Logic: Free Online Course & Practice
Learn the building blocks of clear reasoning: statements, truth values, logical connectives, truth tables, conditionals, logical equivalence, argument validity, quantifiers, counterexamples, sets, diagrams, and proof-ready explanations. A beginner-friendly bridge from math reasoning to algebra, proofs, computer science, and advanced problem solving.
Middle school, high school, homeschool, adult learners, and future computer science students12 units90 lessons + 12 assessments1998 practice & quiz itemsFree
Item counts include repeats and variants. There are 1177 distinct prompt texts after removing variant labels; this is not a count of unique mathematical skills.
Not sure where to start? Take the free 15-minute placement check to identify possible gaps, then practice with instant feedback and worked solutions.
Take the free Logic placement check Open the Logic courseStarting point and intended outcomes
Suggested preparation: Comfort reading short mathematical statements; basic arithmetic.
Learning goals: Analyze statements and quantifiers, test arguments, and write justified mathematical arguments.
These describe the course's intended scope. A score on one screen or final is only one piece of evidence about these goals.
Course outline
Foundations of Logical Thinking
- What Logic Is and Why It Matters
- Statements vs. Non-Statements
- Truth Values: True, False, and Unknown
- Precision in Mathematical Language
- Arguments: Premises and Conclusions
- Validity vs. Truth
- Explanations, Justifications, and Proofs
- Unit 1 Review and Checkpoint
Statements and Negation
- Simple Statements
- Compound Statements
- Negation in Plain English
- Negating Mathematical Claims
- Double Negatives
- Common Negation Mistakes
- Translating "Not" into Symbols
- Unit 2 Review and Checkpoint
And, Or, and Basic Connectives
- Conjunction: "And"
- Disjunction: "Or"
- Inclusive Or vs. Exclusive Or
- Translating And/Or Statements
- Compound Claims in Math Word Problems
- Truth Conditions for And and Or
- Mixed Connective Statements
- Unit 3 Review and Checkpoint
Truth Tables
- What a Truth Table Shows
- Truth Tables for Not
- Truth Tables for And
- Truth Tables for Or
- Truth Tables with Two Variables
- Truth Tables with Three Variables
- Compound Truth Tables
- Reading Truth Tables Back into English
- Unit 4 Review and Checkpoint
Conditionals
- If-Then Statements
- Hypotheses and Conclusions
- Necessary Conditions
- Sufficient Conditions
- Converse Statements
- Inverse Statements
- Contrapositive Statements
- Conditional Statements in Algebra and Geometry
- Unit 5 Review and Checkpoint
Biconditionals and Definitions
- If and Only If
- Biconditionals in Definitions
- Good Definitions vs. Bad Definitions
- Examples and Non-Examples
- Necessary and Sufficient Together
- Translating Biconditionals
- Biconditionals in Math
- Unit 6 Review and Checkpoint
Logical Equivalence
- What Logical Equivalence Means
- Equivalent Negations
- De Morgan's Laws
- Equivalent Conditional Statements
- Using Truth Tables to Test Equivalence
- Simplifying Logical Statements
- Common Equivalence Mistakes
- Unit 7 Review and Checkpoint
Argument Validity
- What Makes an Argument Valid
- Testing Arguments with Truth Tables
- Modus Ponens
- Modus Tollens
- Hypothetical Syllogism
- Disjunctive Syllogism
- Affirming the Consequent
- Denying the Antecedent
- Valid vs. Invalid Reasoning in Word Problems
- Unit 8 Review and Checkpoint
Quantifiers and Predicates
- Predicates and Open Sentences
- Domains of Discourse
- Universal Quantifiers: "All"
- Existential Quantifiers: "Some"
- Translating Quantified Statements
- Negating Universal Statements
- Negating Existential Statements
- Mixed Quantifier Practice
- Unit 9 Review and Checkpoint
Counterexamples, Sets, and Diagrams
- What a Counterexample Does
- Finding Counterexamples
- Sets and Categories
- Venn Diagrams for "All"
- Venn Diagrams for "Some"
- Venn Diagrams for "No"
- Testing Claims with Diagrams
- Logic in Classification Problems
- Unit 10 Review and Checkpoint
Proof Readiness
- What Counts as a Mathematical Explanation
- Direct Reasoning
- Proof by Counterexample
- Reasoning by Cases
- Introduction to Contradiction
- Organizing a Proof-Like Explanation
- Checking Whether an Explanation Is Complete
- Unit 11 Review and Checkpoint
Logic Applications and Final Review
- Logic in Algebra
- Logic in Geometry
- Logic in Statistics and Probability Statements
- Logic in Computer Science and Boolean Thinking
- Logic in AI Prompts and Clear Instructions
- Mixed Logic Review
- Final Logic Readiness Review
- Final Course Assessment
Frequently asked questions
Is the Logic course free?
Yes. The entire Logic course — 102 lessons, 1998 practice and quiz items, including repeats and variants, worked solutions, and the placement check — is free to use at ClearMath Academy.
What does the Logic course cover?
Learn the building blocks of clear reasoning: statements, truth values, logical connectives, truth tables, conditionals, logical equivalence, argument validity, quantifiers, counterexamples, sets, diagrams, and proof-ready explanations. A beginner-friendly bridge from math reasoning to algebra, proofs, computer science, and advanced problem solving. It spans 12 units and 102 lessons, each with a worked example and practice problems with step-by-step solutions.
Who is Logic for?
Logic is aimed at middle school, high school, homeschool, adult learners, and future computer science students. You can take a free 15-minute placement check first to choose lessons to try first.
How do I find my gaps in Logic?
Take the free Logic placement check. It samples selected skills and suggests a practice plan. Skipped or untested topics remain unassessed; this is not a validated readiness certification.