Deductive Reasoning

Show your work for each problem.

1.
What type of reasoning starts with general statements and reaches specific conclusions?
1. a)Deductive reasoning
2. b)Creative thinking
3. c)Inductive reasoning
4. d)Random guessing
Answer: Deductive reasoning
Deductive reasoning starts with general premises and reaches specific conclusions. If the premises are true, the conclusion must be true.
2.
In a syllogism, what do we call the statements that are assumed to be true?
1. a)Questions
2. b)Conclusions
3. c)Premises
4. d)Hypotheses
Answer: Premises
Premises are statements assumed to be true that form the basis of a logical argument. A syllogism typically has two premises that lead to a conclusion.
3.
Premise 1: All birds have feathers. Premise 2: A robin is a bird. Conclusion: A robin has ___.
Answer: feathers
Since all birds have feathers (Premise 1) and a robin is a bird (Premise 2), we can deduce that a robin has feathers.
4.
Premise 1: All even numbers are divisible by 2. Premise 2: 8 is an even number. What can you conclude?
1. a)8 is an odd number
2. b)8 is divisible by 2
3. c)2 is an even number
4. d)8 is divisible by 3
Answer: 8 is divisible by 2
Since all even numbers are divisible by 2 and 8 is an even number, we can deduce that 8 is divisible by 2.
5.
Premise 1: If it snows, school is closed. Premise 2: It is snowing. Conclusion: School is ___.
Answer: closed
If it snows, school is closed (Premise 1). It is snowing (Premise 2). Therefore, school is closed.
6.
Premise 1: All rectangles have four right angles. Premise 2: A square is a rectangle. What can you conclude about a square?
Answer: A square has four right angles
1. What property do all rectangles have (Premise 1)? four right angles
2. What category does a square belong to (Premise 2)? rectangle
3. Since a square is a rectangle, what must a square have? four right angles
7.
Which of the following is a logical fallacy (incorrect reasoning)?
1. a)If A then B. A is true. Therefore, B is true.
2. b)All triangles have 3 sides. This shape has 3 sides. Therefore, this shape is a triangle.
3. c)If it rains, the grass is wet. The grass is wet. Therefore, it rained.
4. d)All dogs are mammals. Rex is a dog. Therefore, Rex is a mammal.
Answer: If it rains, the grass is wet. The grass is wet. Therefore, it rained.
This is called 'affirming the consequent.' Just because rain causes wet grass doesn't mean wet grass was caused by rain. Sprinklers could also cause wet grass!
8.
Premise 1: If a number is divisible by 6, it is divisible by both 2 and 3. Premise 2: 18 is divisible by 6. What can you conclude about 18?
Answer: 18 is divisible by 2 and 3
1. What happens when a number is divisible by 6? It is divisible by 2 and 3
2. Is 18 divisible by 6? Yes
3. Therefore, 18 is divisible by which numbers? 2 and 3
9.
Premise 1: All prime numbers greater than 2 are odd. Premise 2: 7 is a prime number greater than 2. Is 7 odd or even?
Answer: odd
Since all prime numbers greater than 2 are odd, and 7 is a prime number greater than 2, we can deduce that 7 is odd.
10.
A valid deductive argument with true premises will always have:
1. a)A true conclusion
2. b)A false conclusion
3. c)Multiple conclusions
4. d)An uncertain conclusion
Answer: A true conclusion
This is what makes deductive reasoning so powerful: if your logic is valid AND your premises are true, your conclusion MUST be true. This is called a sound argument.
11.
Premise 1: If a polygon is regular, all its sides are equal. Premise 2: If all sides of a polygon are equal and it has 4 sides, it is a rhombus. Premise 3: Figure X is a regular polygon with 4 sides. What is Figure X?
Answer: A rhombus
1. Figure X is regular. What does this mean about its sides? All sides are equal
2. How many sides does Figure X have? 4
3. Figure X has equal sides and 4 sides. What shape is it? rhombus
12.
Premise 1: All mathematicians study logic. Premise 2: Some people who study logic also study philosophy. Premise 3: Dr. Smith is a mathematician. What can we definitely conclude?
1. a)Dr. Smith studies logic
2. b)Some mathematicians study philosophy
3. c)Dr. Smith studies philosophy
4. d)Dr. Smith does not study philosophy
Answer: Dr. Smith studies logic
We can only conclude that Dr. Smith studies logic (from Premises 1 and 3). We cannot conclude anything about philosophy because 'some' doesn't mean 'all' - Dr. Smith might or might not study philosophy.
13.
Premise 1: No reptiles have fur. Premise 2: All mammals have fur. Premise 3: A lizard is a reptile. Does a lizard have fur? (Answer yes or no)
Answer: no
Since no reptiles have fur (Premise 1) and a lizard is a reptile (Premise 3), we can deduce that a lizard does not have fur. Premise 2 about mammals is not needed for this deduction.
14.
Premise 1: If today is Monday, then tomorrow is Tuesday. Premise 2: If tomorrow is Tuesday, then the day after tomorrow is Wednesday. Premise 3: Today is Monday. What day is the day after tomorrow?
Answer: Wednesday
1. What day is today (Premise 3)? Monday
2. Since today is Monday, what day is tomorrow (use Premise 1)? Tuesday
3. Since tomorrow is Tuesday, what day is the day after tomorrow (use Premise 2)? Wednesday
15.
Which of the following arguments is both VALID and SOUND (has true premises and correct logic)?
1. a)Some birds can swim. Penguins are birds. Therefore, penguins can swim.
2. b)All squares have 4 sides. This shape is a square. Therefore, it has 4 sides.
3. c)If it rains, I get wet. I am wet. Therefore, it rained.
4. d)All fish can fly. Salmon is a fish. Therefore, salmon can fly.
Answer: All squares have 4 sides. This shape is a square. Therefore, it has 4 sides.
Only the first argument is sound: the premises are true (squares do have 4 sides) and the logic is valid. The fish argument has a false premise. The bird argument uses 'some' (invalid logic). The rain argument affirms the consequent (invalid logic).