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 units102 lessons3176 practice & quiz problemsFree
Not sure where to start? Take the free 15-minute placement check to pinpoint your gaps, then practice with instant feedback and worked solutions.
Take the free Logic placement check Open the Logic courseCourse 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, 3176 practice and quiz problems, 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 find exactly which lessons to start with.
How do I find my gaps in Logic?
Take the free Logic placement check. It pinpoints the specific skills you're missing in about 15 minutes and builds a practice plan focused on those gaps.