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.
Categorical Logic
The Square of Opposition and Existential Import
Introduces the square of opposition, explains the relationships of contradiction, contrariety, subcontrariety, and subalternation among A, E, I, and O propositions, and distinguishes the traditional and modern treatments of existential import.
Focus on understanding the core distinction first, then use the examples to see how the idea behaves in actual arguments.
Start Here
What this lesson is helping you do
Introduces the square of opposition, explains the relationships of contradiction, contrariety, subcontrariety, and subalternation among A, E, I, and O propositions, and distinguishes the traditional and modern treatments of existential import. The practice in this lesson depends on understanding A, E, I, O Propositions, Square of Opposition, and Existential Import and applying tools such as The Middle Term Must Be Distributed At Least Once and No Term Distributed in Conclusion Unless Distributed in Premises correctly.
How to approach it
Focus on understanding the core distinction first, then use the examples to see how the idea behaves in actual arguments.
What the practice is building
You will put the explanation to work through analysis practice, diagnosis practice, formalization practice, proof construction, evaluation practice, and rapid identification activities, so the goal is not just to recognize the idea but to use it under your own control.
What success should let you do
Given 8 categorical propositions, produce the contradictory of each and determine which square inferences are available under the traditional and the modern reading.
Reading Path
Move through the lesson in this order
The page is designed to teach before it tests. Use this sequence to keep the reading, examples, and practice in the right relationship.
Study
Watch the move in context
Use the worked examples to see how the reasoning behaves when someone else performs it carefully.
Do
Practice with a standard
Only then move into the activities, using the pause-and-check prompts as a final checkpoint before you submit.
Guided Explanation
Read this before you try the activity
These sections give the learner a usable mental model first, so the practice feels like application rather than guesswork.
Orientation
Four propositions, four relationships
When A, E, I, and O propositions share the same subject term and the same predicate term, they stand in four classical relationships to one another. Contradictories are pairs that cannot both be true and cannot both be false: A and O are contradictories, and E and I are contradictories. Contraries cannot both be true but can both be false: A and E are contraries in the traditional reading. Subcontraries cannot both be false but can both be true: I and O are subcontraries in the traditional reading. Subalternation holds between a universal and the corresponding particular: under the traditional reading, A implies I, and E implies O.
These relationships are visualized as a square with A at the top-left, E at the top-right, I at the bottom-left, and O at the bottom-right. Diagonals show contradiction; the top side shows contrariety; the bottom side shows subcontrariety; and the vertical sides show subalternation. The square has been called 'the logician's alphabet' because it lets you read off a dozen immediate inferences at a glance once you memorize it.
What to look for
- Memorize the four corner positions and the four relationships.
- Use contradiction to immediately negate any proposition into its contradictory form.
- Use the other relationships only after you decide whether you are using the traditional or modern reading.
The square of opposition codifies the immediate inferences available among A, E, I, and O propositions sharing subject and predicate terms.
What you can always use
Contradiction is the safest inference
Of the four square relationships, contradiction is the only one that survives both the traditional and modern readings. A and O contradict each other in every interpretation, and so do E and I. That means you can always use contradiction: if A is true, O is false; if I is true, E is false; and so on. These immediate inferences hold regardless of what you assume about the existence of class members.
A common use of contradiction is to negate propositions in argument. If someone asserts 'All swans are white' and you believe this is false, its contradictory is 'Some swans are not white,' and proving that single O proposition refutes the A. This is much more efficient than trying to argue the contrary ('No swans are white'), which is a stronger claim than you need. Good logicians reach for the contradictory because it is the exact denial, and exact denial is the minimal claim that suffices to refute.
What to look for
- Use contradiction to negate any categorical proposition precisely.
- When refuting a universal, reach for the particular negative of the same form.
- Remember that contradiction works in both the traditional and modern readings.
Contradiction is the universal move: always available, always exact. Learn to use it as your default negation tool.
Traditional relationships
Contraries, subcontraries, and subalternation in the traditional reading
Under the traditional reading, universal propositions carry existential import: 'All S are P' implicitly assumes that there are some S. Given that assumption, the other square relationships hold. Contrariety: A and E cannot both be true, because if all S are P, then it is not the case that no S are P. Subcontrariety: I and O cannot both be false, because if some S are P is false, then no S are P, which by subalternation gives us some S are not P. Subalternation: from A you can infer I, and from E you can infer O — the reasoning runs from 'every S is P' to 'at least one S is P.'
These inferences are clean and useful, but they depend on the assumption that the subject class is nonempty. If there are no S at all, subalternation fails: 'All unicorns have one horn' does not entail 'Some unicorn has one horn' because there are no unicorns. Traditional logic handled this by limiting its attention to nonempty classes, which was fine for most practical reasoning but produced trouble in mathematical and hypothetical contexts.
What to look for
- Under the traditional reading, universal propositions imply the corresponding particular via subalternation.
- Contraries cannot both be true but may both be false.
- Subcontraries cannot both be false but may both be true.
The traditional square gives you contrariety, subcontrariety, and subalternation in addition to contradiction, but those extra moves depend on nonempty subject classes.
Modern correction
The modern reading removes existential import from universals
Modern logic, following Frege and Russell, treats universal propositions as not carrying existential import. 'All S are P' is read as 'If anything is S, it is P,' which is vacuously true when there are no S. Under this reading, subalternation fails: from 'All unicorns have one horn' you cannot infer 'Some unicorn has one horn.' Contrariety and subcontrariety also disappear, because the reasoning for them used subalternation implicitly.
What survives is contradiction. A and O still contradict; E and I still contradict. Everything else on the square must be used with care. The modern reading is the default in contemporary logic textbooks, mathematical reasoning, and most computer-science applications. The traditional reading remains useful as a historical touchstone and is perfectly fine when you know the subject class is nonempty, but you should always be clear which reading you are using.
What to look for
- Treat the modern reading as the default unless explicitly told otherwise.
- Remember that only contradiction survives unrestricted in the modern reading.
- When working with nonempty classes, the traditional inferences are still valid if you are careful.
Modern logic removes existential import from universal propositions, which restricts the square but makes it consistent with empty classes.
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.
A, E, I, O Propositions
The four standard forms of categorical proposition: A (universal affirmative), E (universal negative), I (particular affirmative), and O (particular negative).
Why it matters: Every standard-form categorical claim is one of these four, and each form has a characteristic distribution pattern used in syllogistic analysis.
Square of Opposition
A diagram that represents the logical relationships among A, E, I, and O propositions sharing the same subject and predicate terms.
Why it matters: The square codifies inferences of contradiction, contrariety, subcontrariety, and subalternation that have been central to traditional logic since Aristotle.
Existential Import
The question of whether a proposition, especially a universal one, carries the claim that its subject class has at least one member.
Why it matters: Traditional and modern logic disagree about existential import, and the disagreement affects which square-of-opposition inferences are valid.
Reference
Open these only when you need the extra structure
How the lesson is meant to unfold
Concept Intro
The core idea is defined and separated from nearby confusions.
Worked Example
A complete example demonstrates what correct reasoning looks like in context.
Guided Practice
You apply the idea with scaffolding still visible.
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.
The Middle Term Must Be Distributed At Least Once
A valid categorical syllogism must distribute the middle term in at least one premise.
Common failures
- Both premises leave the middle term undistributed, so the subject and predicate of the conclusion are never connected through the whole middle class.
- The student confuses the middle term with the major or minor term and overlooks its distribution.
No Term Distributed in Conclusion Unless Distributed in Premises
No term may be distributed in the conclusion unless it was also distributed in the premise in which it appeared.
Common failures
- Illicit major: the major term is distributed in the conclusion but undistributed in the major premise.
- Illicit minor: the minor term is distributed in the conclusion but undistributed in the minor premise.
No Valid Conclusion from Two Negative Premises
If both premises are negative, no valid conclusion can be drawn.
Common failures
- Two E or O premises appear to support a conclusion but the structural connection is lost.
- Students infer from 'no A are B' and 'no B are C' that 'no A are C,' which is not valid.
Negative Premise Requires Negative Conclusion
If either premise is negative, the conclusion must be negative; if neither premise is negative, the conclusion must be affirmative.
Common failures
- The argument has a negative premise but an affirmative conclusion.
- The argument has two affirmative premises but concludes negatively.
No Particular Conclusion from Two Universal Premises (Modern Reading)
Under the modern (Boolean) reading of existential import, a syllogism with two universal premises cannot yield a particular conclusion, because universal premises do not assert the existence of class members.
Common failures
- The argument infers 'some S are P' from two universal premises under the modern interpretation.
- The student conflates traditional and modern existential import and draws inferences allowed only by the traditional reading.
Patterns
Use these when you need to turn a messy passage into a cleaner logical structure before evaluating it.
Categorical Standard-Form Analysis
Input form
natural_language_categorical_claim
Output form
A_E_I_O_classification
Steps
- Identify the subject class and the predicate class.
- Determine whether the claim is universal (every/no) or particular (some).
- Determine whether the claim is affirmative or negative.
- Classify the proposition as A, E, I, or O.
- Record which terms are distributed under the resulting form.
Watch for
- Misidentifying quantity because of ordinary-language wording ('a' can be universal or particular depending on context).
- Ignoring hidden negation such as 'only' or 'except.'
- Switching subject and predicate when rephrasing into standard form.
Syllogism Term Map
Input form
categorical_syllogism
Output form
major_minor_middle_structure
Steps
- Identify the conclusion and its subject (minor term) and predicate (major term).
- Find the term that appears in both premises but not in the conclusion; that is the middle term.
- Classify each premise as A, E, I, or O.
- Note distribution of every term in every line.
- Apply the five classical rules to determine validity.
Watch for
- Labeling the middle term wrong because the student starts from the first premise instead of the conclusion.
- Checking distribution only for the middle term and missing illicit major or minor.
- Skipping the quality check and missing a negative-premise error.
Three-Circle Venn Validity Test
Input form
categorical_syllogism
Output form
validity_judgment
Steps
- Draw three overlapping circles labeled with the subject, predicate, and middle terms.
- Shade or mark the premises onto the diagram, using shading for universal premises and an 'x' for particular premises.
- After drawing the premises, inspect the diagram to see whether the conclusion is already represented.
- If the conclusion's information is already present in the diagram, the argument is valid; otherwise it is invalid.
- When placing an 'x' that could go in more than one region, place it on the line between regions to represent the ambiguity.
Watch for
- Drawing both premises as shading and then forgetting that the conclusion is a particular claim.
- Placing an 'x' in a single region when the premise does not specify which region.
- Interpreting the shaded diagram as asserting emptiness beyond what the premise actually said.
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
Square Relationships for an A Proposition
Contradiction works in both readings; contrary and subalternation require the traditional reading. When in doubt, use contradiction.
Content
- Proposition: 'All seniors are eligible for graduation.'
- Contradictory (O): 'Some seniors are not eligible for graduation.'
- Contrary (E, traditional only): 'No seniors are eligible for graduation.'
- Subaltern (I, traditional only): 'Some seniors are eligible for graduation.'
Worked Example
Existential Fallacy in Action
The existential fallacy is the move from a universal to a particular without establishing that the subject class is nonempty.
Content
- Argument: 'All trespassers who come within 10 feet of the protected artifact will be fined. Therefore, some trespasser has been fined.'
- Traditional reading: valid via subalternation if you assume there are trespassers.
- Modern reading: invalid — the universal premise does not assert that any trespassers exist.
- Lesson: in modern logic, subalternation is not automatically available.
Pause and Check
Questions to use before you move into practice
Self-check questions
- Which square relationship applies here?
- Am I using the traditional or the modern reading, and is the inference still valid under the reading I am using?
- Is there any doubt that the subject class is nonempty?
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.
Analysis Practice
DeductiveApply the Square
For each proposition, give its contradictory. Then, under the traditional reading, give its contrary (if any), subcontrary (if any), and subaltern (if any).
Propositions to analyze
For each proposition, list its contradictory and the other square relationships under the traditional reading. Note which inferences survive in the modern reading.
Proposition A
All juniors are eligible for the scholarship.
A form. Contradictory is an O proposition; contrary is E; subaltern is I.
Proposition B
No students are late to class.
E form. Contradictory is I; contrary is A; subaltern is O.
Proposition C
Some volunteers are first-year students.
I form. Contradictory is E; subcontrary is O.
Proposition D
Some museum visitors are not season ticket holders.
O form. Contradictory is A; subcontrary is I.
Proof Draft
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Diagnosis Practice
DeductiveSpot the Existential Fallacy
Each argument uses a square-of-opposition inference. Identify whether the inference is valid under the traditional reading, the modern reading, or both.
Inferences to diagnose
Determine the reading under which each inference is valid. Note any inference that depends on existential import.
Inference A
From 'All unicorns have one horn,' therefore 'Some unicorn has one horn.'
Subalternation with empty subject class — invalid under modern reading.
Inference B
From 'No dogs are reptiles,' therefore 'Some dog is not a reptile.'
Subalternation from E to O — valid in the traditional reading, invalid in the modern reading.
Inference C
From 'Some students are athletes,' therefore 'It is not the case that no students are athletes.'
Contradiction from I to denial of E — valid in both readings.
Proof Draft
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Formalization Practice
DeductiveFormalization Drill: The Square of Opposition and Existential Import
Translate each natural-language argument into formal notation. Identify the logical form and check whether the argument is valid.
Practice scenarios
Work through each scenario carefully. Apply the concepts from this lesson.
Argument 1
If the server crashes, then the backup activates. If the backup activates, then an alert is sent. The server crashed. What follows?
Argument 2
Either the contract is valid or the parties must renegotiate. The contract is not valid. What follows?
Argument 3
All databases store records. This system does not store records. What can we conclude about this system?
Choose one of the arguments above, assign sentence letters, and translate the premises and conclusion into symbolic form.
Proof Construction
DeductiveProof Practice: The Square of Opposition and Existential Import
Construct a step-by-step proof or derivation for each argument. Justify every step with the rule you are applying.
Practice scenarios
Work through each scenario carefully. Apply the concepts from this lesson.
Prove
From premises: (1) P -> Q, (2) Q -> R, (3) P. Derive R.
Prove
From premises: (1) A v B, (2) A -> C, (3) B -> C. Derive C.
Prove
From premises: (1) ~(P & Q), (2) P. Derive ~Q.
Proof Draft
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Evaluation Practice
DeductiveValidity Check: The Square of Opposition and Existential Import
Determine whether each argument is deductively valid. If invalid, describe a counterexample where the premises are true but the conclusion is false.
Practice scenarios
Work through each scenario carefully. Apply the concepts from this lesson.
Argument A
All philosophers study logic. Socrates is a philosopher. Therefore, Socrates studies logic.
Argument B
If it rains, the ground is wet. The ground is wet. Therefore, it rained.
Argument C
No birds are mammals. Some mammals fly. Therefore, some things that fly are not birds.
Argument D
Either the door is locked or the alarm is on. The door is not locked. Therefore, the alarm is on.
Proof Draft
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Proof Construction
DeductiveDeep Practice: The Square of Opposition and Existential Import
Work through these challenging exercises. Each one requires careful application of formal reasoning. Show your work step by step.
Challenging derivations
Prove each conclusion from the given premises. Label every inference rule you use.
Challenge 1
Premises: (1) (A & B) -> C, (2) D -> A, (3) D -> B, (4) D. Derive: C.
Challenge 2
Premises: (1) P -> (Q & R), (2) R -> S, (3) ~S. Derive: ~P.
Challenge 3
Premises: (1) A v B, (2) A -> (C & D), (3) B -> (C & E). Derive: C.
Challenge 4
Premises: (1) ~(P & Q), (2) P v Q, (3) P -> R, (4) Q -> S. Derive: R v S.
Proof Draft
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Analysis Practice
DeductiveReal-World Transfer: The Square of Opposition and Existential Import
Apply formal logic to real-world contexts. Translate each scenario into formal notation, determine validity, and explain the practical implications.
Logic in the wild
These scenarios come from law, science, and everyday reasoning. Formalize and evaluate each.
Legal reasoning
A contract states: 'If the product is defective AND the buyer reports within 30 days, THEN a full refund will be issued.' The product was defective. The buyer reported on day 35. The company denies the refund. Is the company's position logically valid?
Medical reasoning
A diagnostic protocol states: 'If the patient has fever AND cough, test for flu. If the flu test is negative AND symptoms persist for 7+ days, test for bacterial infection.' A patient has a cough but no fever. What does the protocol require?
Policy reasoning
A policy reads: 'Students may graduate early IF they complete all required courses AND maintain a 3.5 GPA OR receive special faculty approval.' Due to the ambiguity of OR, identify the two possible readings and explain what difference they make.
Proof Draft
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Rapid Identification
DeductiveTimed Drill: The Square of Opposition and Existential Import
Work through these quickly. For each mini-scenario, identify the logical form, name the rule used, and state whether the inference is valid. Aim for accuracy under time pressure.
Quick-fire logic identification
Identify the logical form and validity of each argument in under 60 seconds per item.
Item 1
If the reactor overheats, the failsafe triggers. The failsafe triggered. Therefore, the reactor overheated.
Item 2
All licensed pilots passed the medical exam. Jenkins is a licensed pilot. Therefore, Jenkins passed the medical exam.
Item 3
Either the encryption key expired or someone changed the password. The encryption key did not expire. Therefore, someone changed the password.
Item 4
If taxes increase, consumer spending decreases. Consumer spending has not decreased. Therefore, taxes have not increased.
Item 5
No insured vehicle was towed. This vehicle was towed. Therefore, this vehicle is not insured.
Item 6
If the sample is contaminated, then the results are unreliable. The sample is contaminated. Therefore, the results are unreliable.
Proof Draft
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Evaluation Practice
DeductivePeer Review: The Square of Opposition and Existential Import
Below are sample student responses to a logic exercise. Evaluate each response: Is the formalization correct? Is the proof valid? Identify specific errors and suggest corrections.
Evaluate student proofs
Each student attempted to prove a conclusion from given premises. Find and correct any mistakes.
Student A's work
Premises: P -> Q, Q -> R, P. Student wrote: (1) P [premise], (2) P -> Q [premise], (3) Q [MP 1,2], (4) Q -> R [premise], (5) R [MP 3,4]. Conclusion: R. Student says: 'Valid proof by two applications of Modus Ponens.'
Student B's work
Premises: A v B, A -> C. Student wrote: (1) A v B [premise], (2) A -> C [premise], (3) A [from 1], (4) C [MP 2,3]. Conclusion: C. Student says: 'Since A or B is true, A must be true, so C follows.'
Student C's work
Premises: ~P v Q, P. Student wrote: (1) ~P v Q [premise], (2) P [premise], (3) ~~P [DN 2], (4) Q [DS 1,3]. Conclusion: Q. Student says: 'I used double negation then disjunctive syllogism.'
Student D's work
Premises: (P & Q) -> R, P, Q. Student wrote: (1) P [premise], (2) Q [premise], (3) (P & Q) -> R [premise], (4) R [MP 1,3]. Conclusion: R. Student says: 'Modus Ponens with P and the conditional.'
Proof Draft
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Proof Construction
DeductiveConstruction Challenge: The Square of Opposition and Existential Import
Build complete proofs or arguments from scratch. You are given only a conclusion and some constraints. Construct valid premises and a rigorous derivation.
Build your own proofs
For each task, create a valid argument with explicit premises and step-by-step derivation.
Task 1
Construct a valid argument with exactly three premises that concludes: 'The network is secure.' Use at least one conditional and one disjunction in your premises.
Task 2
Build a valid syllogistic argument that concludes: 'Some scientists are not wealthy.' Your premises must be universal statements (All X are Y or No X are Y).
Task 3
Create a proof using reductio ad absurdum (indirect proof) that derives ~(P & ~P) from no premises. Show every step and justify each with a rule name.
Task 4
Construct a chain of conditional reasoning with at least four steps that connects 'The satellite detects an anomaly' to 'Emergency protocols are activated.' Make each link realistic and name the domain.
Proof Draft
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Diagnosis Practice
DeductiveCounterexample Challenge: The Square of Opposition and Existential Import
For each invalid argument below, construct a clear counterexample -- a scenario where all premises are true but the conclusion is false. Then explain which logical error the argument commits.
Find counterexamples to invalid arguments
Each argument appears plausible but is invalid. Prove invalidity by constructing a specific counterexample.
Argument 1
If a student studies hard, they pass the exam. Maria passed the exam. Therefore, Maria studied hard.
Argument 2
All roses are flowers. Some flowers are red. Therefore, some roses are red.
Argument 3
No fish can fly. No birds are fish. Therefore, all birds can fly.
Argument 4
If the alarm sounds, there is a fire. The alarm did not sound. Therefore, there is no fire.
Argument 5
All effective medicines have been tested. This substance has been tested. Therefore, this substance is an effective medicine.
Proof Draft
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Analysis Practice
DeductiveIntegration Exercise: The Square of Opposition and Existential Import
These exercises combine deductive logic with other topics and reasoning styles. Apply formal logic alongside empirical evaluation, explanation assessment, or problem-solving frameworks.
Cross-topic deductive exercises
Each scenario requires deductive reasoning combined with at least one other skill area.
Scenario 1
A quality control team uses this rule: 'If a batch fails two consecutive tests, it must be discarded.' Batch 47 failed Test A and passed Test B, then failed Test C. Formally determine whether the rule requires discarding Batch 47, and discuss whether the rule itself is well-designed from a problem-solving perspective.
Scenario 2
A researcher argues: 'All peer-reviewed studies in this meta-analysis show that X reduces Y. This study shows X reduces Y. Therefore, this study will be included in the meta-analysis.' Evaluate the deductive form, then inductively assess whether the meta-analysis conclusion would be strong.
Scenario 3
An insurance policy states: 'Coverage applies if and only if the damage was caused by a covered peril AND the policyholder reported it within 72 hours.' A policyholder reported water damage after 80 hours, claiming the damage was not discoverable sooner. Apply the formal logic of the policy, then consider whether the best explanation supports an exception.
Scenario 4
A hiring algorithm uses: 'If GPA >= 3.5 AND experience >= 2 years, then advance to interview.' Candidate X has GPA 3.8 and 18 months experience. Formally determine the outcome. Then evaluate: is the algorithm's rule inductively justified? What evidence would you want?
Proof Draft
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Diagnosis Practice
DeductiveMisconception Clinic: The Square of Opposition and Existential Import
Each item presents a common misconception about deductive logic. Identify the misconception, explain why it is wrong, and provide a correct version of the reasoning.
Common deductive misconceptions
Diagnose and correct each misconception. Explain the error clearly enough for a fellow student to understand.
Misconception 1
A student claims: 'An argument is valid if its conclusion is true. Since the conclusion "Water is H2O" is obviously true, any argument concluding this must be valid.'
Misconception 2
A student says: 'Modus Tollens and denying the antecedent are the same thing. Both involve negation and a conditional, so they must work the same way.'
Misconception 3
A student writes: 'This argument is invalid because the conclusion is false: All cats are reptiles. All reptiles lay eggs. Therefore, all cats lay eggs.'
Misconception 4
A student argues: 'A sound argument can have a false conclusion, because soundness just means the argument uses correct logical rules.'
Misconception 5
A student claims: 'Since P -> Q is equivalent to ~P v Q, we can derive Q from P -> Q alone, without knowing whether P is true.'
Proof Draft
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Proof Construction
DeductiveScaffolded Proof: The Square of Opposition and Existential Import
Build proofs in stages. Each task gives you a partially completed derivation. Fill in the missing steps, justify each one, and then extend the proof to a further conclusion.
Step-by-step proof building
Complete each partial proof, then extend it. Every step must cite a rule.
Scaffold 1
Premises: (1) (A v B) -> C, (2) D -> A, (3) D. Partial proof: (4) A [MP 2,3]. Your tasks: (a) Complete the proof to derive C. (b) If we add premise (5) C -> E, extend the proof to derive E.
Scaffold 2
Premises: (1) P -> (Q -> R), (2) P, (3) Q. Partial proof: (4) Q -> R [MP 1,2]. Your tasks: (a) Complete the proof to derive R. (b) If we add premise (5) R -> ~S, extend to derive ~S. (c) If we also add (6) S v T, what can you derive?
Scaffold 3
Premises: (1) ~(A & B), (2) A. Your task: Prove ~B step by step. Hint: You may need to use an assumption for indirect proof. Show the subproof structure clearly.
Scaffold 4
Premises: (1) P v Q, (2) P -> R, (3) Q -> S, (4) ~R. Your tasks: (a) Derive ~P from (2) and (4). (b) Using (a), derive Q from (1). (c) Using (b), derive S. (d) Name each rule used.
Proof Draft
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Analysis Practice
DeductiveSynthesis Review: The Square of Opposition and Existential Import
These exercises require you to combine everything you have learned about deductive reasoning. Each scenario tests multiple skills simultaneously: formalization, rule application, validity checking, and proof construction.
Comprehensive deductive review
Each task combines multiple deductive skills. Show all your work.
Comprehensive 1
A software license agreement states: 'The software may be used commercially if and only if the licensee has purchased an enterprise plan and has fewer than 500 employees, or has received written exemption from the vendor.' Formalize this using propositional logic, determine what follows if a company has an enterprise plan and 600 employees with no exemption, and identify any ambiguity in the original text.
Comprehensive 2
Construct a valid argument with four premises and one conclusion about data privacy. Then create an invalid argument about the same topic that looks similar but commits a formal fallacy. Finally, prove the first is valid and show a counterexample for the second.
Proof Draft
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Venn Diagram Builder
Define your sets, then place items into the correct regions to visualize categorical relationships.
Diagram type
Set names
Add an item to a region
2 sets · 0 items placed
Animated Explainers
Step-by-step visual walkthroughs of key concepts. Click to start.
Read the explanation carefully before jumping to activities!
Further Support
Open these only if you need extra help or context
Mistakes to avoid before submitting
- Do not mix traditional and modern reasoning within a single analysis without saying so.
- Do not confuse contradictories with contraries; the contradictory is the exact denial.
Where students usually go wrong
Using subalternation without checking whether the subject class is nonempty.
Confusing contraries with contradictories and overshooting the denial.
Treating subcontraries as if they could not both be true.
Assuming traditional and modern readings agree in all cases.
Historical context for this way of reasoning
Medieval Logicians
Medieval commentators on Aristotle developed the square of opposition in its classic form, adding the terminology of contrariety, subcontrariety, and subalternation and embedding it in the Latin logic textbooks that shaped European education for centuries.