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)Circular reasoning
2. b)Deductive reasoning
3. c)Random reasoning
4. d)Inductive 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)Theories
2. b)Premises
3. c)Hypotheses
4. d)Conclusions
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.
All birds have feathers. A penguin is a bird. Therefore, a penguin has _______.
Answer: feathers
This is deductive reasoning: If all birds have feathers (premise 1), and a penguin is a bird (premise 2), then a penguin must have feathers (conclusion).
4.
Which type of reasoning is guaranteed to be true if the premises are true?
1. a)Deductive reasoning
2. b)Both types
3. c)Neither type
4. d)Inductive 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 the following reasoning deductive or inductive? 'I have seen 50 red apples. Therefore, all apples are red.'
Answer: inductive
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 deductive reasoning
2. b)No, this is affirming the consequent
3. c)No, the premises are false
4. d)Yes, this is valid inductive reasoning
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? (This is an example of _______ reasoning)
Answer: 162
Each number is multiplied by 3: , , , . This is inductive reasoning - we observed a pattern and predicted the next term.
9.
A counterexample can disprove 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.
Every time I water my plant, it grows. I watered it today. What will probably happen? (Use one word: grow/die/shrink)
Answer: grow
This is inductive reasoning: based on past observations (watering caused growth), we predict the same will happen again. However, this is only probable, not guaranteed!
12.
'If it rains, the streets get wet. The streets are wet. Therefore, it rained.' This is an example of:
1. a)Valid deductive reasoning
2. b)Valid inductive reasoning
3. c)A true conclusion
4. d)A logical fallacy (affirming the consequent)
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, 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/no)? yes
14.
Which statement about valid arguments is TRUE?
1. a)A valid argument always has true premises
2. b)A valid argument can have false premises but true conclusion
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 means the conclusion follows logically from the premises. Example: 'All cats are purple. Fluffy is a cat. Therefore, Fluffy is purple.' This is VALID (logic is correct) but UNTRUE (cats aren't purple).
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.
Name one type of reasoning that cannot be disproven by a single counterexample (deductive/inductive):
Answer: deductive
Deductive reasoning cannot be disproven by counterexamples IF the premises are true and the logic is valid. Inductive conclusions, however, can always potentially be disproven by finding one exception.