CMClearMathAcademy

Good Definitions vs. Bad Definitions

A free Logic lesson from the “Biconditionals and Definitions” unit, with a worked example and practice problems including step-by-step solutions.

Definitions should classify examples correctly. A bad definition may include things it should exclude, exclude things it should include, or use the term it is trying to define. Learning objective: Identify definitions that are too broad, too narrow, or circular. 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: 'x is even iff x is divisible by 2' works in both directions. Example 2: A definition that only works one way is too broad or too narrow. 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

Why it matters: Definitions in math and science depend on knowing when a condition works both ways.

Worked example

Problem. Example case A (Good Definitions vs. Bad Definitions): Worked example: Which rule or habit best matches Good Definitions vs. Bad Definitions?

  1. Good Definitions vs. Bad Definitions targets a specific reasoning habit.
  2. Good Definitions vs. Bad Definitions focuses on identify definitions that are too broad, too narrow, or circular.
  3. The other choices either overclaim or change the logical relationship.

Answer: Good Definitions vs. Bad Definitions focuses on identify definitions that are too broad, too narrow, or circular.

Practice problems

1. Practice case A (Good Definitions vs. Bad Definitions): What does p ↔ q mean?

Choices: p implies q and q implies p · p implies q only · p and q are both false · not p or q

Show solution
  1. Warm-up: First identify exactly what the question is asking: Practice case A (Good Definitions vs. Bad Definitions): What does p ↔ q mean?
  2. For symbolic logic, translate the symbols into plain English, then apply the truth condition for the connective.
  3. A biconditional is two-way.
  4. It contains both conditional directions.
  5. So p ↔ q means each statement implies the other.
  6. Verify the selected choice by checking that it preserves the stated logical meaning and that the other choices change the rule or claim.

Answer: p implies q and q implies p

2. Practice case B (Good Definitions vs. Bad Definitions): Which definition works as a biconditional?

Choices: A number is even iff it is divisible by 2. · A number is even iff it is positive. · A rectangle is a square iff it has four sides. · A prime number is any odd number.

Show solution
  1. A good definition works both ways.
  2. Even numbers are exactly the integers divisible by 2.
  3. The other definitions are too broad or false.

Answer: A number is even iff it is divisible by 2.

3. Practice case C (Good Definitions vs. Bad Definitions): A definition is too broad when it:

Choices: includes things that should be excluded · excludes correct examples · uses symbols · has examples

Show solution
  1. Too broad means the category catches extra objects.
  2. For example, defining square as any four-sided figure includes rectangles that are not squares.
  3. So it includes too much.

Answer: includes things that should be excluded

4. Practice case D (Good Definitions vs. Bad Definitions): A definition is too narrow when it:

Choices: excludes things that should be included · includes every possible object · uses if and only if · has a non-example

Show solution
  1. Too narrow means real examples are left out.
  2. A good definition includes all and only the intended objects.
  3. So the first choice is correct.

Answer: excludes things that should be included

5. Practice case E (Good Definitions vs. Bad Definitions): Lesson focus: Which transfer task would show readiness for Good Definitions vs. Bad Definitions?

Choices: Apply identify definitions that are too broad, too narrow, or circular 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
  1. A transfer task keeps the same objective but changes the surface context.
  2. Apply identify definitions that are too broad, too narrow, or circular in a new sentence or context.
  3. That is a better mastery signal than memorizing a practice prompt.

Answer: Apply identify definitions that are too broad, too narrow, or circular in a new sentence or context.

Practice this interactively with instant feedback and an AI tutor.

Practice Good Definitions vs. Bad Definitions Take the free placement check

More Logic lessons