-examples, or counterexamples to check the reasoning
Worked example
Problem. For P(x): x > 22, evaluate P(23).
- 23 exceeds 22.
Answer: True
Lesson preview: this page shows 5 practice items. The interactive activity contains 6 practice items and 6 quiz items (multiple choice). Counts include repeated prompts and variants.
Practice problems
1. For P(x): x > 7, evaluate P(8).
Choices: False · True
Show solution
- 8 exceeds 7.
Answer: True
2. For P(x): x > 7, evaluate P(7).
Choices: False · True
Show solution
- Strict inequality excludes equality.
Answer: False
3. For domain {7, 8, 9} and P(x): x is even, which values satisfy P?
Choices: 7, 9 · All three values · No values · 8
Show solution
- Test the stated predicate separately at every domain value.
Answer: 8
4. Over positive integers, ∃x(x²=2) is false. What happens if the domain changes to positive real numbers?
Choices: It becomes a command · Every real number becomes a witness · It becomes true because √2 is allowed · It stays false because 2 is not a perfect square
Show solution
- Existential truth depends on whether the domain contains a witness.
Answer: It becomes true because √2 is allowed
5. Which operation turns the open sentence P(x) into a closed statement?
Choices: Change the font of P · Assign a particular value to x or bind x with a quantifier · Write x twice · Leave x unspecified
Show solution
- A free variable needs an assignment or quantifier for a definite proposition in the stated interpretation.
Answer: Assign a particular value to x or bind x with a quantifier
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