Introduction to Logical Reasoning

Show your work for each problem.

1.
Which type of reasoning starts with specific observations and forms a general conclusion?
1. a)Inductive reasoning
2. b)Circular reasoning
3. c)Random reasoning
4. d)Deductive reasoning
Answer: Inductive reasoning
Inductive reasoning moves from specific observations to general conclusions. For example, observing that the sun rose every day and concluding it will rise tomorrow.
2.
In an argument, what are statements assumed to be true called?
1. a)Premises
2. b)Conclusions
3. c)Theories
4. d)Hypotheses
Answer: Premises
Premises are statements assumed to be true that form the foundation of an argument. The conclusion is what follows logically from the premises.
3.
Premise 1: Every spider has exactly 8 legs. Premise 2: There are 3 spiders in a box. Using deductive reasoning, how many spider legs are in the box?
Answer: 24
Deductive reasoning: the rule holds for every spider, so each of the 3 spiders has 8 legs. $3 \times 8 = 24$ legs. If both premises are true, the answer must be 24.
4.
Which type of reasoning is guaranteed to be true if the premises are true?
1. a)Inductive reasoning
2. b)Both types
3. c)Neither type
4. d)Deductive reasoning
Answer: Deductive reasoning
Deductive reasoning guarantees the conclusion is true if the premises are true. Inductive reasoning only produces probable conclusions that could be disproven by counterexamples.
5.
Is this reasoning deductive or inductive? 'I have seen 50 red apples. Therefore, all apples are red.'
1. a)Inductive, because it generalizes from the cases it has seen
2. b)Deductive, because all 50 apples are red
3. c)Deductive, because it reaches a rule about all apples
4. d)Neither, it is a proof, so the conclusion must be true
Answer: Inductive, because it generalizes from the cases it has seen
This is inductive reasoning because we observed specific cases (50 red apples) and generalized to all apples. Note: this conclusion could be wrong since green and yellow apples exist!
6.
'All fish live in water. Nemo lives in water. Therefore, Nemo is a fish.' Is this argument valid?
1. a)Yes, this is valid inductive reasoning
2. b)Yes, this is valid deductive reasoning
3. c)No, the premises are false
4. d)No, this is affirming the consequent
Answer: No, this is affirming the consequent
This is the fallacy of 'affirming the consequent'. Nemo could be a dolphin, whale, or even a submarine! Living in water doesn't mean something IS a fish.
7.
All squares are rectangles. All rectangles have four sides. What can we conclude about squares?
Answer: Squares have four sides
1. What is the first premise? All squares are rectangles
2. What is the second premise? All rectangles have four sides
3. If squares are rectangles, and rectangles have four sides, what do squares have? four sides
8.
The pattern continues: 2, 6, 18, 54, ... What is the next number?
Answer: 162
Each number is multiplied by 3: $2 \times 3 = 6$ , $6 \times 3 = 18$ , $18 \times 3 = 54$ , $54 \times 3 = 162$ . This is inductive reasoning - we observed a pattern and predicted the next term.
9.
One counterexample is enough to disprove a conclusion reached by which type of reasoning?
1. a)Both types equally
2. b)Neither type
3. c)Deductive reasoning
4. d)Inductive reasoning
Answer: Inductive reasoning
Inductive conclusions can be disproven by one counterexample. For example, 'All swans are white' is disproven by finding one black swan. Deductive conclusions are guaranteed if premises are true.
10.
The crime happened at 9 PM downtown. Suspect A has a receipt proving they were at a grocery store across town at 9 PM. What can we conclude?
Answer: Suspect A could not have committed the crime
1. What is premise 1 (about where the crime happened)? The crime happened downtown at 9 PM
2. What is premise 2 (about Suspect A)? Suspect A was across town at 9 PM
3. Can someone be in two places at once? no
4. What must we conclude about Suspect A? Suspect A could not have committed the crime
11.
A plant is 10 cm tall on Monday, 12 cm on Tuesday, 14 cm on Wednesday and 16 cm on Thursday. If the pattern continues, how tall will it probably be on Friday? (Answer in cm, number only)
Answer: 18
The plant grows 2 cm every day: 10, 12, 14, 16, so the next value is $16 + 2 = 18$ cm. This is inductive reasoning: we predict from a pattern we observed. It is probable, not guaranteed: the plant could stop growing!
12.
'If it rains, the streets get wet. The streets are wet. Therefore, it rained.' This is an example of:
1. a)Valid inductive reasoning
2. b)A true conclusion
3. c)A logical fallacy (affirming the consequent)
4. d)Valid deductive reasoning
Answer: A logical fallacy (affirming the consequent)
This is 'affirming the consequent' - a fallacy. The streets could be wet from a water main break, street cleaning, or a sprinkler. Wet streets don't prove it rained!
13.
Analyze: 'No reptiles have fur. All snakes are reptiles. Therefore, no snakes have fur.' Is this valid?
Answer: Yes, the argument is valid
1. What is premise 1? No reptiles have fur
2. What is premise 2? All snakes are reptiles
3. If snakes are reptiles, and reptiles have no fur, do snakes have fur? no
4. Is the conclusion logically valid? yes
14.
Which statement about valid arguments is TRUE?
1. a)A valid argument can have false premises but true conclusion
2. b)A valid argument always has true premises
3. c)Valid and true mean the same thing
4. d)A valid argument always has a true conclusion
Answer: A valid argument can have false premises but true conclusion
Validity only means the conclusion follows from the premises. Example: 'All cats can fly. A pigeon is a cat. Therefore, a pigeon can fly.' Both premises are false, yet the argument is valid and the conclusion is true. A valid argument with false premises can also have a false conclusion ('All cats are purple. Fluffy is a cat. Therefore, Fluffy is purple.'), so the other statements are wrong.
15.
Create a valid deductive argument using these premises: 'All students in this class passed the exam' and 'Maria is a student in this class.'
Answer: Maria passed the exam
1. What does premise 1 tell us about ALL students in the class? They passed the exam
2. What does premise 2 tell us about Maria? She is a student in this class
3. If Maria is in the class, and all class members passed, what must be true about Maria? Maria passed the exam
16.
Ana tries the expression $n^2 + n + 41$ for $n = 1, 2, 3, 4, 5$ and gets 43, 47, 53, 61, 71, all prime numbers. She concludes: 'It gives a prime number for every whole number $n$ .' Which statement about her conclusion is true?
1. a)It is proven, because five cases agree
2. b)It can never be checked, because there are infinitely many numbers
3. c)It is an inductive guess, and it is false: for $n = 40$ the result is $41 \times 41$
4. d)It is deductive reasoning, so it must be true
Answer: It is an inductive guess, and it is false: for $n = 40$ the result is $41 \times 41$
Checking a few cases is inductive reasoning: it suggests a rule but does not prove it. For $n = 40$ : $40^2 + 40 + 41 = 1600 + 40 + 41 = 1681 = 41 \times 41$ , which is not prime. One counterexample disproves the conclusion.