Rigorous Reasoning

Natural Deduction

Basic Natural Deduction

Introduces core natural deduction inference rules and trains students to build short, fully justified line-by-line proofs.

Read the explanation sections first, then use the activities to test whether you can apply the idea under pressure.

DeductiveRulesLesson 3 of 50% progress
Previous LessonSymbolizing Propositional ArgumentsBack To UnitNatural Deduction: Validity and Formal ProofNext LessonDiagnosing Invalid Proof Steps

Start Here

What this lesson is helping you do

Introduces core natural deduction inference rules and trains students to build short, fully justified line-by-line proofs. The practice in this lesson depends on understanding Entailment, Proof, and Subproof and applying tools such as Modus Ponens and Modus Tollens correctly.

How to approach it

Read the explanation sections first, then use the activities to test whether you can apply the idea under pressure.

What the practice is building

You will put the explanation to work through proof construction, quiz, formalization practice, evaluation practice, analysis practice, rapid identification, and diagnosis practice 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

Complete 4 short natural-deduction proofs using the core rules, with correct citations on every line.

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.

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.

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

A proof is a justified sequence of lines

A natural-deduction proof is not a single leap from premises to conclusion. It is a numbered sequence of lines, each one justified either as a premise or as the result of a rule applied to earlier lines. The only thing allowed on a line is something the rules say you are allowed to write.

This discipline is what makes a proof checkable. You do not need to agree with the proof writer or share their intuition; you only need to verify that each line follows from cited prior lines by the named rule.

What to look for

  • Number every line of the proof.
  • Justify every non-premise line by a rule and line citations.
  • Do not rely on intuition when a rule has not been applied.
A natural-deduction proof is a public, checkable trail from premises to conclusion, with every step licensed by a rule.

Rule bank

The core rules for conjunctions and conditionals

Start with the smallest toolkit that still lets you build real proofs: modus ponens (from P -> Q and P, infer Q), modus tollens (from P -> Q and ~Q, infer ~P), conjunction introduction (from P and Q, infer P & Q), and conjunction elimination (from P & Q, infer either conjunct).

Add disjunctive syllogism (from P v Q and ~P, infer Q) and hypothetical syllogism (from P -> Q and Q -> R, infer P -> R). These six rules cover most short deductive arguments you will encounter.

What to look for

  • Name the rule you are about to apply before you write the line.
  • Check that the cited lines exactly match the pattern of the rule.
  • Use each rule only in the direction it allows.
Small rule banks produce clean proofs; almost every short deduction can be reached with six well-known rules.

Scope discipline

Subproofs handle assumptions

Some rules require you to assume something temporarily. When you want to prove a conditional P -> Q, you open a subproof by assuming P, derive Q inside that subproof, and then discharge the assumption to conclude P -> Q on the outer level.

Subproofs have scope. Lines inside a closed subproof are no longer available once the assumption has been discharged. A proof that references a line from inside a closed subproof is not merely messy; it is wrong.

What to look for

  • Indent or box each subproof to mark its scope.
  • Treat assumptions as live only until they are discharged.
  • Never cite lines from inside a closed subproof.
Subproofs let you reason under assumptions safely by quarantining those assumptions until they are properly discharged.

Strategy

Plan the proof before you write it

Before starting a proof, read the conclusion and work backward. If the conclusion is P -> Q, you will probably need a subproof that assumes P. If the conclusion is P & Q, you will need both P and Q separately before conjoining them. If the conclusion is ~P, consider indirect proof.

Then look at the premises and ask what they combine to produce. This forward-and-backward pass saves enormous amounts of wasted writing and makes it obvious which rule to apply first.

What to look for

  • Look at the conclusion and decide which rule will close the proof.
  • Look at the premises and ask what they can immediately produce.
  • Bridge between the two before writing the first rule application.
A short plan before the first line prevents most proof failures and reveals which rule belongs where.

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.

Entailment

A relation in which the premises, taken together, guarantee the conclusion.

Why it matters: Entailment explains why a valid deduction gives necessity rather than probability.

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.

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.

Rule Or Standard

This step supports the lesson by moving from explanation toward application.

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

Modus Ponens Proof

Each proof step must be licensed by a rule and matching citations. The shortest proofs are just a direct application of a single rule.

Proof

LineStatementJustification
1P -> QPremise
2PPremise
3QModus Ponens from 1, 2

Worked Example

Chained Conditionals

Longer proofs are usually just several applications of familiar rules, each one building on the last.

Proof

LineStatementJustification
1P -> QPremise
2Q -> RPremise
3PPremise
4QModus Ponens from 1, 3
5RModus Ponens from 2, 4

Pause and Check

Questions to use before you move into practice

Self-check questions

  • Do the cited lines match the rule pattern exactly?
  • Does my derived line contain only what the rule allows?
  • Is every assumption I opened eventually discharged?

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.

Proof Construction

Deductive

Build Short Natural Deduction Proofs

For each symbolic argument, construct a natural-deduction proof. Number each line, state each premise explicitly, and justify every derived line with a rule name and the line numbers it cites.

Proof targets

Work one proof at a time. Start by planning the final rule you intend to apply, then work forward from the premises. Your proof is correct only when every line is justified and the last line matches the conclusion.

Proof A

Premises: P -> Q, P. Conclusion: Q.

Which rule closes this proof in a single step? What are the two citations?

Proof B

Premises: P -> Q, Q -> R, P. Conclusion: R.

Plan two applications of modus ponens. Which order makes the proof cleanest?

Proof C

Premises: (P & Q) -> R, P, Q. Conclusion: R.

You will need to introduce P & Q before you can use the first premise.

Proof D

Premises: P v Q, ~P, Q -> R. Conclusion: R.

Which rule eliminates the disjunction, and what do you then do with the result?

Proof Draft
LineStatementJustificationAction
1
2
3
Not saved yet.

Quiz

Deductive

Scenario Check: Basic Natural Deduction

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: "Choosing a rule that does not fit the cited lines." 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 apply basic inference rules, starting from scratch. Be specific about each step and explain why the order matters.

Can the student transfer the skill of apply basic inference rules 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 "Modus Ponens Proof" 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
LineStatementJustificationAction
1
2
3
Not saved yet.

Formalization Practice

Deductive

Formalization Drill: Basic Natural Deduction

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.

Not saved yet.

Proof Construction

Deductive

Proof Practice: Basic Natural Deduction

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
LineStatementJustificationAction
1
2
3
Not saved yet.

Evaluation Practice

Deductive

Validity Check: Basic Natural Deduction

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
LineStatementJustificationAction
1
2
3
Not saved yet.

Proof Construction

Deductive

Deep Practice: Basic Natural Deduction

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
LineStatementJustificationAction
1
2
3
Not saved yet.

Analysis Practice

Deductive

Real-World Transfer: Basic Natural Deduction

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
LineStatementJustificationAction
1
2
3
Not saved yet.

Rapid Identification

Deductive

Timed Drill: Basic Natural Deduction

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
LineStatementJustificationAction
1
2
3
Not saved yet.

Evaluation Practice

Deductive

Peer Review: Basic Natural Deduction

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
LineStatementJustificationAction
1
2
3
Not saved yet.

Proof Construction

Deductive

Construction Challenge: Basic Natural Deduction

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
LineStatementJustificationAction
1
2
3
Not saved yet.

Diagnosis Practice

Deductive

Counterexample Challenge: Basic Natural Deduction

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
LineStatementJustificationAction
1
2
3
Not saved yet.

Analysis Practice

Deductive

Integration Exercise: Basic Natural Deduction

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
LineStatementJustificationAction
1
2
3
Not saved yet.

Diagnosis Practice

Deductive

Misconception Clinic: Basic Natural Deduction

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
LineStatementJustificationAction
1
2
3
Not saved yet.

Proof Construction

Deductive

Scaffolded Proof: Basic Natural Deduction

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.

Proof Draft
LineStatementJustificationAction
1
2
3
Not saved yet.

Analysis Practice

Deductive

Synthesis Review: Basic Natural Deduction

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
LineStatementJustificationAction
1
2
3
Not saved yet.

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!

Riko

Further Support

Open these only if you need extra help or context

Mistakes to avoid before submitting
  • Copying the structure of a previous proof without checking citations.
  • Treating any plausible-looking line as a legal rule application.
Where students usually go wrong

Choosing a rule that does not fit the cited lines.

Writing the right conclusion with the wrong justification.

Skipping intermediate lines so the rule cannot be verified.

Citing a line from inside a closed subproof.

Historical context for this way of reasoning

Gerhard Gentzen

Modern proof editors often inherit the line-by-line and subproof-oriented style of natural deduction associated with Gentzen's 1934 system.