Deductive Reasoning

Show your work for each problem.

1.
What type of reasoning starts with general statements and reaches specific conclusions?
1. a)Random guessing
2. b)Creative thinking
3. c)Inductive reasoning
4. d)Deductive reasoning
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)Premises
3. c)Conclusions
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: Every spider has 8 legs. Premise 2: Charlotte is a spider. How many legs does Charlotte have?
Answer: 8
Every spider has 8 legs (Premise 1) and Charlotte is a spider (Premise 2), so Charlotte has 8 legs.
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)2 is an even number
3. c)8 is divisible by 2
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: All triangles have 3 sides. Premise 2: Shapes A, B and C are triangles. How many sides do the three shapes have in total?
Answer: 9
By Premises 1 and 2, each of the shapes A, B and C has 3 sides. In total: $3 + 3 + 3 = 9$ sides.
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)If it rains, the grass is wet. The grass is wet. Therefore, it rained.
3. c)All triangles have 3 sides. This shape is a triangle. Therefore, this shape has 3 sides.
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?
1. a)It cannot be determined
2. b)Odd
3. c)Both odd and even
4. d)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)Some mathematicians study philosophy
2. b)Dr. Smith studies logic
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, a snake and a turtle are reptiles; a dog and a rabbit are mammals. How many of these five animals have fur?
Answer: 2
The lizard, the snake and the turtle are reptiles, so by Premise 1 they have no fur. The dog and the rabbit are mammals, so by Premise 2 they have fur. That makes 2 animals with fur.
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 fish can fly. Salmon is a fish. Therefore, salmon can fly.
3. c)If it rains, I get wet. I am wet. Therefore, it rained.
4. d)All squares have 4 sides. This shape is a square. Therefore, it has 4 sides.
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).