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Logical Deduction

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Logical deduction tests measure your ability to draw valid conclusions from given statements. You'll be presented with premises (facts assumed to be true) and must determine what can be logically concluded.

These tests assess critical thinking, the ability to identify valid arguments, and skill in avoiding logical fallacies—essential for legal, analytical, and strategic roles. Our procedurally generated syllogisms use a rich taxonomy of categories and relationships, ensuring you'll never see the same question twice and providing unlimited unique practice.

All questions are generated locally in your browser. Your progress and answers are stored only on your device — we never collect any personal data.

Key Patterns & Formats

Barbara Syllogism

The most fundamental valid syllogism: All A are B, All B are C, therefore All A are C.

All M are P, All S are M ⊢ All S are P

Celarent Syllogism

A valid syllogism with negative conclusion: No A are B, All C are A, therefore No C are B.

No M are P, All S are M ⊢ No S are P

Darii Syllogism

A valid syllogism with particular conclusion: All A are B, Some C are A, therefore Some C are B.

All M are P, Some S are M ⊢ Some S are P

Ferio Syllogism

A valid syllogism: No A are B, Some C are A, therefore Some C are not B.

No M are P, Some S are M ⊢ Some S are not P

Modus Ponens

The 'affirming mode': If P then Q, P is true, therefore Q is true.

P → Q, P ⊢ Q

Modus Tollens

The 'denying mode': If P then Q, Q is false, therefore P is false.

P → Q, ¬Q ⊢ ¬P

Hypothetical Syllogism

Chain reasoning: If A then B, If B then C, therefore If A then C.

P → Q, Q → R ⊢ P → R

Disjunctive Syllogism

Either/Or reasoning: Either A or B, not A, therefore B.

P ∨ Q, ¬P ⊢ Q

Contrapositive

Logical equivalence: 'If A then B' is equivalent to 'If not B then not A'.

P → Q ≡ ¬Q → ¬P

Transitive Relation

If A > B and B > C, then A > C. Applies to ordering relations.

R(a,b) ∧ R(b,c) → R(a,c)

Biconditional

'A if and only if B' means A and B are either both true or both false.

P ↔ Q ≡ (P → Q) ∧ (Q → P)

Solving Strategies

  • Look for two premises that both use 'All' (universal affirmatives)
  • The middle term appears as subject in one premise and predicate in another
  • The conclusion links the two end terms with 'All'
  • One premise uses 'No' (universal negative) and one uses 'All' (universal affirmative)
  • The conclusion is always 'No' (universal negative)
  • The middle term connects the chain through negation
  • Major premise uses 'All' (universal affirmative)
  • Minor premise uses 'Some' (particular affirmative)

Common Pitfalls

  • Confusing Barbara with invalid forms like 'All A are B, All C are B, therefore All A are C' (undistributed middle)
  • Reversing the chain direction
  • Confusing the direction: 'No A are B' is not the same as 'No B are A' in context
  • Missing that both terms in the conclusion inherit the negative relationship
  • Over-generalizing: concluding 'All' when only 'Some' is warranted
  • Confusing with invalid forms where the middle term is not distributed

Frequently Asked Questions

A syllogism is a logical argument with two premises and a conclusion. Classic example: 'All men are mortal. Socrates is a man. Therefore, Socrates is mortal.' The conclusion must follow necessarily from the premises.
A valid argument means the conclusion follows logically from the premises—if the premises are true, the conclusion must be true. But the premises themselves might be false. 'All fish can fly. Salmon are fish. Therefore salmon can fly' is valid but not true.
Learn common fallacies to avoid them. Watch for words like 'all', 'some', 'no', 'if...then'. Remember: you can only conclude what MUST be true, not what MIGHT be true. Practice diagramming premises with Venn diagrams.
Each syllogism is procedurally generated using a rich taxonomy of nearly 1,000 category chains. The system selects terms, applies valid logical forms (Barbara, Celarent, Darii, and others), and generates distractors that contain specific logical fallacies. Root overlap filtering ensures premises always sound natural. This provides unlimited unique practice questions.

Sources & References

Barbara Syllogism

Barbara Syllogism
All mammals are animals. All dogs are mammals. What can we conclude?
  1. 1
    Identify the premises: 'All mammals are animals' and 'All dogs are mammals'Two premises
  2. 2
    Find the middle term that appears in both: 'mammals'Middle term: mammals
  3. 3
    Chain the relationship: Dogs → Mammals → AnimalsDogs → Animals
  4. 4
    Conclude with universal affirmative: All dogs are animalsAll S are P
Refs:Aristotle, Prior Analytics, Book I, Chapter 4Smith, Robin. 'Aristotle's Logic', Stanford Encyclopedia of Philosophy

Celarent Syllogism

Celarent Syllogism
No M are P, All S are M ⊢ No S are P
No reptiles are mammals. All snakes are reptiles. Therefore, no snakes are mammals.
No even numbers are odd. All multiples of 4 are even. Therefore, no multiples of 4 are odd.
No fish can breathe air. All salmon are fish. Therefore, no salmon can breathe air.
No reptiles are mammals. All snakes are reptiles. What can we conclude?
  1. 1
    Identify the negative premise: 'No reptiles are mammals'No M are P
  2. 2
    Identify the affirmative premise: 'All snakes are reptiles'All S are M
  3. 3
    Since snakes are reptiles, and reptiles are excluded from mammals...Chain through middle term
  4. 4
    Conclude with universal negative: No snakes are mammalsNo S are P
Refs:Aristotle, Prior Analytics, Book I, Chapter 4Smith, Robin. 'Aristotle's Logic', Stanford Encyclopedia of Philosophy

Darii Syllogism

Darii Syllogism
All M are P, Some S are M ⊢ Some S are P
All birds can fly. Some pets are birds. Therefore, some pets can fly.
All prime numbers greater than 2 are odd. Some integers are prime numbers greater than 2. Therefore, some integers are odd.
All doctors have medical degrees. Some volunteers are doctors. Therefore, some volunteers have medical degrees.
All doctors have medical degrees. Some volunteers are doctors. What can we conclude?
  1. 1
    Identify universal premise: 'All doctors have medical degrees'All M are P
  2. 2
    Identify particular premise: 'Some volunteers are doctors'Some S are M
  3. 3
    Those volunteers who are doctors must have medical degreesPartial overlap
  4. 4
    Conclude with 'Some' (not 'All'): Some volunteers have medical degreesSome S are P
Refs:Aristotle, Prior Analytics, Book I, Chapter 4Smith, Robin. 'Aristotle's Logic', Stanford Encyclopedia of Philosophy

Ferio Syllogism

Ferio Syllogism
No M are P, Some S are M ⊢ Some S are not P
No reptiles are warm-blooded. Some pets are reptiles. Therefore, some pets are not warm-blooded.
No vegetables contain cholesterol. Some foods I eat are vegetables. Therefore, some foods I eat do not contain cholesterol.
No beginners pass the advanced test. Some students are beginners. Therefore, some students do not pass the advanced test.
No reptiles are warm-blooded. Some pets are reptiles. What can we conclude?
  1. 1
    Identify universal negative: 'No reptiles are warm-blooded'No M are P
  2. 2
    Identify particular affirmative: 'Some pets are reptiles'Some S are M
  3. 3
    Those pets that are reptiles cannot be warm-bloodedPartial exclusion
  4. 4
    Conclude: Some pets are not warm-bloodedSome S are not P
Refs:Aristotle, Prior Analytics, Book I, Chapter 4Smith, Robin. 'Aristotle's Logic', Stanford Encyclopedia of Philosophy

Modus Ponens

Modus Ponens
P → Q, P ⊢ Q
If it is raining, then the ground is wet. It is raining. Therefore, the ground is wet.
If you study hard, you will pass the exam. You study hard. Therefore, you will pass the exam.
If a number is divisible by 4, it is divisible by 2. 16 is divisible by 4. Therefore, 16 is divisible by 2.
If it rains, the ground is wet. It is raining. What can we conclude?
  1. 1
    Identify the conditional: 'If it rains, the ground is wet'P → Q
  2. 2
    Check if the antecedent (if-part) is true: 'It is raining' = YESP is true
  3. 3
    Apply Modus Ponens: Since P is true, Q must be trueP → Q, P ⊢ Q
  4. 4
    Conclude: The ground is wetQ is true
Refs:Klement, Kevin. 'Propositional Logic', Internet Encyclopedia of Philosophy

Modus Tollens

Modus Tollens
P → Q, ¬Q ⊢ ¬P
If it is raining, the ground is wet. The ground is not wet. Therefore, it is not raining.
If the battery is charged, the phone will turn on. The phone does not turn on. Therefore, the battery is not charged.
If a number is prime and greater than 2, it is odd. 8 is not odd. Therefore, 8 is not a prime number greater than 2.
If the battery is charged, the phone turns on. The phone does not turn on. What can we conclude?
  1. 1
    Identify the conditional: 'If battery charged, phone turns on'P → Q
  2. 2
    Check the consequent (then-part): 'Phone does not turn on' = Q is false¬Q
  3. 3
    Apply Modus Tollens: If Q is false, P must be falseP → Q, ¬Q ⊢ ¬P
  4. 4
    Conclude: The battery is not charged¬P
Refs:Klement, Kevin. 'Propositional Logic', Internet Encyclopedia of Philosophy

Hypothetical Syllogism

Hypothetical Syllogism
P → Q, Q → R ⊢ P → R
If it rains, the streets get wet. If the streets get wet, traffic slows down. Therefore, if it rains, traffic slows down.
If you exercise regularly, you build stamina. If you build stamina, you can run longer. Therefore, if you exercise regularly, you can run longer.
If a shape is a square, it is a rectangle. If a shape is a rectangle, it is a quadrilateral. Therefore, if a shape is a square, it is a quadrilateral.
If it rains, the streets get wet. If the streets get wet, traffic slows. What can we conclude?
  1. 1
    Identify first conditional: 'If rain → streets wet'P → Q
  2. 2
    Identify second conditional: 'If streets wet → traffic slows'Q → R
  3. 3
    The consequent of first (Q) matches antecedent of second (Q)Chain link: Q
  4. 4
    Combine: If it rains, traffic slowsP → R
Refs:Klement, Kevin. 'Propositional Logic', Internet Encyclopedia of Philosophy

Disjunctive Syllogism

Disjunctive Syllogism
P ∨ Q, ¬P ⊢ Q
Either the package was delivered or it was lost. It was not delivered. Therefore, it was lost.
Either the answer is 42 or 56. The answer is not 42. Therefore, the answer is 56.
Either John took the bus or he walked. John did not take the bus. Therefore, John walked.
Either the answer is 42 or 56. The answer is not 42. What can we conclude?
  1. 1
    Identify the disjunction: 'Either 42 or 56'P ∨ Q
  2. 2
    One option is ruled out: 'Not 42'¬P
  3. 3
    Apply Disjunctive Syllogism: If P is false, Q must be trueP ∨ Q, ¬P ⊢ Q
  4. 4
    Conclude: The answer is 56Q
Refs:Klement, Kevin. 'Propositional Logic', Internet Encyclopedia of Philosophy

Contrapositive

Contrapositive
P → Q ≡ ¬Q → ¬P
'If it is a dog, then it is an animal' is equivalent to 'If it is not an animal, then it is not a dog'
'If n² is even, then n is even' is equivalent to 'If n is odd, then n² is odd'
'If you pass the exam, you studied' is equivalent to 'If you did not study, you did not pass the exam'
Find the contrapositive of: 'If it is a dog, then it is an animal'
  1. 1
    Identify the original: 'If dog → animal'P → Q
  2. 2
    Negate both parts: 'not dog' and 'not animal'¬P and ¬Q
  3. 3
    Swap their positions: 'If not animal → not dog'¬Q → ¬P
  4. 4
    State the contrapositive: 'If it is not an animal, then it is not a dog'Equivalent statement
Refs:Klement, Kevin. 'Propositional Logic', Internet Encyclopedia of Philosophy

Transitive Relation

Transitive Relation
R(a,b) ∧ R(b,c) → R(a,c)
Alice is taller than Bob. Bob is taller than Carol. Therefore, Alice is taller than Carol.
5 > 3 and 3 > 1, therefore 5 > 1.
New York is larger than Chicago. Chicago is larger than Houston. Therefore, New York is larger than Houston.
Alice is taller than Bob. Bob is taller than Carol. Who is tallest?
  1. 1
    Identify first comparison: Alice > Bob (height)A > B
  2. 2
    Identify second comparison: Bob > Carol (height)B > C
  3. 3
    Find the common element: Bob appears in bothMiddle term: B
  4. 4
    Apply transitivity: Alice > Bob > Carol, so Alice > CarolA > C
Refs:Mendelson, Elliott. 'Mathematical Logic', in various logic textbooks

Biconditional

Biconditional
P ↔ Q ≡ (P → Q) ∧ (Q → P)
A triangle is equilateral if and only if all its angles are 60°. (Having equal sides is both necessary and sufficient for having 60° angles)
A number is even if and only if it is divisible by 2.
You will pass the course if and only if you score above 70%. (Scoring above 70% guarantees passing, and passing requires scoring above 70%)
A number is even if and only if it is divisible by 2. Is 8 even?
  1. 1
    Identify the biconditional: Even ↔ Divisible by 2P ↔ Q
  2. 2
    This means: Even → Divisible by 2 AND Divisible by 2 → EvenBoth directions
  3. 3
    Check: Is 8 divisible by 2? Yes (8 ÷ 2 = 4)Q is true
  4. 4
    Since divisible by 2 → even, and 8 is divisible by 2, 8 is evenP is true
Refs:Klement, Kevin. 'Propositional Logic', Internet Encyclopedia of Philosophy