Rigorous Reasoning

Propositional Logic

Building Short Formal Proofs

Introduces the basic inference rules of propositional proof (modus ponens, modus tollens, hypothetical syllogism, disjunctive syllogism, simplification, conjunction, and addition) and teaches students to build short formal proofs step by step.

Treat the lesson like coached reps. Compare each move you make with the worked examples and common mistakes before saving a response.

DeductiveGuided PracticeLesson 5 of 60% progress
Previous LessonTruth Tables, Equivalence, and ValidityBack To UnitPropositional Logic: Form, Connectives, and Valid InferenceNext LessonCapstone: Symbolizing, Proving, and Refuting Propositional Arguments

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What this lesson is helping you do

Introduces the basic inference rules of propositional proof (modus ponens, modus tollens, hypothetical syllogism, disjunctive syllogism, simplification, conjunction, and addition) and teaches students to build short formal proofs step by step. The practice in this lesson depends on understanding Validity and Inference Rule and applying tools such as Respect the Main Connective and Assign Sentence Letters Consistently 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 proof construction, 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

Construct correct formal proofs for 4 propositional arguments, using at least three different inference rules across the set, with every step properly justified.

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.

Motivation

Why proofs complement truth tables

Truth tables are complete: they can test any propositional argument. But they grow quickly. An argument with five atomic letters needs 32 rows; ten atomic letters needs 1,024. Formal proofs avoid this explosion by proceeding step by step using rules that preserve truth. Instead of enumerating possibilities, you derive the conclusion directly from the premises.

Proof construction also teaches a different kind of thinking. Where truth tables ask you to exhaust cases, proofs ask you to see structure. Each step must be justified by a rule applied to earlier lines, and the whole proof must end at the goal. Learning to think in this forward-moving way is a skill that carries into predicate logic and beyond.

What to look for

  • Use truth tables when the argument is small and you need an exhaustive check.
  • Use proofs when the argument has many letters or when you want to see the derivation structure.
  • Do not treat proofs and truth tables as competitors; they are complementary tools.
Proofs trade exhaustive case analysis for step-by-step derivation. Both methods establish validity, but they build different skills.

Rule catalog

The basic rules and what they mean

Every propositional proof uses a small number of rules. Modus ponens lets you derive Q from P → Q and P. Modus tollens lets you derive ¬P from P → Q and ¬Q. Hypothetical syllogism lets you derive P → R from P → Q and Q → R. Disjunctive syllogism lets you derive Q from P ∨ Q and ¬P (or P from P ∨ Q and ¬Q). These four rules handle the majority of short proofs and correspond directly to argument forms students already recognize.

Three more rules round out the basic set. Simplification lets you derive either conjunct from a conjunction: P from P ∧ Q, or Q from P ∧ Q. Conjunction lets you combine two independently derived lines P and Q into P ∧ Q. Addition lets you derive P ∨ Q from P (for any Q), which sounds trivial but is sometimes the only way to set up a disjunctive syllogism later. Memorize these seven rules and practice recognizing which one fits a given step.

What to look for

  • Memorize modus ponens, modus tollens, hypothetical syllogism, and disjunctive syllogism.
  • Memorize simplification, conjunction, and addition.
  • For each rule, memorize both the schema and when to use it.
Seven basic rules will carry you through most propositional proofs. Memorize them as patterns you can recognize on sight.

Strategic thinking

Proof strategy: work from the goal backwards

Beginners usually stare at the premises hoping inspiration will strike. A more productive strategy is to start from the goal and ask what rule could produce it. If the goal is Q and there is a premise P → Q, you need to derive P and then apply modus ponens. If the goal is ¬P and there is a premise P → Q, you need to derive ¬Q and then apply modus tollens. Each goal suggests the last rule you will use, which tells you the subgoal you need to reach first.

Keep doing this recursively. Each subgoal becomes a new goal, and you work backwards until you reach something that is already a premise. Then you write the proof forward, line by line. This reverse-planning approach takes practice but turns proof construction from guesswork into method. It is the same technique used in geometry proofs and more advanced logic.

What to look for

  • Start from the goal and ask which rule could produce it.
  • Identify the subgoals needed to apply that rule.
  • Recurse until your subgoals become premises, then write the proof forward.
Working backwards from the goal is the single most valuable strategy for building propositional proofs.

What not to do

Common mistakes that waste time

Beginners often derive many lines that look useful but do not advance toward the goal. Every line in a proof should serve a purpose, even if you only see how at the end. If you find yourself deriving things 'just in case,' stop and plan. A short proof built with intention is always better than a long proof built by flailing.

The other frequent mistake is citing the wrong lines. Modus ponens requires a line of the form P → Q and a line of the form P. If you cite a line whose antecedent does not appear earlier, the proof is broken regardless of the plausibility of the derived line. Read each justification carefully and make sure the cited lines actually match the rule's schema.

What to look for

  • Derive only what you can use; stop if you cannot see how a step advances the goal.
  • Double-check every citation against the rule's schema.
  • If you are stuck, rewrite the goal and restart the backwards search.
Good proofs are intentional. Every line should have a purpose, and every citation must actually fit the rule.

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.

Validity

The property of an argument whose conclusion cannot be false while all its premises are true.

Why it matters: Validity is the central standard of deductive evaluation, and in propositional logic it can be mechanically tested.

Inference Rule

A schematic pattern that licenses the derivation of a conclusion from one or more premises, such as modus ponens or disjunctive syllogism.

Why it matters: Inference rules are the building blocks of formal proof and turn validity checking into step-by-step derivation.

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.

Respect the Main Connective

A symbolization is acceptable only if the main connective of the symbolic form matches the main connective of the natural-language statement.

Common failures

  • A conditional is symbolized as a conjunction because the student saw two claims joined.
  • A negation is applied to one conjunct when the whole conjunction was intended to be negated.
  • A biconditional is read as a one-directional conditional.

Assign Sentence Letters Consistently

Use the same sentence letter for every occurrence of the same atomic claim, and use different letters for distinct claims.

Common failures

  • The same claim is given two different letters in the same argument.
  • Two different claims are given the same letter because they share a topic.
  • A negated and an unnegated version of the same claim are assigned different letters.

Truth-Functional Validity Standard

A propositional argument is valid if and only if there is no truth-value assignment on which the premises are all true and the conclusion is false.

Common failures

  • The student treats one favorable row as proof of validity.
  • The student reads off the truth of the conclusion without checking whether the premises are all true in that row.
  • The student mistakes invalidity for falsehood or vice versa.

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.

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.

Patterns

Use these when you need to turn a messy passage into a cleaner logical structure before evaluating it.

Argument Symbolization Schema

Input form

natural_language_argument

Output form

propositional_argument_form

Steps

  • Identify the distinct atomic claims used in the argument.
  • Assign a sentence letter to each distinct atomic claim and record the key.
  • For each premise and the conclusion, find the main connective.
  • Symbolize each statement using the assigned letters and the main connective.
  • Verify that the final symbolization preserves both scope and inferential role.

Watch for

  • Using one sentence letter for two different claims that merely share a topic.
  • Failing to identify the main connective and symbolizing from the wrong structural layer.
  • Applying negation to the wrong part of a compound statement.

Truth-Table Validity Test

Input form

propositional_argument_form

Output form

validity_judgment

Steps

  • List the atomic letters used in the premises and conclusion.
  • Enumerate every truth-value assignment to those letters (2^n rows).
  • Compute the truth value of each premise and the conclusion in every row.
  • Find any row in which all premises are true and the conclusion is false.
  • Classify the argument as valid if no such row exists and invalid otherwise.

Watch for

  • Omitting rows and failing to cover all assignments.
  • Mislabeling a row as a counterexample without checking that every premise is true in it.
  • Confusing the shape of the truth table with the shape of the argument.

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

Chained Modus Ponens

Chained modus ponens peels off conditionals one at a time. The middle line Q is the bridge that lets you move from P to R.

Content

  • Premises: P → Q, Q → R, P.
  • Goal: R.
  • Line 1: P → Q (premise).
  • Line 2: Q → R (premise).
  • Line 3: P (premise).
  • Line 4: Q (modus ponens from 1, 3).
  • Line 5: R (modus ponens from 2, 4).

Worked Example

Disjunctive Syllogism Into Modus Ponens

Each rule consumes specific earlier lines. Disjunctive syllogism needs a disjunction and a negation; modus ponens needs a conditional and its antecedent. Planning the sequence backwards from the goal makes the whole proof clear before you write it.

Content

  • Premises: P ∨ Q, ¬P, Q → R.
  • Goal: R.
  • Line 1: P ∨ Q (premise).
  • Line 2: ¬P (premise).
  • Line 3: Q → R (premise).
  • Line 4: Q (disjunctive syllogism from 1, 2).
  • Line 5: R (modus ponens from 3, 4).

Pause and Check

Questions to use before you move into practice

Self-check questions

  • Does every derived line match the schema of the rule I cited?
  • Does my final line match the goal exactly, including any negations?
  • Can I explain why each step was necessary for reaching the goal?

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

Construct a Short Proof

For each argument, construct a formal proof using the basic inference rules. Number each line, state the formula, and justify each step with the rule and cited lines.

Proofs to construct

Write out a proof with numbered lines. For each line that is not a premise, cite the rule and the lines it depends on.

Proof A

Premises: P → Q, P. Goal: Q.

Single-step modus ponens.

Proof B

Premises: P → Q, Q → R, P. Goal: R.

Chain two modus ponens steps, or use hypothetical syllogism then modus ponens.

Proof C

Premises: P ∨ Q, ¬P, Q → R. Goal: R.

Use disjunctive syllogism to derive Q, then modus ponens to derive R.

Proof D

Premises: P → Q, ¬Q, P ∨ R. Goal: R.

Use modus tollens to derive ¬P, then disjunctive syllogism.

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Proof Construction

Deductive

Plan Backwards, Write Forwards

For each argument, write out the backwards plan first (what rule produces the goal, what subgoals you need) and then write the proof forwards.

Plan-and-prove

For each argument, write a short plan: 'Goal is G; to derive G, I will use rule X, which requires subgoals Y and Z.' Then write the formal proof.

Argument A

Premises: (P ∧ Q) → R, P, Q. Goal: R.

Use conjunction to combine P and Q, then modus ponens.

Argument B

Premises: P → (Q → R), P, Q. Goal: R.

Apply modus ponens twice to peel off the nested conditional.

Argument C

Premises: P → Q, R → ¬Q, R. Goal: ¬P.

Derive ¬Q from modus ponens, then apply modus tollens.

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Formalization Practice

Deductive

Formalization Drill: Building Short Formal Proofs

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.

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Proof Construction

Deductive

Proof Practice: Building Short Formal Proofs

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.

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Evaluation Practice

Deductive

Validity Check: Building Short Formal Proofs

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.

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Proof Construction

Deductive

Deep Practice: Building Short Formal Proofs

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.

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Analysis Practice

Deductive

Real-World Transfer: Building Short Formal Proofs

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.

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Rapid Identification

Deductive

Timed Drill: Building Short Formal Proofs

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.

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Evaluation Practice

Deductive

Peer Review: Building Short Formal Proofs

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.'

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Proof Construction

Deductive

Construction Challenge: Building Short Formal Proofs

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.

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Diagnosis Practice

Deductive

Counterexample Challenge: Building Short Formal Proofs

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.

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Analysis Practice

Deductive

Integration Exercise: Building Short Formal Proofs

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?

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Diagnosis Practice

Deductive

Misconception Clinic: Building Short Formal Proofs

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.'

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Proof Construction

Deductive

Scaffolded Proof: Building Short Formal Proofs

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

Deductive

Synthesis Review: Building Short Formal Proofs

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.

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Truth-Table Builder

Enter a propositional formula using variables (P, Q, R...) and connectives. Separate multiple formulas with commas to compare them.

~ or ¬& or ∧| or ∨-> or →<-> or ↔
PQP → Q
FFT
FTT
TFF
TTT
Contingent2 variables · 4 rows · 3 true

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
  • Do not skip the planning step; backwards reasoning from the goal is the fastest way to find the proof.
  • Do not apply modus ponens to a line that is not actually a conditional of the right shape.
Where students usually go wrong

Citing a rule that does not actually fit the cited lines.

Trying to derive the conclusion directly without intermediate steps.

Deriving lines that look useful but do not advance toward the goal.

Confusing modus ponens with affirming the consequent or modus tollens with denying the antecedent.

Historical context for this way of reasoning

Chrysippus

The rules taught in this lesson, including modus ponens, modus tollens, and disjunctive syllogism, were catalogued by Chrysippus and the Stoics and have been central to logical training ever since.