Which sentence is a logical statement?
A statement is a sentence whose truth can be established.
Text version of this lesson
Draw strict conclusions from the evidence without confusing a claim's truth with the speaker's habits.
Key takeaway: A statement's truth depends on the evidence; who said it is a separate fact.
Question 1
A statement is a sentence whose truth can be established.
Which sentence is a logical statement?
Options
- Close the window!
- Tashkent is the capital of Uzbekistan.
- What wonderful weather today!
- Would you like some tea?
Show answer and explanation
Answer: Tashkent is the capital of Uzbekistan.
Why: Right. This sentence's truth can be checked, so it is a statement.
Hint: Ask the sentence: is this true or false?
Question 2
What is the negation of "All the books are on the shelf"?
Options
- No book is on the shelf.
- At least one book is not on the shelf.
- Every shelf has books on it.
- The books are off the shelf.
Show answer and explanation
Answer: At least one book is not on the shelf.
Why: Yes. One exception is enough to refute an "all" claim.
Hint: How many exceptions does it take to break an "all" claim?
Question 3
Only one of them tells the truth. Who is it?
Show answer and explanation
Answer: Aziz
Why: Aziz tells the truth. If Bek's "we are both lying" were true, Bek would be lying and telling the truth at once — a contradiction.
Hint: Suppose Bek is telling the truth.
Question 4
Exactly one of the three friends is telling the truth. Which box holds the prize?
Options
- Red box
- Blue box
- Green box
Show answer and explanation
Answer: Blue box
Why: The blue box. Only Nodir speaks truly there: the prize really is not in the red box.
Hint: Try each box in turn as a hypothesis.
Question 5
Which pair cannot both be true at the same time?
Options
- Some students work. / Some students do not work.
- All the doors are shut. / At least one door is open.
- Aziz likes football. / Bek likes football.
- It is hot today. / It may be cold tomorrow.
Show answer and explanation
Answer: All the doors are shut. / At least one door is open.
Why: Right. "All the doors are shut" and "at least one door is open" cannot both hold.
Hint: What does the first statement claim without exception?
Question 6
In an interview one of two candidates gave false information. Kamol: "Madina gave false information." Madina: "Kamol did not give false information." Which conclusion is forced?
Options
- Kamol must be the one who gave false information.
- Madina must be the one who gave false information.
- There is no way to tell which of them it was.
- Both of them told the truth.
Show answer and explanation
Answer: Madina must be the one who gave false information.
Why: Madina must be the one. She vouches for Kamol, so the single false record can only be hers.
Hint: If Madina speaks truly, then Kamol gave no false information.