Worksheet for identifying countersamples in logical reasoning.

To solve the problem, we need to evaluate each statement and determine whether it is TRUE or FALSE. If a statement is FALSE, we must provide a counterexample to demonstrate why it is incorrect. Let's go through each statement step by step.

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Statement 15: No two things can be at two places at the same time.

- Analysis: This statement refers to the principle of non-simultaneous occupation of space. In classical physics, two distinct objects cannot occupy the exact same space at the same time.
- Conclusion: TRUE
- Counterexample: Not applicable (since the statement is true).

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Statement 16: Vermont is the only state that ends in the letter T.

- Analysis: Vermont is indeed a state that ends with the letter "T." However, there are other states that also end with "T," such as Montana.
- Conclusion: FALSE
- Counterexample: Montana

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Statement 17: All cats have four legs.

- Analysis: While most cats have four legs, there are exceptions, such as cats born with birth defects or those that have lost limbs due to injury.
- Conclusion: FALSE
- Counterexample: A three-legged cat

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Statement 18: All precipitation is made up of water.

- Analysis: Precipitation generally consists of water in various forms (rain, snow, sleet, etc.). However, in extreme conditions, such as on other planets, precipitation can consist of substances other than water (e.g., methane rain on Titan).
- Conclusion: FALSE
- Counterexample: Methane rain on Titan

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Statement 19: Every mother loves her children.

- Analysis: This is a subjective statement about human emotions. While many mothers love their children, it is not universally true for all mothers due to psychological, emotional, or situational factors.
- Conclusion: FALSE
- Counterexample: A mother who has abandoned her child due to neglect or abuse

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Statement 20: All sounds can be heard.

- Analysis: Sounds are vibrations that travel through a medium (air, water, etc.) and can be detected by ears or instruments. However, some sounds are outside the range of human hearing (e.g., ultrasonic waves above 20 kHz or infrasonic waves below 20 Hz).
- Conclusion: FALSE
- Counterexample: Ultrasonic waves (above 20 kHz)

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Statement 21: A compass always points north.

- Analysis: A standard magnetic compass typically points toward Earth's magnetic north pole. However, this is not always true because the magnetic north pole is not fixed and can vary slightly over time. Additionally, local magnetic anomalies can affect a compass's accuracy.
- Conclusion: FALSE
- Counterexample: A compass near a strong magnetic anomaly or during a geomagnetic storm

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Statement 22: All birds can fly.

- Analysis: Most birds can fly, but there are notable exceptions, such as ostriches, emus, and penguins, which are flightless birds.
- Conclusion: FALSE
- Counterexample: Ostrich

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Statement 23: All natural fruit have seeds.

- Analysis: Most natural fruits contain seeds as part of their reproductive process. However, there are exceptions, such as seedless grapes or certain cultivated varieties of bananas.
- Conclusion: FALSE
- Counterexample: Seedless grapes

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Statement 24: What goes up must come down.

- Analysis: This statement is generally true for objects under the influence of gravity on Earth. However, if an object reaches escape velocity (approximately 11.2 km/s), it can leave Earth's gravitational pull and not come back down.
- Conclusion: FALSE
- Counterexample: A rocket launched into space with enough velocity to escape Earth's gravity

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Statement 25: Every day ends in day.

- Analysis: This statement is ambiguous. If interpreted literally, it suggests that every day concludes with daylight, which is not true for locations experiencing polar night (e.g., areas near the North or South Poles during winter).
- Conclusion: FALSE
- Counterexample: Polar regions during winter (e.g., Longyearbyen, Norway, during December)

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Statement 26: All state names have at least 2 syllables.

- Analysis: Most state names have at least two syllables, but there are exceptions, such as "Ohio" and "Utah," which have only one syllable.
- Conclusion: FALSE
- Counterexample: Ohio

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Final Answers:

1. Statement 15: TRUE
2. Statement 16: FALSE, Counterexample: Montana
3. Statement 17: FALSE, Counterexample: A three-legged cat
4. Statement 18: FALSE, Counterexample: Methane rain on Titan
5. Statement 19: FALSE, Counterexample: A mother who has abandoned her child
6. Statement 20: FALSE, Counterexample: Ultrasonic waves
7. Statement 21: FALSE, Counterexample: A compass near a magnetic anomaly
8. Statement 22: FALSE, Counterexample: Ostrich
9. Statement 23: FALSE, Counterexample: Seedless grapes
10. Statement 24: FALSE, Counterexample: A rocket launched into space
11. Statement 25: FALSE, Counterexample: Polar regions during winter
12. Statement 26: FALSE, Counterexample: Ohio

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Boxed Final Answer:

\[
\boxed{
\begin{array}{ll}
\text{15: TRUE} & \text{16: FALSE, Counterexample: Montana} \\
\text{17: FALSE, Counterexample: A three-legged cat} & \text{18: FALSE, Counterexample: Methane rain on Titan} \\
\text{19: FALSE, Counterexample: A mother who has abandoned her child} & \text{20: FALSE, Counterexample: Ultrasonic waves} \\
\text{21: FALSE, Counterexample: A compass near a magnetic anomaly} & \text{22: FALSE, Counterexample: Ostrich} \\
\text{23: FALSE, Counterexample: Seedless grapes} & \text{24: FALSE, Counterexample: A rocket launched into space} \\
\text{25: FALSE, Counterexample: Polar regions during winter} & \text{26: FALSE, Counterexample: Ohio} \\
\end{array}
}
\]