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): Worked example: Which rule or habit best matches Explanations, Justifications, and Proofs?
- Explanations, Justifications, and Proofs targets a specific reasoning habit.
- Explanations, Justifications, and Proofs focuses on recognize what makes an explanation proof-ready.
- The other choices either overclaim or change the logical relationship.
Answer: Explanations, Justifications, and Proofs focuses on recognize what makes an explanation proof-ready.
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 choice with the stated logical rule, and eliminate choices that change the claim's meaning.
- 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 preserves the stated logical meaning and that the other choices change the rule or claim.
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 choice with the stated logical rule, and eliminate choices that change the claim's meaning.
- 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 preserves the stated logical meaning and that the other choices change the rule or claim.
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?
- For quantified statements, identify the domain first, then decide whether the claim is about all objects or at least one object.
- 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 preserves the stated logical meaning and that the other choices change the rule or claim.
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): Lesson focus: Which transfer task would show readiness for Explanations, Justifications, and Proofs?
Choices: Apply recognize what makes an explanation proof-ready in a new sentence or context. · Repeat the exact worked example without changing the context. · Pick an answer before identifying the claim parts. · Use a rule from another lesson because the words sound close.
Show solution
- A transfer task keeps the same objective but changes the surface context.
- Apply recognize what makes an explanation proof-ready in a new sentence or context.
- That is a better mastery signal than memorizing a practice prompt.
Answer: Apply recognize what makes an explanation proof-ready in a new sentence or context.
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