Rigorous Reasoning
Think clearly. Argue precisely. Reason rigorously.
Guided lessons in proof, explanation, probability, fallacies, and problem solving, with instant AI feedback on every response you write. A complete logic education for $5/month.
Built for serious students and instructors who want more than quiz apps
Formal Systems
Symbolic structure and proof
Move from propositional form to categorical structure, quantified reasoning, and rigorous proof practice. Build fluency in the foundations that every other reasoning mode depends on.
Reasoning Under Uncertainty
Evidence, explanation, and probability
Compare inductive support, abductive inference, arguments to the best explanation, and Bayesian updating without collapsing them into deduction. Learn when each tool applies and why.
Conceptual Control
Definitions, fallacies, and problem solving
Diagnose bad reasoning, construct precise definitions, and evaluate strategies for solving real intellectual problems. The skills that tie formal logic to everyday thinking.
How it works
Three steps to better reasoning
1
Study the concepts
Each lesson walks you through the ideas with careful exposition, worked examples, and formalization patterns, not just a list of rules to memorize.
2
Practice with feedback
Write proofs, construct explanations, evaluate arguments, and get AI-powered feedback that tells you exactly where your reasoning breaks down.
3
Track your mastery
Progress tracking across 69+ lessons and 14 units shows you what you know, what needs work, and what to study next.
Why Rigorous Reasoning?
Depth that other platforms skip
Not just quizzes
Full instructional sequences with exposition, worked examples, formalization patterns, and guided activities. Every unit teaches before it tests.
Cross-modal reasoning
Deductive, inductive, abductive, Bayesian, definitional, analogical, and more. You learn each mode on its own terms, then see how they connect.
Built for depth
Historical context, formalization patterns, mastery checks, and conceptual links between units. This is a curriculum, not a flashcard deck.
Pricing
Plans from $0/month
Start with free access to sample lessons. Unlock the full curriculum, AI feedback, and progress tracking for $5/month.
Curriculum Catalog
Open a unit with real instructional depth
Search by topic, reasoning mode, lesson content, or concepts. Every card previews the actual material inside the unit before you open it.
Foundations | 1 unitDefinitions And Concepts | 1 unitNatural Deduction | 1 unitCategorical Logic | 1 unitPropositional Logic | 1 unitPredicate Logic | 1 unit
Foundations
5 lessonsFoundations of Logical Reasoning
What reasoning is and why structure matters
Students learn what arguments are, how to identify their parts, distinguish three fundamental modes of reasoning, evaluate argument quality, spot arguments in everyday language, and understand why formalization is valuable.
Inside This Unit
- Arguments and Their Parts
- Three Modes of Reasoning
- Evaluating Arguments
ArgumentPremiseConclusionInference
Open unitDefinitions And Concepts
4 lessonsDefinitions and Conceptual Precision
How to construct, classify, and evaluate definitions rigorously
Students learn what definitions do, how different kinds of definitions serve different purposes, and how to assess definitions for scope, circularity, vagueness, and practical usefulness. The unit equips learners to both build better definitions and critique sloppy ones.
Inside This Unit
- What Definitions Are For
- Kinds of Definitions and When to Use Them
- Constructing and Evaluating Definitions
DefiniendumDefiniensFunction of a DefinitionCircularity
Open unitNatural Deduction
5 lessonsNatural Deduction: Validity and Formal Proof
Why some conclusions follow necessarily
Students learn to distinguish validity from truth, translate ordinary arguments into symbolic form, construct short natural-deduction proofs, and diagnose exactly where a flawed derivation breaks down.
Inside This Unit
- Validity vs Truth
- Symbolizing Propositional Arguments
- Basic Natural Deduction
ValiditySoundnessEntailmentProof
Open unitCategorical Logic
6 lessonsCategorical Logic: Terms, Classes, and Syllogistic Form
How class statements support traditional deductive reasoning
Students learn to analyze standard-form categorical propositions, master quantity and quality, use the square of opposition, represent class claims with Venn diagrams, map syllogisms by major/minor/middle terms, and evaluate categorical syllogisms by distribution rules and diagrammatic tests.
Inside This Unit
- Categorical Propositions and the A, E, I, O Forms
- The Square of Opposition and Existential Import
- Venn Diagrams for Categorical Propositions
Categorical PropositionSubject TermPredicate TermQuantity and Quality
Open unitPropositional Logic
6 lessonsPropositional Logic: Form, Connectives, and Valid Inference
How whole statements combine into logically assessable structures
Students learn to identify atomic and compound statements, master the five connectives of propositional logic, symbolize natural-language arguments, evaluate validity with truth tables, and construct formal proofs using basic inference rules.
Inside This Unit
- Atomic and Compound Statements
- The Five Connectives in Depth
- Symbolizing Whole Arguments
Atomic StatementCompound StatementLogical ConnectiveMain Connective
Open unitPredicate Logic
5 lessonsPredicate Logic: Quantifiers, Predicates, and Formal Structure
How internal sentence structure changes formal reasoning
Students move beyond sentence-level structure to analyze internal form with predicates, variables, and quantifiers. They learn to translate quantified claims, manage quantifier scope and variable binding, and construct short proofs using universal and existential instantiation and generalization.
Inside This Unit
- Why Predicate Logic Extends Propositional Logic
- Universal and Existential Quantifiers
- Multiple Quantifiers and Scope
PredicateIndividual ConstantVariableDomain of Discourse
Open unitModal And Intensional Logic
5 lessonsModal Logic: Necessity, Possibility, and Counterfactuals
Reasoning about what must, might, and could have been
Students learn to reason about necessity, possibility, and counterfactual conditionals using the box and diamond operators, possible-world semantics, and the Lewis-Stalnaker treatment of would-conditionals. They also learn to formalize de dicto and de re modal claims and to apply modal reasoning to philosophical, ethical, and scientific arguments.
Inside This Unit
- What Is Modal Logic?
- The Modal Operators and Their Rules
- Formalizing Modal Arguments
NecessityPossibilityPossible WorldAccessibility Relation
Open unitFallacies And Errors
5 lessonsFallacies and Reasoning Errors
From fundamental inferential failures to specific species across modes of reasoning
Students learn a systematic framework for fallacies by grouping common errors under fundamental patterns of reasoning failure, then tracing how those patterns generate distinct species in deductive, inductive, abductive, best-explanation, Bayesian, and problem-solving contexts. The unit emphasizes diagnosis and repair rather than label memorization.
Inside This Unit
- Fundamental Fallacy Families
- Deductive and Formal Fallacies
- Inductive, Abductive, and Best-Explanation Fallacies
Fundamental FallacySpecies of a FallacyRelevance FailureSupport Miscalibration
Open unitInductive Logic
5 lessonsInductive Logic: Evidence, Generalization, and Causal Support
How to reason well under uncertainty
Students learn how inductive reasoning differs from deductive reasoning, how to measure inductive strength, and how to assess generalizations, analogies, and causal claims responsibly. The unit builds from the idea of defeasible support through sampling, analogy, and Mill's methods for disentangling causation from correlation.
Inside This Unit
- What Makes an Inductive Argument Strong?
- Generalization and Sample Quality
- Analogical Reasoning
Inductive StrengthDefeasibilityRepresentativenessSample Size
Open unitBayesian Probability
4 lessonsBayesian Probability: Updating Belief with Evidence
How priors, likelihoods, and evidence interact in rational belief revision
Students learn the logic of Bayesian reasoning, including priors, likelihoods, posterior updating, base rates, and the disciplined use of probabilistic evidence in inquiry and decision making. The unit emphasizes qualitative Bayesian discipline first and then builds toward simple quantitative updates.
Inside This Unit
- Priors, Evidence, and Posterior Belief
- Base Rates and Conditional Probability
- Bayesian Comparison of Rival Hypotheses
Prior ProbabilityLikelihoodPosterior ProbabilityBase Rate
Open unitAbductive Logic
5 lessonsAbductive Logic: Arguments to the Best Explanation
How to compare explanations without confusing them with proofs
Students learn how arguments to the best explanation work, how to compare competing hypotheses using explanatory virtues, and how to state abductive conclusions with the right level of confidence. The unit emphasizes that abduction is a comparative discipline, not a search for single winning stories.
Inside This Unit
- What Is Abductive Reasoning?
- Formalizing an Argument to the Best Explanation
- Explanatory Virtues and Hypothesis Comparison
ObservationHypothesisArgument to the Best ExplanationExplanatory Scope
Open unitProblem Solving Logic
4 lessonsLogic of Problem Solving: Goals, Constraints, and Strategy
How to reason from a problem state to a workable next move
Students learn to formalize practical problems, identify goals and constraints, compare candidate strategies, and justify a plan of action without pretending that every problem has a single certain answer. The unit builds from problem analysis through strategy selection to disciplined revision under new information.
Inside This Unit
- Understanding the Problem Before Solving It
- Building a Structured Problem Map
- Choosing and Revising a Strategy
Problem StateGoal StateConstraintResource
Open unitDecision And Rational Choice
5 lessonsDecision Theory: Choosing Under Uncertainty
How rational agents choose when outcomes depend on chance
Students learn the logic of rational decision making under risk and uncertainty, including expected value, utility, decision matrices, dominance, and the systematic biases that cause real decisions to depart from the normative ideal. The unit builds from qualitative preference reasoning through quantitative expected-utility analysis to the integrated evaluation of complex real-world choices.
Inside This Unit
- What Makes a Decision Rational?
- Expected Value and Expected Utility
- Decision Matrices and Dominance
Expected ValueUtilityRisk versus UncertaintyDominance
Open unitMathematical Foundations
5 lessonsSet Theory and Relations: The Structure Underneath Logic
The mathematical scaffolding that makes formal reasoning possible
Students learn the language of sets, relations, and functions that underlies most of modern formal logic. They master set operations, relational properties, equivalence classes, partitions, order relations, functions, cardinality, and the pigeonhole principle, and they use set theory as the semantic foundation for categorical and predicate logic.
Inside This Unit
- What Is a Set?
- Operations on Sets
- Relations and Their Properties
SetElement (Member)SubsetPower Set
Open unitNot sure where to start?
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