Read
Build the mental model
Move through the guided explanation first so the central distinction and purpose are clear before you evaluate your own work.
Natural Deduction
Diagnosing Invalid Proof Steps
Students inspect flawed derivations and explain exactly why the steps fail, naming the mismatched condition of the rule being misused.
Treat the lesson like coached reps. Compare each move you make with the worked examples and common mistakes before saving a response.
Start Here
What this lesson is helping you do
Students inspect flawed derivations and explain exactly why the steps fail, naming the mismatched condition of the rule being misused. The practice in this lesson depends on understanding Proof, Subproof, and Counterexample and applying tools such as Modus Ponens and Modus Tollens correctly.
How to approach it
Treat the lesson like coached reps. Compare each move you make with the worked examples and common mistakes before saving a response.
What the practice is building
You will put the explanation to work through diagnosis practice, quiz, formalization practice, proof construction, evaluation practice, analysis practice, and rapid identification activities, so the goal is not just to recognize the idea but to use it under your own control.
What success should let you do
Explain at least 3 invalid proof steps in precise language, naming the rule, the failed condition, and the error category.
Reading Path
Move through the lesson in this order
The page is designed to teach before it tests. Use this sequence to keep the reading, examples, and practice in the right relationship.
Study
Watch the move in context
Use the worked examples to see how the reasoning behaves when someone else performs it carefully.
Do
Practice with a standard
Only then move into the activities, using the pause-and-check prompts as a final checkpoint before you submit.
Guided Explanation
Read this before you try the activity
These sections give the learner a usable mental model first, so the practice feels like application rather than guesswork.
Framing
An invalid step is a specific rule failure
When a proof step is wrong, it is wrong for a precise reason. Either the cited lines do not match the pattern of the rule, the derived line is not what the rule actually licenses, or the citation reaches into a closed subproof. Your job in diagnosis is to name that precise reason.
Saying 'this step is invalid' is not enough. You should be able to say which rule was claimed, which condition of that rule failed, and what the correct application (if any) would look like.
What to look for
- Identify the rule the student claimed to apply.
- State the exact condition of the rule that failed.
- Show what the correct derivation would look like if one exists.
Diagnosis is a structural skill: you match the failed step against the rule's pattern and locate the specific mismatch.
Pattern recognition
Common categories of rule misuse
Most invalid steps fall into a few categories. The first is wrong-direction conditional errors: affirming the consequent (from P -> Q and Q, wrongly inferring P) and denying the antecedent (from P -> Q and ~P, wrongly inferring ~Q). These look like modus ponens or modus tollens but fail the pattern.
The second is conjunction or disjunction confusion: eliminating a conjunction to get something that was never a conjunct, or applying disjunctive syllogism without actually negating one disjunct. The third is scope errors: pulling a line out of a closed subproof to use on the outer level.
What to look for
- Watch for affirming the consequent and denying the antecedent.
- Confirm that each conjunction or disjunction rule actually matches the cited structure.
- Check that every citation is in scope.
Invalid steps cluster around a short list of recognizable categories; once you know them, you can diagnose most errors quickly.
Communication
Explain, do not just correct
A good diagnosis does not just fix the proof. It explains what went wrong in language the original proof writer can learn from. If you simply replace the bad line with a correct one, the student may not learn why their attempt was wrong.
Aim for a two-sentence explanation: 'The student cited rule X, but rule X requires a line of form Y. Line Z is not of that form, so the step is not licensed.' This habit makes you both a better logician and a better teacher.
What to look for
- Name the rule the student claimed.
- State the condition of the rule that was violated.
- Describe what a correct application would look like.
A diagnosis is a teaching move, not just a correction; precise language turns a caught error into a durable lesson.
Judgment
Distinguish typo errors from structural errors
Not every broken proof is a deep error. Sometimes a student copied a letter wrong, or cited line 5 when they meant line 4. Those are typo errors: the intended step is legal, but the writing of it is sloppy.
Structural errors are different. A structural error is one where, even if you fix the citations, the rule cannot license the move. Separate these two types in your diagnosis. For typo errors, say 'the intended step is modus ponens from lines X and Y'. For structural errors, say 'no rearrangement of citations will make this step legal.'
What to look for
- Ask whether fixing citations would make the step legal.
- Label typo errors differently from structural ones.
- Propose a repair only when the intended step is actually legal.
Separating typo-level mistakes from structural rule failures makes your diagnoses both accurate and fair.
Core Ideas
The main concepts to keep in view
Use these as anchors while you read the example and draft your response. If the concepts blur together, the practice usually blurs too.
Proof
A rule-governed derivation showing that a conclusion follows from a set of premises.
Why it matters: Proof is the main formal tool used to establish deductive validity.
Subproof
A nested section of a proof used to track assumptions and scope in conditional or indirect derivations.
Why it matters: Subproof structure is essential for handling assumptions correctly in natural deduction.
Counterexample
A situation in which the premises of an argument are all true while the conclusion is false.
Why it matters: Producing a counterexample is one of the cleanest ways to show an argument is invalid.
Reference
Open these only when you need the extra structure
How the lesson is meant to unfold
Concept Intro
The core idea is defined and separated from nearby confusions.
Worked Example
A complete example demonstrates what correct reasoning looks like in context.
Guided Practice
You apply the idea with scaffolding still visible.
Independent Practice
You work more freely, with less support, to prove the idea is sticking.
Assessment Advice
Use these prompts to judge whether your reasoning meets the standard.
Mastery Check
The final target tells you what successful understanding should enable you to do.
Reasoning tools and formal patterns
Rules and standards
These are the criteria the unit uses to judge whether your reasoning is actually sound.
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.
Conjunction Elimination
From P & Q, infer either P or Q.
Common failures
- The cited line is not a conjunction.
- The derived statement is not one of the conjuncts.
Conditional Introduction
If assuming P lets you derive Q within a subproof, you may discharge the assumption and infer P -> Q.
Common failures
- Discharging the assumption before Q has actually been derived.
- Using lines from inside a closed subproof after its assumption has been discharged.
Necessity Standard
A deductive conclusion must follow necessarily from the premises, not merely appear plausible.
Common failures
- The student defends a conclusion on the basis of plausibility alone.
- The student confuses a likely conclusion with a logically guaranteed one.
Patterns
Use these when you need to turn a messy passage into a cleaner logical structure before evaluating it.
Sentence-Letter Translation
Input form
natural_language_argument
Output form
symbolic_argument
Steps
- Identify the atomic statements in the argument.
- Assign a sentence letter to each distinct atomic statement.
- Identify the logical connectives in each premise and the conclusion.
- Translate each premise and the conclusion into symbolic form.
- Check that the symbolic form preserves the original logical structure.
Watch for
- Using different letters for the same atomic statement in different places.
- Mistranslating conditionals by reading 'only if' as 'if'.
- Representing surface grammar instead of underlying logical form.
Natural Deduction Proof Format
Input form
symbolic_argument
Output form
line_by_line_proof
Steps
- List the premises as the first numbered lines.
- State the target conclusion.
- Add justified lines one at a time, citing the rule and the prior lines used.
- Open a subproof whenever you need to make an assumption.
- Close each subproof by discharging its assumption with the appropriate rule.
- Verify that the final line matches the intended conclusion and that every citation is in scope.
Watch for
- Citing lines that do not match the pattern of the chosen rule.
- Using a line inside a closed subproof after the assumption has been discharged.
- Skipping intermediate steps so that the rule cannot be verified.
Counterexample Construction
Input form
symbolic_argument
Output form
row_of_truth_values
Steps
- List the atomic letters that appear in the argument.
- Search for an assignment of truth values that makes every premise true.
- Check whether that same assignment also makes the conclusion false.
- If such an assignment exists, present it as the counterexample.
Watch for
- Claiming a counterexample without actually satisfying all premises.
- Overlooking rows that make the conclusion false by coincidence of truth values.
Worked Through
Examples that model the standard before you try it
Do not skim these. A worked example earns its place when you can point to the exact move it is modeling and the mistake it is trying to prevent.
Worked Example
Misused Rule
Diagnosis
The step is not licensed. Disjunction elimination requires two subproofs, one assuming each disjunct, each reaching the same conclusion. You cannot extract a single disjunct from a bare disjunction; that would need an additional premise like ~Q and would be disjunctive syllogism, not disjunction elimination.
Claimed Rule
Disjunction Elimination
Student Lines
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P v Q2
PWorked Example
Affirming the Consequent
Diagnosis
Modus ponens requires the antecedent of the conditional, not the consequent. Here the student cited the consequent (Q) and tried to infer the antecedent (P). This is the classic fallacy of affirming the consequent. There is no legal rearrangement of citations that makes this inference go through.
Claimed Rule
Modus Ponens
Student Lines
LineStatementJustification
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P -> Q2
Q3
PPause and Check
Questions to use before you move into practice
Self-check questions
- Can I state which condition of the rule failed?
- Can I identify the exact mismatch between the cited lines and the derived line?
- Have I decided whether a legal repair is actually possible?
Practice
Now apply the idea yourself
Move into practice only after you can name the standard you are using and the structure you are trying to preserve or evaluate.
Diagnosis Practice
DeductiveDiagnose the Broken Proof
For each flawed derivation, identify the rule the student claimed to use, explain exactly which condition of that rule failed, and state whether the error is a typo-level slip or a genuine structural failure. Where possible, describe what a correct application would look like.
Flawed derivations
Each case shows the cited lines, the derived line, and the rule the student named. Your task is to say why the step is not legal and what category of error it represents.
Case A
Lines: 1. P -> Q (premise), 2. Q (premise). Derived: 3. P, cited rule: Modus Ponens from 1, 2.
Does modus ponens run from the consequent back to the antecedent? Name the classic fallacy.
Case B
Lines: 1. P -> Q (premise), 2. ~P (premise). Derived: 3. ~Q, cited rule: Modus Tollens from 1, 2.
Modus tollens requires the denial of the consequent, not the antecedent. Which fallacy is this?
Case C
Lines: 1. P v Q (premise). Derived: 2. P, cited rule: Disjunction Elimination from 1.
Disjunction elimination cannot extract a single disjunct from a bare disjunction. What does the rule actually require?
Case D
Lines: 1. P & Q (premise). Derived: 2. P v R, cited rule: Conjunction Elimination from 1.
Conjunction elimination yields a conjunct of the cited line. Is P v R a conjunct of P & Q?
Proof Draft
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Quiz
DeductiveScenario Check: Diagnosing Invalid Proof Steps
Each question presents a scenario or challenge. Answer in two to four sentences. Focus on showing that you can use what you learned, not just recall it.
Scenario questions
Work through each scenario. Precise, specific answers are better than long vague ones.
Question 1 — Diagnose
A student makes the following mistake: "Saying a step is wrong without explaining why." Explain specifically what is wrong with this reasoning and what the student should have done instead.
Can the student identify the flaw and articulate the correction?
Question 2 — Apply
You encounter a new argument that you have never seen before. Walk through exactly how you would diagnose invalid step, starting from scratch. Be specific about each step and explain why the order matters.
Can the student transfer the skill of diagnose invalid step to a genuinely new case?
Question 3 — Distinguish
Someone confuses proof with subproof. Write a short explanation that would help them see the difference, and give one example where getting them confused leads to a concrete mistake.
Does the student understand the boundary between the two concepts?
Question 4 — Transfer
The worked example "Misused Rule" showed one way to handle a specific case. Describe a situation where the same method would need to be adjusted, and explain what you would change and why.
Can the student adapt the demonstrated method to a variation?
Proof Draft
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Formalization Practice
DeductiveFormalization Drill: Diagnosing Invalid Proof Steps
Translate each natural-language argument into formal notation. Identify the logical form and check whether the argument is valid.
Practice scenarios
Work through each scenario carefully. Apply the concepts from this lesson.
Argument 1
If the server crashes, then the backup activates. If the backup activates, then an alert is sent. The server crashed. What follows?
Argument 2
Either the contract is valid or the parties must renegotiate. The contract is not valid. What follows?
Argument 3
All databases store records. This system does not store records. What can we conclude about this system?
Choose one of the arguments above, assign sentence letters, and translate the premises and conclusion into symbolic form.
Proof Construction
DeductiveProof Practice: Diagnosing Invalid Proof Steps
Construct a step-by-step proof or derivation for each argument. Justify every step with the rule you are applying.
Practice scenarios
Work through each scenario carefully. Apply the concepts from this lesson.
Prove
From premises: (1) P -> Q, (2) Q -> R, (3) P. Derive R.
Prove
From premises: (1) A v B, (2) A -> C, (3) B -> C. Derive C.
Prove
From premises: (1) ~(P & Q), (2) P. Derive ~Q.
Proof Draft
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Evaluation Practice
DeductiveValidity Check: Diagnosing Invalid Proof Steps
Determine whether each argument is deductively valid. If invalid, describe a counterexample where the premises are true but the conclusion is false.
Practice scenarios
Work through each scenario carefully. Apply the concepts from this lesson.
Argument A
All philosophers study logic. Socrates is a philosopher. Therefore, Socrates studies logic.
Argument B
If it rains, the ground is wet. The ground is wet. Therefore, it rained.
Argument C
No birds are mammals. Some mammals fly. Therefore, some things that fly are not birds.
Argument D
Either the door is locked or the alarm is on. The door is not locked. Therefore, the alarm is on.
Proof Draft
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Proof Construction
DeductiveDeep Practice: Diagnosing Invalid Proof Steps
Work through these challenging exercises. Each one requires careful application of formal reasoning. Show your work step by step.
Challenging derivations
Prove each conclusion from the given premises. Label every inference rule you use.
Challenge 1
Premises: (1) (A & B) -> C, (2) D -> A, (3) D -> B, (4) D. Derive: C.
Challenge 2
Premises: (1) P -> (Q & R), (2) R -> S, (3) ~S. Derive: ~P.
Challenge 3
Premises: (1) A v B, (2) A -> (C & D), (3) B -> (C & E). Derive: C.
Challenge 4
Premises: (1) ~(P & Q), (2) P v Q, (3) P -> R, (4) Q -> S. Derive: R v S.
Proof Draft
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Analysis Practice
DeductiveReal-World Transfer: Diagnosing Invalid Proof Steps
Apply formal logic to real-world contexts. Translate each scenario into formal notation, determine validity, and explain the practical implications.
Logic in the wild
These scenarios come from law, science, and everyday reasoning. Formalize and evaluate each.
Legal reasoning
A contract states: 'If the product is defective AND the buyer reports within 30 days, THEN a full refund will be issued.' The product was defective. The buyer reported on day 35. The company denies the refund. Is the company's position logically valid?
Medical reasoning
A diagnostic protocol states: 'If the patient has fever AND cough, test for flu. If the flu test is negative AND symptoms persist for 7+ days, test for bacterial infection.' A patient has a cough but no fever. What does the protocol require?
Policy reasoning
A policy reads: 'Students may graduate early IF they complete all required courses AND maintain a 3.5 GPA OR receive special faculty approval.' Due to the ambiguity of OR, identify the two possible readings and explain what difference they make.
Proof Draft
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Rapid Identification
DeductiveTimed Drill: Diagnosing Invalid Proof Steps
Work through these quickly. For each mini-scenario, identify the logical form, name the rule used, and state whether the inference is valid. Aim for accuracy under time pressure.
Quick-fire logic identification
Identify the logical form and validity of each argument in under 60 seconds per item.
Item 1
If the reactor overheats, the failsafe triggers. The failsafe triggered. Therefore, the reactor overheated.
Item 2
All licensed pilots passed the medical exam. Jenkins is a licensed pilot. Therefore, Jenkins passed the medical exam.
Item 3
Either the encryption key expired or someone changed the password. The encryption key did not expire. Therefore, someone changed the password.
Item 4
If taxes increase, consumer spending decreases. Consumer spending has not decreased. Therefore, taxes have not increased.
Item 5
No insured vehicle was towed. This vehicle was towed. Therefore, this vehicle is not insured.
Item 6
If the sample is contaminated, then the results are unreliable. The sample is contaminated. Therefore, the results are unreliable.
Proof Draft
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Evaluation Practice
DeductivePeer Review: Diagnosing Invalid Proof Steps
Below are sample student responses to a logic exercise. Evaluate each response: Is the formalization correct? Is the proof valid? Identify specific errors and suggest corrections.
Evaluate student proofs
Each student attempted to prove a conclusion from given premises. Find and correct any mistakes.
Student A's work
Premises: P -> Q, Q -> R, P. Student wrote: (1) P [premise], (2) P -> Q [premise], (3) Q [MP 1,2], (4) Q -> R [premise], (5) R [MP 3,4]. Conclusion: R. Student says: 'Valid proof by two applications of Modus Ponens.'
Student B's work
Premises: A v B, A -> C. Student wrote: (1) A v B [premise], (2) A -> C [premise], (3) A [from 1], (4) C [MP 2,3]. Conclusion: C. Student says: 'Since A or B is true, A must be true, so C follows.'
Student C's work
Premises: ~P v Q, P. Student wrote: (1) ~P v Q [premise], (2) P [premise], (3) ~~P [DN 2], (4) Q [DS 1,3]. Conclusion: Q. Student says: 'I used double negation then disjunctive syllogism.'
Student D's work
Premises: (P & Q) -> R, P, Q. Student wrote: (1) P [premise], (2) Q [premise], (3) (P & Q) -> R [premise], (4) R [MP 1,3]. Conclusion: R. Student says: 'Modus Ponens with P and the conditional.'
Proof Draft
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Proof Construction
DeductiveConstruction Challenge: Diagnosing Invalid Proof Steps
Build complete proofs or arguments from scratch. You are given only a conclusion and some constraints. Construct valid premises and a rigorous derivation.
Build your own proofs
For each task, create a valid argument with explicit premises and step-by-step derivation.
Task 1
Construct a valid argument with exactly three premises that concludes: 'The network is secure.' Use at least one conditional and one disjunction in your premises.
Task 2
Build a valid syllogistic argument that concludes: 'Some scientists are not wealthy.' Your premises must be universal statements (All X are Y or No X are Y).
Task 3
Create a proof using reductio ad absurdum (indirect proof) that derives ~(P & ~P) from no premises. Show every step and justify each with a rule name.
Task 4
Construct a chain of conditional reasoning with at least four steps that connects 'The satellite detects an anomaly' to 'Emergency protocols are activated.' Make each link realistic and name the domain.
Proof Draft
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Diagnosis Practice
DeductiveCounterexample Challenge: Diagnosing Invalid Proof Steps
For each invalid argument below, construct a clear counterexample -- a scenario where all premises are true but the conclusion is false. Then explain which logical error the argument commits.
Find counterexamples to invalid arguments
Each argument appears plausible but is invalid. Prove invalidity by constructing a specific counterexample.
Argument 1
If a student studies hard, they pass the exam. Maria passed the exam. Therefore, Maria studied hard.
Argument 2
All roses are flowers. Some flowers are red. Therefore, some roses are red.
Argument 3
No fish can fly. No birds are fish. Therefore, all birds can fly.
Argument 4
If the alarm sounds, there is a fire. The alarm did not sound. Therefore, there is no fire.
Argument 5
All effective medicines have been tested. This substance has been tested. Therefore, this substance is an effective medicine.
Proof Draft
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Analysis Practice
DeductiveIntegration Exercise: Diagnosing Invalid Proof Steps
These exercises combine deductive logic with other topics and reasoning styles. Apply formal logic alongside empirical evaluation, explanation assessment, or problem-solving frameworks.
Cross-topic deductive exercises
Each scenario requires deductive reasoning combined with at least one other skill area.
Scenario 1
A quality control team uses this rule: 'If a batch fails two consecutive tests, it must be discarded.' Batch 47 failed Test A and passed Test B, then failed Test C. Formally determine whether the rule requires discarding Batch 47, and discuss whether the rule itself is well-designed from a problem-solving perspective.
Scenario 2
A researcher argues: 'All peer-reviewed studies in this meta-analysis show that X reduces Y. This study shows X reduces Y. Therefore, this study will be included in the meta-analysis.' Evaluate the deductive form, then inductively assess whether the meta-analysis conclusion would be strong.
Scenario 3
An insurance policy states: 'Coverage applies if and only if the damage was caused by a covered peril AND the policyholder reported it within 72 hours.' A policyholder reported water damage after 80 hours, claiming the damage was not discoverable sooner. Apply the formal logic of the policy, then consider whether the best explanation supports an exception.
Scenario 4
A hiring algorithm uses: 'If GPA >= 3.5 AND experience >= 2 years, then advance to interview.' Candidate X has GPA 3.8 and 18 months experience. Formally determine the outcome. Then evaluate: is the algorithm's rule inductively justified? What evidence would you want?
Proof Draft
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Diagnosis Practice
DeductiveMisconception Clinic: Diagnosing Invalid Proof Steps
Each item presents a common misconception about deductive logic. Identify the misconception, explain why it is wrong, and provide a correct version of the reasoning.
Common deductive misconceptions
Diagnose and correct each misconception. Explain the error clearly enough for a fellow student to understand.
Misconception 1
A student claims: 'An argument is valid if its conclusion is true. Since the conclusion "Water is H2O" is obviously true, any argument concluding this must be valid.'
Misconception 2
A student says: 'Modus Tollens and denying the antecedent are the same thing. Both involve negation and a conditional, so they must work the same way.'
Misconception 3
A student writes: 'This argument is invalid because the conclusion is false: All cats are reptiles. All reptiles lay eggs. Therefore, all cats lay eggs.'
Misconception 4
A student argues: 'A sound argument can have a false conclusion, because soundness just means the argument uses correct logical rules.'
Misconception 5
A student claims: 'Since P -> Q is equivalent to ~P v Q, we can derive Q from P -> Q alone, without knowing whether P is true.'
Proof Draft
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Proof Construction
DeductiveScaffolded Proof: Diagnosing Invalid Proof Steps
Build proofs in stages. Each task gives you a partially completed derivation. Fill in the missing steps, justify each one, and then extend the proof to a further conclusion.
Step-by-step proof building
Complete each partial proof, then extend it. Every step must cite a rule.
Scaffold 1
Premises: (1) (A v B) -> C, (2) D -> A, (3) D. Partial proof: (4) A [MP 2,3]. Your tasks: (a) Complete the proof to derive C. (b) If we add premise (5) C -> E, extend the proof to derive E.
Scaffold 2
Premises: (1) P -> (Q -> R), (2) P, (3) Q. Partial proof: (4) Q -> R [MP 1,2]. Your tasks: (a) Complete the proof to derive R. (b) If we add premise (5) R -> ~S, extend to derive ~S. (c) If we also add (6) S v T, what can you derive?
Scaffold 3
Premises: (1) ~(A & B), (2) A. Your task: Prove ~B step by step. Hint: You may need to use an assumption for indirect proof. Show the subproof structure clearly.
Scaffold 4
Premises: (1) P v Q, (2) P -> R, (3) Q -> S, (4) ~R. Your tasks: (a) Derive ~P from (2) and (4). (b) Using (a), derive Q from (1). (c) Using (b), derive S. (d) Name each rule used.
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Analysis Practice
DeductiveSynthesis Review: Diagnosing Invalid Proof Steps
These exercises require you to combine everything you have learned about deductive reasoning. Each scenario tests multiple skills simultaneously: formalization, rule application, validity checking, and proof construction.
Comprehensive deductive review
Each task combines multiple deductive skills. Show all your work.
Comprehensive 1
A software license agreement states: 'The software may be used commercially if and only if the licensee has purchased an enterprise plan and has fewer than 500 employees, or has received written exemption from the vendor.' Formalize this using propositional logic, determine what follows if a company has an enterprise plan and 600 employees with no exemption, and identify any ambiguity in the original text.
Comprehensive 2
Construct a valid argument with four premises and one conclusion about data privacy. Then create an invalid argument about the same topic that looks similar but commits a formal fallacy. Finally, prove the first is valid and show a counterexample for the second.
Proof Draft
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Proof Constructor
Build a formal proof step by step. Add premises, apply inference rules, cite earlier lines, and derive your conclusion.
Add premises and derived steps above, or load a template to get started.
\u2588 Premise\u2588 Derived step
Symbols:\u00AC not\u2227 and\u2228 or\u2192 if-then\u2194 iff\u2234 therefore
Animated Explainers
Step-by-step visual walkthroughs of key concepts. Click to start.
Read the explanation carefully before jumping to activities!
Further Support
Open these only if you need extra help or context
Mistakes to avoid before submitting
- Replacing the bad line with a correct one without explaining the original failure.
- Naming a rule by sound rather than by pattern.
Where students usually go wrong
Saying a step is wrong without explaining why.
Confusing the names of rules, especially modus ponens and modus tollens.
Ignoring scope or citation details in subproofs.
Labeling a structural failure as a typo and proposing a repair that does not exist.
Historical context for this way of reasoning
Gerhard Gentzen
Gentzen's natural-deduction system was designed so that rule violations are locally visible: you can check a proof step by inspecting only the line itself and the lines it cites.