Rigorous Reasoning

Propositional Logic

Symbolizing Whole Arguments

Teaches students to move from natural-language arguments to complete symbolic forms, assigning sentence letters consistently and preserving the inferential structure of the whole argument.

Read for structure, not just vocabulary. The goal is to learn how natural-language claims are converted into a cleaner formal shape.

DeductiveFormalizationLesson 3 of 60% progress
Previous LessonThe Five Connectives in DepthBack To UnitPropositional Logic: Form, Connectives, and Valid InferenceNext LessonTruth Tables, Equivalence, and Validity

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What this lesson is helping you do

Teaches students to move from natural-language arguments to complete symbolic forms, assigning sentence letters consistently and preserving the inferential structure of the whole argument. The practice in this lesson depends on understanding Atomic Statement, Logical Connective, and Main Connective and applying tools such as Respect the Main Connective and Assign Sentence Letters Consistently correctly.

How to approach it

Read for structure, not just vocabulary. The goal is to learn how natural-language claims are converted into a cleaner formal shape.

What the practice is building

You will put the explanation to work through formalization practice, quiz, proof construction, 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

Symbolize 6 natural-language arguments with a complete key, including at least one argument involving chained conditionals, one disjunctive syllogism, and one argument containing a biconditional.

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.

Context

From single statements to whole arguments

Symbolizing a single statement is a valuable skill, but the payoff of propositional logic comes when you symbolize whole arguments. An argument is a sequence of premises leading to a conclusion, and its validity depends on the logical relationships among all of those symbolic forms. A consistent key is therefore not optional — it is what makes the argument analyzable at all.

When you symbolize an argument, write out the key first. List every distinct claim and the letter assigned to it. This is the single best habit you can develop in this unit. A clear key takes thirty seconds to write and saves you from almost every catastrophic error downstream.

What to look for

  • Write the key (sentence letter to English claim) before you start symbolizing any premise.
  • Reuse the same letter every time the same claim reappears.
  • Symbolize premises and conclusion using the same key, without introducing new letters mid-argument.
Argument symbolization is a two-step process: build a key, then symbolize every statement consistently against that key.

Discipline

Handling claims that almost repeat

Natural-language arguments often state claims in slightly different words across premises. 'The server is down' in one premise may appear as 'the server has failed' in another. If these really refer to the same state of affairs, they should receive the same sentence letter. If they refer to different states, they should not, even if the wording is close. Your job is to decide which and be consistent.

When in doubt, write out both English claims side by side and ask whether accepting one commits you to accepting the other. If yes, they are the same atomic claim. If not, they are distinct and need different letters. Do not let surface paraphrase multiply the letters in your key.

What to look for

  • Ask whether two differently-worded claims actually describe the same state of affairs.
  • Give the same letter to identical claims, no matter how the words change.
  • Give different letters to genuinely different claims, even on the same topic.
Consistent letter assignment is a judgment call about claim identity, not a matter of exact wording.

Method

Translating the whole argument in a single pass

Once the key is written, work through the premises and conclusion one at a time. For each statement, identify the main connective, symbolize the parts, and write down the full symbolic form. Keep the premises and the conclusion visually separated so you can see the overall inference.

At the end, do a sanity check: read each symbolic form back as English and compare it to the original. If the translation is faithful, you are ready to test validity. If something sounds wrong on the way back, the symbolization probably has an error. This check catches the most common mistakes without any extra machinery.

What to look for

  • Symbolize each premise and the conclusion one at a time.
  • Keep the structure of the whole argument visible on the page.
  • Read every symbolic form back into English as a final check.
A one-pass symbolization, followed by a read-back check, catches most beginner errors before they propagate into validity testing.

Principle

Why structural fidelity matters more than elegance

Beginners sometimes try to simplify as they symbolize: flattening a nested conditional, collapsing a conjunction, or dropping a clause they think is redundant. Resist that urge. Your job at the symbolization step is to capture the structure of the original argument, not to improve it. Simplification and equivalence transformations come later, and they work correctly only on faithful translations.

The same principle applies to making the argument look 'nicer.' A long, ugly symbolic form that mirrors the original is more useful than a short, clean form that has quietly changed the argument. You can always simplify later using equivalences. You cannot recover a structure you threw away on your first pass.

What to look for

  • Do not simplify or clean up the symbolization on the first pass.
  • Preserve nested structure even when it looks clumsy.
  • Use equivalence transformations later, not during the initial translation.
Fidelity beats elegance. Symbolize the argument as written, then simplify once you are sure the translation is correct.

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.

Atomic Statement

A declarative sentence that is not further analyzed at the propositional level and is represented by a single sentence letter.

Why it matters: Atomic statements are the basic building blocks from which compound propositions are constructed.

Logical Connective

An operator such as negation, conjunction, disjunction, conditional, or biconditional that forms a compound proposition from simpler ones.

Why it matters: Connectives are the joints of propositional logic; the truth value of a compound depends entirely on them.

Main Connective

The connective with the widest scope in a compound statement, which determines the statement's overall logical form.

Why it matters: The main connective tells you the basic shape of the proposition and therefore governs how it participates in inference.

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.

Formalization Demo

The lesson shows how the same reasoning looks once its structure is made explicit.

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

Modus Ponens in the Wild

This is the textbook form of modus ponens. The key lets you see the pattern clearly, and the symbolization shows that the topic is irrelevant to the validity of the form.

Content

  • Argument: If the lab report is submitted on time, then Nora will receive full credit. The lab report was submitted on time. Therefore, Nora will receive full credit.
  • Key: L = the lab report is submitted on time; N = Nora will receive full credit.
  • Premise 1: L → N.
  • Premise 2: L.
  • Conclusion: N.

Worked Example

Chaining Two Conditionals

This is hypothetical syllogism. The symbolization makes the chaining structure obvious, and the middle letter F drops out of the conclusion exactly as the rule predicts.

Content

  • Argument: If the server is down, the service will fail. If the service fails, users will complain. Therefore, if the server is down, users will complain.
  • Key: D = the server is down; F = the service will fail; C = users will complain.
  • Premise 1: D → F.
  • Premise 2: F → C.
  • Conclusion: D → C.

Pause and Check

Questions to use before you move into practice

Self-check questions

  • Is my key complete and consistent with every premise and the conclusion?
  • If I read each symbolic form back into English, does it match the original?
  • Have I preserved the main connective and the scope of every compound?

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.

Formalization Practice

Deductive

Symbolize the Whole Argument

For each argument, write out the key of sentence letters and symbolize the premises and the conclusion.

Arguments to symbolize

Select an argument. Write the key first, then symbolize each premise and the conclusion. Confirm that your symbolization preserves the argument's structure.

Argument A

If the lab report is submitted on time, then Nora will receive full credit. The lab report was submitted on time. Therefore, Nora will receive full credit.

Track the conditional and keep the recurring claim consistent across premises and conclusion.

Argument B

Either the debate starts late or the final speech is shortened. The debate does not start late. Therefore, the final speech is shortened.

Notice the disjunction and the role played by the negated second premise.

Argument C

If the archive is open, then Maya can study in the quiet room. Maya cannot study in the quiet room. Therefore, the archive is not open.

Preserve the direction of the conditional and the scope of the negation in the second premise.

Argument D

If the server is down, the service will fail. If the service fails, users will complain. Therefore, if the server is down, users will complain.

Chain the two conditionals without collapsing them into a single claim.

Choose one of the arguments above, assign sentence letters, and translate the premises and conclusion into symbolic form.

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Quiz

Deductive

Scenario Check: Symbolizing Whole Arguments

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: "Writing a key that is incomplete or inconsistent with the symbolized premises." 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 assign sentence letters consistently, starting from scratch. Be specific about each step and explain why the order matters.

Can the student transfer the skill of assign sentence letters consistently to a genuinely new case?

Question 3 — Distinguish

Someone confuses atomic statement with logical connective. 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 in the Wild" 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?

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Formalization Practice

Deductive

Formalization Drill: Symbolizing Whole Arguments

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: Symbolizing Whole Arguments

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: Symbolizing Whole Arguments

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: Symbolizing Whole Arguments

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: Symbolizing Whole Arguments

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: Symbolizing Whole Arguments

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: Symbolizing Whole Arguments

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: Symbolizing Whole Arguments

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: Symbolizing Whole Arguments

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: Symbolizing Whole Arguments

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: Symbolizing Whole Arguments

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: Symbolizing Whole Arguments

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: Symbolizing Whole Arguments

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 writing the key; it is the cheapest insurance against symbolization errors.
  • Do not simplify or clean up the structure during translation; do it later if at all.
Where students usually go wrong

Writing a key that is incomplete or inconsistent with the symbolized premises.

Inventing new letters for claims that already have letters assigned.

Simplifying the symbolization at the translation stage and losing the original structure.

Writing symbolic forms that do not actually match the direction or scope of the English.

Historical context for this way of reasoning

Gottlob Frege

Frege was the first logician to insist on fully explicit symbolic translation as a precondition for rigorous argument analysis. The habit of writing a key and symbolizing premises before testing validity is part of his legacy.