← Back to curriculumNatural Deduction·Intermediate·5 lessons·~300 min
Natural Deduction: Validity and Formal Proof
Why some conclusions follow necessarily
What you'll learn
By the end of this unit, you can…
- Distinguish validity from truth.
- Symbolize argument.
- Construct short proofs.
- Diagnose invalid step.
Lessons
Lesson sequence
- 1
Validity vs Truth
Introduces the difference between validity, truth, and soundness, and trains students to judge the form of a deductive argument separately from the truth of its claims.
15 activities5 worked examples
- 2
Symbolizing Propositional Arguments
Teaches students how to translate short arguments from ordinary language into propositional notation with a clear sentence-letter key.
15 activities5 worked examples
- 3
Basic Natural Deduction
Introduces core natural deduction inference rules and trains students to build short, fully justified line-by-line proofs.
15 activities5 worked examples
- 4
Diagnosing Invalid Proof Steps
Students inspect flawed derivations and explain exactly why the steps fail, naming the mismatched condition of the rule being misused.
15 activities5 worked examples
- 5
Capstone: Building and Defending a Complete Deductive Argument
An integrative lesson that asks students to move through the full cycle of deductive evaluation: read an argument in ordinary language, symbolize it, classify its validity, either prove it or refute it with a counterexample, and then explain the result in plain English.
2 activities1 worked example
How to study
Three moves that work for this unit
1Read the explanation
Each lesson opens with a guided walkthrough — read it before the activity.
2Study the worked example
Look at why each step follows, not just what the answer is.
3Practice with the target in mind
Know which rule applies and what would make the response weak before you start.
Reference materials
Optional context for the unit. Each lesson surfaces the concepts and rules it uses — these are here when you want the bigger picture.
Concept map (6 terms)
Validity
The property of an argument whose conclusion cannot be false while all its premises are true.
Soundness
A deductive argument is sound when it is valid and all of its premises are true.
Entailment
A relation in which the premises, taken together, guarantee the conclusion.
Proof
A rule-governed derivation showing that a conclusion follows from a set of premises.
Subproof
A nested section of a proof used to track assumptions and scope in conditional or indirect derivations.
Counterexample
A situation in which the premises of an argument are all true while the conclusion is false.
Rules and standards (8)
- Modus Ponens. From 'P → Q' and 'P', one may derive 'Q'. Common failures: The student affirms the consequent by deriving 'P' from 'P → Q' and 'Q'.; The student derives 'Q' from 'Q → P' and 'P' after confusing the direction of the conditional..
- Modus Tollens. From 'P → Q' and '¬Q', one may derive '¬P'. Common failures: The student denies the antecedent by deriving '¬Q' from 'P → Q' and '¬P'.; The student ignores the conditional's direction when the negation is placed on the consequent..
- Hypothetical Syllogism. From P -> Q and Q -> R, infer P -> R. Common failures: The chained conditionals do not actually share a middle term.; The derived conditional swaps antecedent and consequent..
- Disjunctive Syllogism. From 'P ∨ Q' and '¬P', one may derive 'Q'; similarly from 'P ∨ Q' and '¬Q', one may derive 'P'. Common failures: The student derives the negated disjunct instead of the remaining disjunct.; The student assumes an exclusive disjunction and draws an unlicensed inference about the second disjunct..
- Conjunction Introduction. From P and Q, infer P & Q. Common failures: One of the conjuncts was not established on a prior line.; The conjunction changes the content of the cited lines..