All mammals are vertebrates. All whales are mammals. Which conclusion must be true?

Correct: B) All whales are vertebrates.

Transitive inclusion: whale ⊆ mammal ⊆ vertebrate, so every whale is a vertebrate (B). A reverses the relation, C contradicts the premise, and D over-restricts — none of these is guaranteed.

“If you press the button, the lamp turns on.” You pressed the button. What must be true?

Correct: A) The lamp turns on.

This is affirming the antecedent (modus ponens): the antecedent 'pressed' holds, so the consequent 'turns on' must follow (A). C affirms the consequent, an invalid direction that is not guaranteed.

An orange is heavier than an apple. An apple is heavier than a grape. Which must be true?

Correct: C) The orange is heavier than the grape.

'Heavier than' is transitive: orange > apple > grape, so orange > grape (C). A reverses it, B is wrong since the orange is heaviest, and D has no basis in the premises.

What is the logical negation of “All the balls in this box are red”?

Correct: D) At least one ball is not red.

The negation of 'all are red' is 'at least one is not red' (D). A and B state the contrary (none is red), not the negation, and C is consistent with the original claim, so it does not negate it.

All firefighters are brave. Some citizens are firefighters. Which must be true?

Correct: A) Some citizens are brave.

Some citizens are firefighters, and firefighters are brave, so some citizens are brave (A). B over-strengthens to 'all', C reverses the relation, and D contradicts the premises — all exceed what is guaranteed.

“If the key is correct, the door opens.” The door did not open. What must be true?

Correct: C) The key was not correct.

Contrapositive: 'did not open' implies 'key not correct'. Had the key been correct, the door would have opened; since it did not, the key was not correct (C). B guesses at a cause not in the premises, so it is not guaranteed.

Taro is taller than Jiro. Saburo is taller than Taro. Jiro is taller than Shiro. Who is the tallest?

Correct: C) Saburo

Saburo > Taro > Jiro and Jiro > Shiro, so overall Saburo > Taro > Jiro > Shiro. The tallest is Saburo (C). Shiro is the shortest, so D is wrong.

What is the logical negation of “Some employees work from home”?

Correct: B) No employee works from home.

'Some do (at least one exists)' is negated by 'none do (all do not)' (B). A merely says some others do not, which is compatible with the original, and D is the same; C moves in the wrong direction.

All metals conduct electricity. Rubber does not conduct electricity. Which must be true?

Correct: D) Rubber is not a metal.

This uses the contrapositive: 'metal → conducts' gives 'does not conduct → not metal'. Rubber does not conduct, so rubber is not a metal (D). C is the converse and fails because non-metals can also conduct.

“If you pass the exam, you can enroll. If you can enroll, you can apply for a scholarship.” Mina passed the exam. What must be true?

Correct: B) Mina can apply for a scholarship.

Chain the conditionals: pass → enroll → can apply, so Mina can apply (B). A confuses 'can apply' with 'will receive'; being awarded the scholarship is not guaranteed.

Five people stand in a row. A is immediately to the right of B. C is somewhere to the left of B. D is immediately to the right of A. Which ordering statement must be true? (left to right)

Correct: A) C is always to the left of A.

The fixed block is C … B, A, D (A right of B, D right of A, C left of B), so C is always left of A (A). E could sit right of D, so B is unsettled; E could sit left of C, so C is unsettled; B sits between A and C, so D is unsettled too.

It turns out that “everyone in this class is good at both math and English” is false. Which must be true?

Correct: C) At least one person is bad at math or bad at English (at least one of the two).

Negating 'everyone is good at (math AND English)' gives 'at least one person is not good at (math AND English)', i.e. that person is bad at math or bad at English (C). A, B, and D over-reach to 'everyone' or a blanket denial, which is not guaranteed.

“All planets shine by their own light. Earth is a planet. Therefore Earth shines by its own light.” Which best evaluates this argument?

Correct: C) The form is valid, but the premise 'all planets shine by their own light' is factually false.

The syllogistic form is valid — if the premises were true the conclusion would follow. The flaw is that the premise 'all planets shine by their own light' is factually false; validity (form) and truth of premises are separate (C). Earth is a planet, so D is wrong.

“If it is genuine, it has a hallmark.” An item is examined and it has a hallmark. From this fact alone, what must be true?

Correct: D) This fact alone cannot decide whether it is genuine.

From 'genuine → hallmark', the mere presence of a hallmark cannot establish genuineness (a fake can be stamped too — affirming the consequent is invalid). So it cannot be decided (D). A commits that fallacy, and C mangles the contrapositive (the correct one is 'no hallmark → not genuine').

About the ranking of four teams we know: 'Red ranks above Blue', 'Blue ranks above Green', 'Yellow is not ranked below Red'. No ties. Which team must be in 1st place?

Correct: B) Yellow

Red > Blue > Green (above = better rank). 'Yellow not below Red' with no ties means Yellow > Red. So Yellow > Red > Blue > Green, and 1st must be Yellow (B). Red is below Yellow, so A is wrong, and the order is fully forced, so D is wrong.

What is the correct negation of “there exists a book that every student in the class has read”? (mind the scope)

Correct: B) For every book, at least one student has not read it.

The claim is existential: 'some book exists that all students read' (∃book ∀student read). Its negation is 'no such book exists' — for every book, at least one student has not read it (B). A only fixes a single book and negates the 'every student' part, leaving the 'exists' claim un-negated, so it is wrong. C is a stronger, unrelated claim, and D is irrelevant.

All mammals are vertebrates. All whales are mammals. Which conclusion must be true?

Correct: B) All whales are vertebrates.

Transitive inclusion: whale ⊆ mammal ⊆ vertebrate, so every whale is a vertebrate (B). A reverses the relation, C contradicts the premise, and D over-restricts — none of these is guaranteed.

“If you press the button, the lamp turns on.” You pressed the button. What must be true?

Correct: A) The lamp turns on.

This is affirming the antecedent (modus ponens): the antecedent 'pressed' holds, so the consequent 'turns on' must follow (A). C affirms the consequent, an invalid direction that is not guaranteed.

An orange is heavier than an apple. An apple is heavier than a grape. Which must be true?

Correct: C) The orange is heavier than the grape.

'Heavier than' is transitive: orange > apple > grape, so orange > grape (C). A reverses it, B is wrong since the orange is heaviest, and D has no basis in the premises.

What is the logical negation of “All the balls in this box are red”?

Correct: D) At least one ball is not red.

The negation of 'all are red' is 'at least one is not red' (D). A and B state the contrary (none is red), not the negation, and C is consistent with the original claim, so it does not negate it.

All firefighters are brave. Some citizens are firefighters. Which must be true?

Correct: A) Some citizens are brave.

Some citizens are firefighters, and firefighters are brave, so some citizens are brave (A). B over-strengthens to 'all', C reverses the relation, and D contradicts the premises — all exceed what is guaranteed.

“If the key is correct, the door opens.” The door did not open. What must be true?

Correct: C) The key was not correct.

Contrapositive: 'did not open' implies 'key not correct'. Had the key been correct, the door would have opened; since it did not, the key was not correct (C). B guesses at a cause not in the premises, so it is not guaranteed.

Taro is taller than Jiro. Saburo is taller than Taro. Jiro is taller than Shiro. Who is the tallest?

Correct: C) Saburo

Saburo > Taro > Jiro and Jiro > Shiro, so overall Saburo > Taro > Jiro > Shiro. The tallest is Saburo (C). Shiro is the shortest, so D is wrong.

What is the logical negation of “Some employees work from home”?

Correct: B) No employee works from home.

'Some do (at least one exists)' is negated by 'none do (all do not)' (B). A merely says some others do not, which is compatible with the original, and D is the same; C moves in the wrong direction.

All metals conduct electricity. Rubber does not conduct electricity. Which must be true?

Correct: D) Rubber is not a metal.

This uses the contrapositive: 'metal → conducts' gives 'does not conduct → not metal'. Rubber does not conduct, so rubber is not a metal (D). C is the converse and fails because non-metals can also conduct.

“If you pass the exam, you can enroll. If you can enroll, you can apply for a scholarship.” Mina passed the exam. What must be true?

Correct: B) Mina can apply for a scholarship.

Chain the conditionals: pass → enroll → can apply, so Mina can apply (B). A confuses 'can apply' with 'will receive'; being awarded the scholarship is not guaranteed.

Five people stand in a row. A is immediately to the right of B. C is somewhere to the left of B. D is immediately to the right of A. Which ordering statement must be true? (left to right)

Correct: A) C is always to the left of A.

The fixed block is C … B, A, D (A right of B, D right of A, C left of B), so C is always left of A (A). E could sit right of D, so B is unsettled; E could sit left of C, so C is unsettled; B sits between A and C, so D is unsettled too.

It turns out that “everyone in this class is good at both math and English” is false. Which must be true?

Correct: C) At least one person is bad at math or bad at English (at least one of the two).

Negating 'everyone is good at (math AND English)' gives 'at least one person is not good at (math AND English)', i.e. that person is bad at math or bad at English (C). A, B, and D over-reach to 'everyone' or a blanket denial, which is not guaranteed.

“All planets shine by their own light. Earth is a planet. Therefore Earth shines by its own light.” Which best evaluates this argument?

Correct: C) The form is valid, but the premise 'all planets shine by their own light' is factually false.

The syllogistic form is valid — if the premises were true the conclusion would follow. The flaw is that the premise 'all planets shine by their own light' is factually false; validity (form) and truth of premises are separate (C). Earth is a planet, so D is wrong.

“If it is genuine, it has a hallmark.” An item is examined and it has a hallmark. From this fact alone, what must be true?

Correct: D) This fact alone cannot decide whether it is genuine.

From 'genuine → hallmark', the mere presence of a hallmark cannot establish genuineness (a fake can be stamped too — affirming the consequent is invalid). So it cannot be decided (D). A commits that fallacy, and C mangles the contrapositive (the correct one is 'no hallmark → not genuine').

About the ranking of four teams we know: 'Red ranks above Blue', 'Blue ranks above Green', 'Yellow is not ranked below Red'. No ties. Which team must be in 1st place?

Correct: B) Yellow

Red > Blue > Green (above = better rank). 'Yellow not below Red' with no ties means Yellow > Red. So Yellow > Red > Blue > Green, and 1st must be Yellow (B). Red is below Yellow, so A is wrong, and the order is fully forced, so D is wrong.

What is the correct negation of “there exists a book that every student in the class has read”? (mind the scope)

Correct: B) For every book, at least one student has not read it.

The claim is existential: 'some book exists that all students read' (∃book ∀student read). Its negation is 'no such book exists' — for every book, at least one student has not read it (B). A only fixes a single book and negates the 'every student' part, leaving the 'exists' claim un-negated, so it is wrong. C is a stronger, unrelated claim, and D is irrelevant.

All mammals are vertebrates. All whales are mammals. Which conclusion must be true?

Correct: B) All whales are vertebrates.

Transitive inclusion: whale ⊆ mammal ⊆ vertebrate, so every whale is a vertebrate (B). A reverses the relation, C contradicts the premise, and D over-restricts — none of these is guaranteed.

“If you press the button, the lamp turns on.” You pressed the button. What must be true?

Correct: A) The lamp turns on.

This is affirming the antecedent (modus ponens): the antecedent 'pressed' holds, so the consequent 'turns on' must follow (A). C affirms the consequent, an invalid direction that is not guaranteed.

An orange is heavier than an apple. An apple is heavier than a grape. Which must be true?

Correct: C) The orange is heavier than the grape.

'Heavier than' is transitive: orange > apple > grape, so orange > grape (C). A reverses it, B is wrong since the orange is heaviest, and D has no basis in the premises.

What is the logical negation of “All the balls in this box are red”?

Correct: D) At least one ball is not red.

The negation of 'all are red' is 'at least one is not red' (D). A and B state the contrary (none is red), not the negation, and C is consistent with the original claim, so it does not negate it.

All firefighters are brave. Some citizens are firefighters. Which must be true?

Correct: A) Some citizens are brave.

Some citizens are firefighters, and firefighters are brave, so some citizens are brave (A). B over-strengthens to 'all', C reverses the relation, and D contradicts the premises — all exceed what is guaranteed.

“If the key is correct, the door opens.” The door did not open. What must be true?

Correct: C) The key was not correct.

Contrapositive: 'did not open' implies 'key not correct'. Had the key been correct, the door would have opened; since it did not, the key was not correct (C). B guesses at a cause not in the premises, so it is not guaranteed.

Taro is taller than Jiro. Saburo is taller than Taro. Jiro is taller than Shiro. Who is the tallest?

Correct: C) Saburo

Saburo > Taro > Jiro and Jiro > Shiro, so overall Saburo > Taro > Jiro > Shiro. The tallest is Saburo (C). Shiro is the shortest, so D is wrong.

What is the logical negation of “Some employees work from home”?

Correct: B) No employee works from home.

'Some do (at least one exists)' is negated by 'none do (all do not)' (B). A merely says some others do not, which is compatible with the original, and D is the same; C moves in the wrong direction.

All metals conduct electricity. Rubber does not conduct electricity. Which must be true?

Correct: D) Rubber is not a metal.

This uses the contrapositive: 'metal → conducts' gives 'does not conduct → not metal'. Rubber does not conduct, so rubber is not a metal (D). C is the converse and fails because non-metals can also conduct.

“If you pass the exam, you can enroll. If you can enroll, you can apply for a scholarship.” Mina passed the exam. What must be true?

Correct: B) Mina can apply for a scholarship.

Chain the conditionals: pass → enroll → can apply, so Mina can apply (B). A confuses 'can apply' with 'will receive'; being awarded the scholarship is not guaranteed.

Five people stand in a row. A is immediately to the right of B. C is somewhere to the left of B. D is immediately to the right of A. Which ordering statement must be true? (left to right)

Correct: A) C is always to the left of A.

The fixed block is C … B, A, D (A right of B, D right of A, C left of B), so C is always left of A (A). E could sit right of D, so B is unsettled; E could sit left of C, so C is unsettled; B sits between A and C, so D is unsettled too.

It turns out that “everyone in this class is good at both math and English” is false. Which must be true?

Correct: C) At least one person is bad at math or bad at English (at least one of the two).

Negating 'everyone is good at (math AND English)' gives 'at least one person is not good at (math AND English)', i.e. that person is bad at math or bad at English (C). A, B, and D over-reach to 'everyone' or a blanket denial, which is not guaranteed.

“All planets shine by their own light. Earth is a planet. Therefore Earth shines by its own light.” Which best evaluates this argument?

Correct: C) The form is valid, but the premise 'all planets shine by their own light' is factually false.

The syllogistic form is valid — if the premises were true the conclusion would follow. The flaw is that the premise 'all planets shine by their own light' is factually false; validity (form) and truth of premises are separate (C). Earth is a planet, so D is wrong.

“If it is genuine, it has a hallmark.” An item is examined and it has a hallmark. From this fact alone, what must be true?

Correct: D) This fact alone cannot decide whether it is genuine.

From 'genuine → hallmark', the mere presence of a hallmark cannot establish genuineness (a fake can be stamped too — affirming the consequent is invalid). So it cannot be decided (D). A commits that fallacy, and C mangles the contrapositive (the correct one is 'no hallmark → not genuine').

About the ranking of four teams we know: 'Red ranks above Blue', 'Blue ranks above Green', 'Yellow is not ranked below Red'. No ties. Which team must be in 1st place?

Correct: B) Yellow

Red > Blue > Green (above = better rank). 'Yellow not below Red' with no ties means Yellow > Red. So Yellow > Red > Blue > Green, and 1st must be Yellow (B). Red is below Yellow, so A is wrong, and the order is fully forced, so D is wrong.

What is the correct negation of “there exists a book that every student in the class has read”? (mind the scope)

Correct: B) For every book, at least one student has not read it.

The claim is existential: 'some book exists that all students read' (∃book ∀student read). Its negation is 'no such book exists' — for every book, at least one student has not read it (B). A only fixes a single book and negates the 'every student' part, leaving the 'exists' claim un-negated, so it is wrong. C is a stronger, unrelated claim, and D is irrelevant.