Explanations, Justifications, and Proofs
A free Logic lesson from the “Foundations of Logical Thinking” unit, with a worked example and practice problems including step-by-step solutions.
A proof-ready explanation does more than report an answer. It states the claim, gives reasons, and connects each step so another person can follow why the conclusion must be true. Learning objective: Recognize what makes an explanation proof-ready. Prerequisite: No formal prerequisite. Work in this lesson starts with ordinary language, then connects the idea to symbols only after the meaning is clear. Example 1: A direct explanation starts with definitions and moves forward to the conclusion. Example 2: A counterexample must satisfy the setup and break the conclusion. A common misconception is to treat familiar wording as proof; instead, check exactly what the statement says and what follows from it.
What you'll learn
- Recognize what makes an explanation proof-ready
- Explain the idea in plain English before using symbols
- Use examples, non-examples, or counterexamples to check the reasoning
Worked example
Problem. Example case A (Explanations, Justifications, and Proofs): Which explanation is most complete?
- Worked Example: First identify exactly what the question is asking: Example case A (Explanations, Justifications, and Proofs): Which explanation is most complete?
- Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
- A complete explanation uses a definition.
- It shows the algebraic step.
Answer: Because x is even, x = 2k for an integer k, so x + 2 = 2(k + 1), which is even.
Practice problems
1. Practice case A (Explanations, Justifications, and Proofs): Which explanation is most complete?
Choices: Because x is even, x = 2k for an integer k, so x + 2 = 2(k + 1), which is even. · It stays even. · I tried x = 4. · The answer looks right.
Show solution
- Warm-up: First identify exactly what the question is asking: Practice case A (Explanations, Justifications, and Proofs): Which explanation is most complete?
- Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
- A complete explanation uses a definition.
- It shows the algebraic step.
- It connects the result back to evenness.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: Because x is even, x = 2k for an integer k, so x + 2 = 2(k + 1), which is even.
2. Practice case B (Explanations, Justifications, and Proofs): Direct reasoning usually starts with:
Choices: the given information and definitions · the opposite of the conclusion · a random answer choice · a diagram with no labels
Show solution
- Warm-up: First identify exactly what the question is asking: Practice case B (Explanations, Justifications, and Proofs): Direct reasoning usually starts with:
- Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
- Direct reasoning moves forward.
- It begins from what is given.
- Definitions and known facts justify each step.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: the given information and definitions
3. Practice case C (Explanations, Justifications, and Proofs): To disprove "All multiples of 4 are multiples of 8," which counterexample works?
Choices: 4 · 8 · 16 · 24
Show solution
- Warm-up: First identify exactly what the question is asking: Practice case C (Explanations, Justifications, and Proofs): To disprove "All multiples of 4 are multiples of 8," which counterexample works?
- Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
- A counterexample must be a multiple of 4.
- 4 is not a multiple of 8.
- So 4 disproves the claim.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: 4
4. Practice case D (Explanations, Justifications, and Proofs): Reasoning by cases is appropriate when:
Choices: the cases cover all possibilities · only one example is checked · the conclusion is ignored · the domain is unknown
Show solution
- Casework splits a problem into possibilities.
- The proof is complete only if every possibility is covered.
- Then each case can be handled separately.
Answer: the cases cover all possibilities
5. Practice case E (Explanations, Justifications, and Proofs): In a contradiction argument, you begin by:
Choices: assuming the opposite of what you want to prove · assuming the conclusion is already true · checking only easy numbers · deleting the hypothesis
Show solution
- Contradiction starts with the opposite assumption.
- If that assumption leads to an impossibility, it cannot hold.
- Then the original claim is supported.
Answer: assuming the opposite of what you want to prove
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