Categorical Propositions and the A, E, I, O Forms

Notice what categorical propositions do not do: they do not express conditional relationships ('if ... then'), they do not express degrees of probability, and they do not talk about individual named objects as such. The sentence 'Socrates is mortal' can be forced into categorical form only by treating 'Socrates' as a one-member class, which is a trick that works but obscures what categorical logic does best. Categorical logic shines when the subject and predicate both name genuine classes with multiple members: physicians, reptiles, prime numbers, licensed drivers. Keep the class-based reading front of mind as you work through the forms.

A useful orientation exercise is to take any declarative sentence and ask two questions: What is the subject class? What is the predicate class? If you can answer both, the sentence is a candidate for categorical analysis. If you cannot — for instance, because the sentence says 'It is raining' or 'Democracy is the best system' — then categorical logic is not the right tool, and you should look to propositional or predicate logic instead.

• Read the proposition as a claim about class membership, not as a claim about individual objects.
• Identify the two classes involved — the subject class and the predicate class.
• Decide whether the proposition is including, excluding, or partially overlapping them.
• Verify that both terms name genuine classes, not conditionals, probabilities, or bare predicates.

1. 01Scenario Check: Categorical Propositions and the A, E, I, O FormsQuiz−

   Quiz

   Deductive

   Scenario Check: Categorical Propositions and the A, E, I, O Forms

   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.

   Answer each question in one or two sentences from memory. Your answers stay on this page — nothing is sent or graded.

   1. Question 1 — Diagnose

      A student makes the following mistake: "Confusing universal with affirmative, or particular with negative; quantity and quality are independent." Explain specifically what is wrong with this reasoning and what the student should have done instead.

      Your answer to Question 1 — Diagnose
      What a strong answer covers

      Can the student identify the flaw and articulate the correction?

   2. Question 2 — Apply

      You encounter a new argument that you have never seen before. Walk through exactly how you would identify subject and predicate terms, starting from scratch. Be specific about each step and explain why the order matters.

      Your answer to Question 2 — Apply
      What a strong answer covers

      Can the student transfer the skill of identify subject and predicate terms to a genuinely new case?

   3. Question 3 — Distinguish

      Someone confuses categorical proposition with subject term. Write a short explanation that would help them see the difference, and give one example where getting them confused leads to a concrete mistake.

      Your answer to Question 3 — Distinguish
      What a strong answer covers

      Does the student understand the boundary between the two concepts?

   4. Question 4 — Transfer

      The worked example "Translating 'Only'" 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.

      Your answer to Question 4 — Transfer
      What a strong answer covers

      Can the student adapt the demonstrated method to a variation?

2. 02Classify into A, E, I, OClassification Practice+

   Classification Practice

   Deductive

   Classify into A, E, I, O

   For each proposition, rewrite it in standard form (if needed), identify the subject and predicate terms, and classify it as A, E, I, or O.

   Propositions to classify

   Rewrite each proposition in standard form with clear subject and predicate classes, then label it A, E, I, or O.

   Proposition A

   All registered voters received a ballot.

   Straightforward A proposition — check the subject and predicate terms.

   Proposition B

   No reptiles are warm-blooded.

   Universal negative form. Note that both terms will end up distributed.

   Proposition C

   Some artists are perfectionists.

   Particular affirmative. Note that neither term is distributed.

   Proposition D

   Some politicians are not truthful.

   Particular negative. Only the predicate is distributed.

   Proposition E

   Only registered students may use the library.

   Watch for 'only' — it flips the direction of the class relation.

   Proposition F

   Not all senators attended the vote.

   Translate to a particular negative O form.

   Type these symbols: ~ = ¬ not · & = ∧ and · | = ∨ or · -> = → if-then · <-> = ↔ iff · variables: P, Q, R...

   Show me an example

   How to fill this out

   1. Premises (top box): note the assumptions for your own thinking. The graded record is the rows below — repeat each premise there as line 1, 2, ... with Justification = "Premise".
   2. Goal: the formula you intend to derive on the final line.
   3. Proof rows: one row per step. Statement is the formula. Justification says how you got it:
      • For an assumption: type Premise
      • For a derived line, name the rule and the lines it uses:Modus Ponens from 1, 2 · MT 3, 4 · DeM 5
   4. Watch the badges next to each row. Verified means the inference checks out. An error names what didn't fit.
   5. Click Show me an example if you want a complete sample loaded into the form.

   Premises (scratch space)Goal (formula to derive)

   Proof Draft

   LineStatementJustificationAction

   1Remove

   2Remove

   3Remove

   Add Proof Line

   Save responseNot saved yet.
3. 03Timed Drill: Categorical Propositions and the A, E, I, O FormsRapid Identification+

   Rapid Identification

   Deductive

   Timed Drill: Categorical Propositions and the A, E, I, O Forms

   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.

   Type these symbols: ~ = ¬ not · & = ∧ and · | = ∨ or · -> = → if-then · <-> = ↔ iff · variables: P, Q, R...

   Show me an example

   How to fill this out

   1. Premises (top box): note the assumptions for your own thinking. The graded record is the rows below — repeat each premise there as line 1, 2, ... with Justification = "Premise".
   2. Goal: the formula you intend to derive on the final line.
   3. Proof rows: one row per step. Statement is the formula. Justification says how you got it:
      • For an assumption: type Premise
      • For a derived line, name the rule and the lines it uses:Modus Ponens from 1, 2 · MT 3, 4 · DeM 5
   4. Watch the badges next to each row. Verified means the inference checks out. An error names what didn't fit.
   5. Click Show me an example if you want a complete sample loaded into the form.

   Premises (scratch space)Goal (formula to derive)

   Proof Draft

   LineStatementJustificationAction

   1Remove

   2Remove

   3Remove

   Add Proof Line

   Save responseNot saved yet.
4. 04Formalization Drill: Categorical Propositions and the A, E, I, O FormsFormalization Practice+

   Formalization Practice

   Deductive

   Formalization Drill: Categorical Propositions and the A, E, I, O Forms

   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.

   Sentence-letter key

   Symbolic premisesSymbolic conclusion

   Main connective or structure note

   Save responseNot saved yet.
5. 05Real-World Transfer: Categorical Propositions and the A, E, I, O FormsAnalysis Practice+

   Analysis Practice

   Deductive

   Real-World Transfer: Categorical Propositions and the A, E, I, O Forms

   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.

   Type these symbols: ~ = ¬ not · & = ∧ and · | = ∨ or · -> = → if-then · <-> = ↔ iff · variables: P, Q, R...

   Show me an example

   How to fill this out

   1. Premises (top box): note the assumptions for your own thinking. The graded record is the rows below — repeat each premise there as line 1, 2, ... with Justification = "Premise".
   2. Goal: the formula you intend to derive on the final line.
   3. Proof rows: one row per step. Statement is the formula. Justification says how you got it:
      • For an assumption: type Premise
      • For a derived line, name the rule and the lines it uses:Modus Ponens from 1, 2 · MT 3, 4 · DeM 5
   4. Watch the badges next to each row. Verified means the inference checks out. An error names what didn't fit.
   5. Click Show me an example if you want a complete sample loaded into the form.

   Premises (scratch space)Goal (formula to derive)

   Proof Draft

   LineStatementJustificationAction

   1Remove

   2Remove

   3Remove

   Add Proof Line

   Save responseNot saved yet.
6. 06Integration Exercise: Categorical Propositions and the A, E, I, O FormsAnalysis Practice+

   Analysis Practice

   Deductive

   Integration Exercise: Categorical Propositions and the A, E, I, O Forms

   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?

   Type these symbols: ~ = ¬ not · & = ∧ and · | = ∨ or · -> = → if-then · <-> = ↔ iff · variables: P, Q, R...

   Show me an example

   How to fill this out

   1. Premises (top box): note the assumptions for your own thinking. The graded record is the rows below — repeat each premise there as line 1, 2, ... with Justification = "Premise".
   2. Goal: the formula you intend to derive on the final line.
   3. Proof rows: one row per step. Statement is the formula. Justification says how you got it:
      • For an assumption: type Premise
      • For a derived line, name the rule and the lines it uses:Modus Ponens from 1, 2 · MT 3, 4 · DeM 5
   4. Watch the badges next to each row. Verified means the inference checks out. An error names what didn't fit.
   5. Click Show me an example if you want a complete sample loaded into the form.

   Premises (scratch space)Goal (formula to derive)

   Proof Draft

   LineStatementJustificationAction

   1Remove

   2Remove

   3Remove

   Add Proof Line

   Save responseNot saved yet.
7. 07Synthesis Review: Categorical Propositions and the A, E, I, O FormsAnalysis Practice+

   Analysis Practice

   Deductive

   Synthesis Review: Categorical Propositions and the A, E, I, O Forms

   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.

   Type these symbols: ~ = ¬ not · & = ∧ and · | = ∨ or · -> = → if-then · <-> = ↔ iff · variables: P, Q, R...

   Show me an example

   How to fill this out

   1. Premises (top box): note the assumptions for your own thinking. The graded record is the rows below — repeat each premise there as line 1, 2, ... with Justification = "Premise".
   2. Goal: the formula you intend to derive on the final line.
   3. Proof rows: one row per step. Statement is the formula. Justification says how you got it:
      • For an assumption: type Premise
      • For a derived line, name the rule and the lines it uses:Modus Ponens from 1, 2 · MT 3, 4 · DeM 5
   4. Watch the badges next to each row. Verified means the inference checks out. An error names what didn't fit.
   5. Click Show me an example if you want a complete sample loaded into the form.

   Premises (scratch space)Goal (formula to derive)

   Proof Draft

   LineStatementJustificationAction

   1Remove

   2Remove

   3Remove

   Add Proof Line

   Save responseNot saved yet.
8. 08Validity Check: Categorical Propositions and the A, E, I, O FormsEvaluation Practice+

   Evaluation Practice

   Deductive

   Validity Check: Categorical Propositions and the A, E, I, O Forms

   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.

   Type these symbols: ~ = ¬ not · & = ∧ and · | = ∨ or · -> = → if-then · <-> = ↔ iff · variables: P, Q, R...

   Show me an example

   How to fill this out

   1. Premises (top box): note the assumptions for your own thinking. The graded record is the rows below — repeat each premise there as line 1, 2, ... with Justification = "Premise".
   2. Goal: the formula you intend to derive on the final line.
   3. Proof rows: one row per step. Statement is the formula. Justification says how you got it:
      • For an assumption: type Premise
      • For a derived line, name the rule and the lines it uses:Modus Ponens from 1, 2 · MT 3, 4 · DeM 5
   4. Watch the badges next to each row. Verified means the inference checks out. An error names what didn't fit.
   5. Click Show me an example if you want a complete sample loaded into the form.

   Premises (scratch space)Goal (formula to derive)

   Proof Draft

   LineStatementJustificationAction

   1Remove

   2Remove

   3Remove

   Add Proof Line

   Save responseNot saved yet.
9. 09Peer Review: Categorical Propositions and the A, E, I, O FormsEvaluation Practice+

   Evaluation Practice

   Deductive

   Peer Review: Categorical Propositions and the A, E, I, O Forms

   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.'

   Type these symbols: ~ = ¬ not · & = ∧ and · | = ∨ or · -> = → if-then · <-> = ↔ iff · variables: P, Q, R...

   Show me an example

   How to fill this out

   1. Premises (top box): note the assumptions for your own thinking. The graded record is the rows below — repeat each premise there as line 1, 2, ... with Justification = "Premise".
   2. Goal: the formula you intend to derive on the final line.
   3. Proof rows: one row per step. Statement is the formula. Justification says how you got it:
      • For an assumption: type Premise
      • For a derived line, name the rule and the lines it uses:Modus Ponens from 1, 2 · MT 3, 4 · DeM 5
   4. Watch the badges next to each row. Verified means the inference checks out. An error names what didn't fit.
   5. Click Show me an example if you want a complete sample loaded into the form.

   Premises (scratch space)Goal (formula to derive)

   Proof Draft

   LineStatementJustificationAction

   1Remove

   2Remove

   3Remove

   Add Proof Line

   Save responseNot saved yet.
10. 10Counterexample Challenge: Categorical Propositions and the A, E, I, O FormsDiagnosis Practice+

    Diagnosis Practice

    Deductive

    Counterexample Challenge: Categorical Propositions and the A, E, I, O Forms

    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.

    Type these symbols: ~ = ¬ not · & = ∧ and · | = ∨ or · -> = → if-then · <-> = ↔ iff · variables: P, Q, R...

    Show me an example

    How to fill this out

    1. Premises (top box): note the assumptions for your own thinking. The graded record is the rows below — repeat each premise there as line 1, 2, ... with Justification = "Premise".
    2. Goal: the formula you intend to derive on the final line.
    3. Proof rows: one row per step. Statement is the formula. Justification says how you got it:
       • For an assumption: type Premise
       • For a derived line, name the rule and the lines it uses:Modus Ponens from 1, 2 · MT 3, 4 · DeM 5
    4. Watch the badges next to each row. Verified means the inference checks out. An error names what didn't fit.
    5. Click Show me an example if you want a complete sample loaded into the form.

    Premises (scratch space)Goal (formula to derive)

    Proof Draft

    LineStatementJustificationAction

    1Remove

    2Remove

    3Remove

    Add Proof Line

    Save responseNot saved yet.
11. 11Misconception Clinic: Categorical Propositions and the A, E, I, O FormsDiagnosis Practice+

    Diagnosis Practice

    Deductive

    Misconception Clinic: Categorical Propositions and the A, E, I, O Forms

    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.'

    Type these symbols: ~ = ¬ not · & = ∧ and · | = ∨ or · -> = → if-then · <-> = ↔ iff · variables: P, Q, R...

    Show me an example

    How to fill this out

    1. Premises (top box): note the assumptions for your own thinking. The graded record is the rows below — repeat each premise there as line 1, 2, ... with Justification = "Premise".
    2. Goal: the formula you intend to derive on the final line.
    3. Proof rows: one row per step. Statement is the formula. Justification says how you got it:
       • For an assumption: type Premise
       • For a derived line, name the rule and the lines it uses:Modus Ponens from 1, 2 · MT 3, 4 · DeM 5
    4. Watch the badges next to each row. Verified means the inference checks out. An error names what didn't fit.
    5. Click Show me an example if you want a complete sample loaded into the form.

    Premises (scratch space)Goal (formula to derive)

    Proof Draft

    LineStatementJustificationAction

    1Remove

    2Remove

    3Remove

    Add Proof Line

    Save responseNot saved yet.
12. 12Proof Practice: Categorical Propositions and the A, E, I, O FormsProof Construction+

    Proof Construction

    Deductive

    Proof Practice: Categorical Propositions and the A, E, I, O Forms

    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.

    Type these symbols: ~ = ¬ not · & = ∧ and · | = ∨ or · -> = → if-then · <-> = ↔ iff · variables: P, Q, R...

    Show me an example

    How to fill this out

    1. Premises (top box): note the assumptions for your own thinking. The graded record is the rows below — repeat each premise there as line 1, 2, ... with Justification = "Premise".
    2. Goal: the formula you intend to derive on the final line.
    3. Proof rows: one row per step. Statement is the formula. Justification says how you got it:
       • For an assumption: type Premise
       • For a derived line, name the rule and the lines it uses:Modus Ponens from 1, 2 · MT 3, 4 · DeM 5
    4. Watch the badges next to each row. Verified means the inference checks out. An error names what didn't fit.
    5. Click Show me an example if you want a complete sample loaded into the form.

    Premises (scratch space)Goal (formula to derive)

    Proof Draft

    LineStatementJustificationAction

    1Remove

    2Remove

    3Remove

    Add Proof Line

    Save responseNot saved yet.
13. 13Deep Practice: Categorical Propositions and the A, E, I, O FormsProof Construction+

    Proof Construction

    Deductive

    Deep Practice: Categorical Propositions and the A, E, I, O Forms

    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.

    Type these symbols: ~ = ¬ not · & = ∧ and · | = ∨ or · -> = → if-then · <-> = ↔ iff · variables: P, Q, R...

    Show me an example

    How to fill this out

    1. Premises (top box): note the assumptions for your own thinking. The graded record is the rows below — repeat each premise there as line 1, 2, ... with Justification = "Premise".
    2. Goal: the formula you intend to derive on the final line.
    3. Proof rows: one row per step. Statement is the formula. Justification says how you got it:
       • For an assumption: type Premise
       • For a derived line, name the rule and the lines it uses:Modus Ponens from 1, 2 · MT 3, 4 · DeM 5
    4. Watch the badges next to each row. Verified means the inference checks out. An error names what didn't fit.
    5. Click Show me an example if you want a complete sample loaded into the form.

    Premises (scratch space)Goal (formula to derive)

    Proof Draft

    LineStatementJustificationAction

    1Remove

    2Remove

    3Remove

    Add Proof Line

    Save responseNot saved yet.
14. 14Construction Challenge: Categorical Propositions and the A, E, I, O FormsProof Construction+

    Proof Construction

    Deductive

    Construction Challenge: Categorical Propositions and the A, E, I, O Forms

    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.

    Type these symbols: ~ = ¬ not · & = ∧ and · | = ∨ or · -> = → if-then · <-> = ↔ iff · variables: P, Q, R...

    Show me an example

    How to fill this out

    1. Premises (top box): note the assumptions for your own thinking. The graded record is the rows below — repeat each premise there as line 1, 2, ... with Justification = "Premise".
    2. Goal: the formula you intend to derive on the final line.
    3. Proof rows: one row per step. Statement is the formula. Justification says how you got it:
       • For an assumption: type Premise
       • For a derived line, name the rule and the lines it uses:Modus Ponens from 1, 2 · MT 3, 4 · DeM 5
    4. Watch the badges next to each row. Verified means the inference checks out. An error names what didn't fit.
    5. Click Show me an example if you want a complete sample loaded into the form.

    Premises (scratch space)Goal (formula to derive)

    Proof Draft

    LineStatementJustificationAction

    1Remove

    2Remove

    3Remove

    Add Proof Line

    Save responseNot saved yet.
15. 15Scaffolded Proof: Categorical Propositions and the A, E, I, O FormsProof Construction+

    Proof Construction

    Deductive

    Scaffolded Proof: Categorical Propositions and the A, E, I, O Forms

    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.

    Type these symbols: ~ = ¬ not · & = ∧ and · | = ∨ or · -> = → if-then · <-> = ↔ iff · variables: P, Q, R...

    Show me an example

    How to fill this out

    1. Premises (top box): note the assumptions for your own thinking. The graded record is the rows below — repeat each premise there as line 1, 2, ... with Justification = "Premise".
    2. Goal: the formula you intend to derive on the final line.
    3. Proof rows: one row per step. Statement is the formula. Justification says how you got it:
       • For an assumption: type Premise
       • For a derived line, name the rule and the lines it uses:Modus Ponens from 1, 2 · MT 3, 4 · DeM 5
    4. Watch the badges next to each row. Verified means the inference checks out. An error names what didn't fit.
    5. Click Show me an example if you want a complete sample loaded into the form.

    Premises (scratch space)Goal (formula to derive)

    Proof Draft

    LineStatementJustificationAction

    1Remove

    2Remove

    3Remove

    Add Proof Line

    Save responseNot saved yet.