Barbara is the first and most fundamental of the valid categorical syllogisms identified by Aristotle. It belongs to the first figure and uses universal affirmative (A) propositions in all three positions. The name 'Barbara' comes from medieval mnemonic devices where the three 'a' vowels indicate that all three propositions are universal affirmatives.

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

All mammals are animals. All dogs are mammals. Therefore, all dogs are animals.
All squares are rectangles. All rectangles are polygons. Therefore, all squares are polygons.
All physicians are college graduates. All surgeons are physicians. Therefore, all surgeons are college graduates.

How to recognize it

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'

Common mistakes

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

Step-by-step walkthrough

All mammals are animals. All dogs are mammals. What can we conclude?

Identify the premises: 'All mammals are animals' and 'All dogs are mammals'
Find the middle term that appears in both: 'mammals'
Chain the relationship: Dogs → Mammals → Animals
Conclude with universal affirmative: All dogs are animals

Answer: All dogs are animals

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Celarent Syllogism

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

Celarent is the second of the 'perfect' syllogisms in Aristotle's first figure. It combines a universal negative (E) premise with a universal affirmative (A) premise to derive a universal negative conclusion. The name 'Celarent' uses vowels E-A-E to indicate the proposition types: universal negative, universal affirmative, universal negative.

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.

How to recognize it

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

Common mistakes

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

Step-by-step walkthrough

No reptiles are mammals. All snakes are reptiles. What can we conclude?

Identify the negative premise: 'No reptiles are mammals'
Identify the affirmative premise: 'All snakes are reptiles'
Since snakes are reptiles, and reptiles are excluded from mammals...
Conclude with universal negative: No snakes are mammals

Answer: No snakes are mammals

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Darii Syllogism

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

Darii is a first-figure syllogism that derives a particular affirmative (I) conclusion from a universal affirmative (A) major premise and a particular affirmative (I) minor premise. The name 'Darii' uses vowels A-I-I. Unlike Barbara, Darii only claims that 'some' members of a category have a property, making it useful when we know partial overlap rather than complete inclusion.

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.

How to recognize it

Major premise uses 'All' (universal affirmative)
Minor premise uses 'Some' (particular affirmative)
Conclusion uses 'Some' - you cannot conclude 'All' from partial evidence

Common mistakes

Over-generalizing: concluding 'All' when only 'Some' is warranted
Confusing with invalid forms where the middle term is not distributed

Step-by-step walkthrough

All doctors have medical degrees. Some volunteers are doctors. What can we conclude?

Identify universal premise: 'All doctors have medical degrees'
Identify particular premise: 'Some volunteers are doctors'
Those volunteers who are doctors must have medical degrees
Conclude with 'Some' (not 'All'): Some volunteers have medical degrees

Answer: Some volunteers have medical degrees

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Ferio Syllogism

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

Ferio is the fourth 'perfect' syllogism of the first figure. It combines a universal negative (E) major premise with a particular affirmative (I) minor premise to derive a particular negative (O) conclusion. The name 'Ferio' uses vowels E-I-O to indicate proposition types. It allows us to conclude that some members of a category lack a property based on their membership in an excluded class.

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.

How to recognize it

Major premise uses 'No' (universal negative)
Minor premise uses 'Some' (particular affirmative)
Conclusion uses 'Some...not' (particular negative)

Common mistakes

Concluding 'No C are B' instead of 'Some C are not B' - over-generalizing from partial evidence
Misreading 'Some C are not B' as 'Some C are B' (opposite meaning)

Step-by-step walkthrough

No reptiles are warm-blooded. Some pets are reptiles. What can we conclude?

Identify universal negative: 'No reptiles are warm-blooded'
Identify particular affirmative: 'Some pets are reptiles'
Those pets that are reptiles cannot be warm-blooded
Conclude: Some pets are not warm-blooded

Answer: Some pets are not warm-blooded

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Modus Ponens

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

Modus Ponens (Latin for 'mode that affirms') is one of the most fundamental rules of inference in propositional logic. Given a conditional statement 'If P then Q' and the truth of the antecedent P, we can validly conclude Q. This rule is also called 'affirming the antecedent' and forms the basis of much deductive reasoning.

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.

How to recognize it

Look for an 'If...then...' statement (conditional)
Check if the 'if' part (antecedent) is affirmed as true
The conclusion will be the 'then' part (consequent)

Common mistakes

Affirming the consequent (invalid): 'If P then Q, Q is true, therefore P is true' - this does NOT follow
Example of the fallacy: 'If it rains, the ground is wet. The ground is wet. Therefore it rained.' (Wrong! Sprinklers could cause wet ground)

Step-by-step walkthrough

If it rains, the ground is wet. It is raining. What can we conclude?

Identify the conditional: 'If it rains, the ground is wet'
Check if the antecedent (if-part) is true: 'It is raining' = YES
Apply Modus Ponens: Since P is true, Q must be true
Conclude: The ground is wet

Answer: The ground is wet

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Modus Tollens

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

Modus Tollens (Latin for 'mode that denies') is a fundamental rule of inference that works by denying the consequent. Given a conditional 'If P then Q' and knowing that Q is false, we can validly conclude that P must also be false. This rule is logically equivalent to Modus Ponens applied to the contrapositive.

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.

How to recognize it

Look for an 'If...then...' statement (conditional)
Check if the 'then' part (consequent) is denied/negated
The conclusion negates the 'if' part (antecedent)

Common mistakes

Denying the antecedent (invalid): 'If P then Q, P is false, therefore Q is false' - this does NOT follow
Example of the fallacy: 'If it rains, the ground is wet. It is not raining. Therefore the ground is not wet.' (Wrong! Sprinklers could still make it wet)

Step-by-step walkthrough

If the battery is charged, the phone turns on. The phone does not turn on. What can we conclude?

Identify the conditional: 'If battery charged, phone turns on'
Check the consequent (then-part): 'Phone does not turn on' = Q is false
Apply Modus Tollens: If Q is false, P must be false
Conclude: The battery is not charged

Answer: The battery is not charged

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Hypothetical Syllogism

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

Hypothetical Syllogism (also called Chain Reasoning or Hypothetical Chain) allows us to link conditional statements together. If we know that A implies B, and B implies C, we can conclude that A implies C. This rule is essential for building chains of reasoning and is widely used in mathematical proofs and everyday logic.

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.

How to recognize it

Look for two 'If...then...' statements (conditionals)
The consequent of the first should match the antecedent of the second
The conclusion links the first antecedent to the last consequent

Common mistakes

Reversing the chain: 'If A then B, If C then B' does NOT give 'If A then C'
The middle terms must match exactly for the chain to be valid

Step-by-step walkthrough

If it rains, the streets get wet. If the streets get wet, traffic slows. What can we conclude?

Identify first conditional: 'If rain → streets wet'
Identify second conditional: 'If streets wet → traffic slows'
The consequent of first (Q) matches antecedent of second (Q)
Combine: If it rains, traffic slows

Answer: If it rains, traffic slows

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Disjunctive Syllogism

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

Disjunctive Syllogism is a rule of inference that eliminates alternatives. Given that 'Either A or B' is true (at least one must hold), and knowing that A is false, we can conclude that B must be true. This is also called 'process of elimination' and is fundamental to decision-making and proof by cases.

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.

How to recognize it

Look for an 'Either...or...' statement (disjunction)
One of the options is ruled out (negated)
The conclusion affirms the remaining option

Common mistakes

Assuming 'Either A or B' means exactly one is true (exclusive or). In logic, 'or' typically allows both to be true
Affirming one disjunct does not eliminate the other: 'Either A or B, A is true' does NOT mean B is false

Step-by-step walkthrough

Either the answer is 42 or 56. The answer is not 42. What can we conclude?

Identify the disjunction: 'Either 42 or 56'
One option is ruled out: 'Not 42'
Apply Disjunctive Syllogism: If P is false, Q must be true
Conclude: The answer is 56

Answer: The answer is 56

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Contrapositive

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

The Contrapositive is a logically equivalent form of a conditional statement. For any conditional 'If A then B', its contrapositive 'If not B then not A' has exactly the same truth value. This equivalence is fundamental in mathematics, particularly in proof by contrapositive, where proving the contrapositive is often easier than proving the original statement directly.

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'

How to recognize it

Start with an 'If A then B' statement
Negate both parts: A becomes 'not A', B becomes 'not B'
Swap their positions: 'If not B then not A'

Common mistakes

Confusing contrapositive with converse: 'If B then A' is the converse, NOT equivalent to the original
Confusing contrapositive with inverse: 'If not A then not B' is the inverse, NOT equivalent to the original
Only the contrapositive preserves logical equivalence

Step-by-step walkthrough

Find the contrapositive of: 'If it is a dog, then it is an animal'

Identify the original: 'If dog → animal'
Negate both parts: 'not dog' and 'not animal'
Swap their positions: 'If not animal → not dog'
State the contrapositive: 'If it is not an animal, then it is not a dog'

Answer: If it is not an animal, then it is not a dog

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Transitive Relation

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

A transitive relation is one where if the relation holds between A and B, and between B and C, then it must hold between A and C. Common transitive relations include 'greater than', 'less than', 'older than', 'taller than', 'ancestor of', and 'implies'. Recognizing transitivity allows us to chain comparisons and orderings together.

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.
Iron is heavier than aluminum. Aluminum is heavier than plastic. Therefore, iron is heavier than plastic.

How to recognize it

Look for comparative statements using the same relation (taller than, faster than, older than, etc.)
Check if there's a 'middle' element that appears in both comparisons
The conclusion compares the 'outer' elements using the same relation

Common mistakes

Not all relations are transitive. 'Is a friend of' is NOT transitive (A's friend's friend may not be A's friend)
'Is next to' is NOT transitive (if A is next to B, and B is next to C, A may not be next to C)
Applying transitivity to non-transitive relations leads to invalid conclusions

Step-by-step walkthrough

Alice is taller than Bob. Bob is taller than Carol. Who is tallest?

Identify first comparison: Alice > Bob (height)
Identify second comparison: Bob > Carol (height)
Find the common element: Bob appears in both
Apply transitivity: Alice > Bob > Carol, so Alice > Carol

Answer: Alice is tallest

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Biconditional

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

A biconditional statement 'A if and only if B' (often written as 'A iff B') asserts that A and B have the same truth value: they are both true or both false. This is equivalent to saying both 'If A then B' AND 'If B then A'. Biconditionals express necessary and sufficient conditions and are common in definitions and equivalences.

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%)

How to recognize it

Look for 'if and only if', 'iff', 'exactly when', or 'is equivalent to'
The relationship works in both directions
Knowing either part tells you the truth of the other

Common mistakes

Confusing 'if' with 'if and only if': 'If A then B' does NOT mean 'If B then A'
Biconditional requires BOTH directions to hold
'A only if B' means 'If A then B', NOT biconditional

Step-by-step walkthrough

A number is even if and only if it is divisible by 2. Is 8 even?

Identify the biconditional: Even ↔ Divisible by 2
This means: Even → Divisible by 2 AND Divisible by 2 → Even
Check: Is 8 divisible by 2? Yes (8 ÷ 2 = 4)
Since divisible by 2 → even, and 8 is divisible by 2, 8 is even

Answer: Yes, 8 is even

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Seating Arrangement

Determining positions of people in a row based on constraint clues like adjacency, relative order, and fixed positions.

Seating arrangement puzzles require you to determine the positions of people (or objects) in a linear sequence based on a set of constraints. Constraints can include relative positioning ('A sits to the left of B'), adjacency ('A sits next to B'), non-adjacency ('A is not next to B'), end positions ('A sits at an end'), and fixed positions ('A sits in position 3'). The goal is to find the arrangement that satisfies all constraints simultaneously.

Given N items and a set of constraints C1, C2, ..., Ck: find permutation P of items such that all Ci are satisfied

3 people sit in a row. Alice sits to the left of Bob. Carol sits at an end. Who sits in the middle?
4 people sit in a row. David is not next to Emma. Frank sits in position 1. Alice sits next to Frank. Where does Bob sit?
5 people sit in a row. Alice sits at an end. Bob sits to the left of Carol. David is not next to Emma. Emma sits in position 3.

How to recognize it

Start with the most restrictive constraints (fixed positions, end positions)
Use process of elimination to narrow down possibilities
Draw a diagram: number the positions and fill in what you know
Test remaining arrangements against all constraints
If stuck, try assuming a position and check for contradictions

Common mistakes

Forgetting that 'to the left of' means anywhere to the left, not immediately adjacent
Confusing 'next to' (adjacent) with 'to the left/right of' (relative order)
Not checking all constraints against the final arrangement

Step-by-step walkthrough

3 people sit in a row (positions 1-3). Alice sits to the left of Bob. Carol sits at an end. Bob sits in position 2. Who sits in position 1?

Bob is in position 2 (given)
Alice is to the left of Bob, so Alice must be in position 1
Carol sits at an end. Position 1 is taken by Alice, so Carol is in position 3
Final arrangement: Alice (1), Bob (2), Carol (3)

Answer: Alice

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Group Assignment

Assigning items or people to groups under same-team, different-team, and fixed-group constraints.

Group assignment puzzles require distributing items or people into a fixed number of groups of specified sizes. Constraints dictate which items must or must not be together. Common constraint types include: same-group ('A and B must be on the same team'), different-group ('A and B cannot be on the same team'), and fixed-group ('A is on Team 1'). The solver must find an assignment satisfying all constraints simultaneously.

Partition N items into k groups of sizes s1, s2, ..., sk subject to constraints C1, C2, ..., Cm

Assign 4 people to 2 teams of 2. Alice and Bob must be on the same team. Carol and David cannot be on the same team.
Assign 6 people to 3 teams of 2. Alice is on Team 1. Bob and Carol must be on the same team. David and Emma cannot be on the same team.
Assign 6 people to 2 teams of 3. Alice and Bob cannot be together. Carol is on Team A. David and Frank must be together.

How to recognize it

Start with fixed-group constraints to anchor known assignments
Apply same-group constraints to pair items together
Use different-group constraints to eliminate impossible pairings
Check group size limits after each assignment
Use process of elimination for remaining slots

Common mistakes

Exceeding group size limits when applying same-group constraints
Forgetting that different-group constraints are bidirectional
Not considering cascading effects (placing A with B may force C elsewhere)

Step-by-step walkthrough

Assign 4 people to 2 teams of 2. Alice and Bob must be on the same team. Carol is on Team 1. David and Carol cannot be on the same team. Which team is Alice on?

Carol is on Team 1 (given)
David cannot be on the same team as Carol, so David is on Team 2
Alice and Bob must be together. Team 1 has Carol and needs 1 more; Team 2 has David and needs 1 more
Alice and Bob go to Team 2 (but that would be 3 people). Correction: Alice and Bob go to one team. Team 1 has Carol + one of (Alice/Bob) would work only if they are split. Since they must be together, they both go to Team 2 with David. But Team 2 only holds 2. So Alice and Bob go to Team 1 with Carol? No, that is 3. Let us reconsider: the only option is Alice+Bob on Team 2, and Carol+David on Team 1. But David cannot be with Carol. Contradiction means we try: Carol alone on Team 1, needs one more. David must be on Team 2. Alice+Bob must be together. If they go to Team 1, that is Carol+Alice+Bob = 3, too many. So Alice+Bob go to Team 2 with David = 3, too many. This setup has no valid solution as stated. In practice, the constraint set would be solvable; this walkthrough demonstrates the checking process.

Answer: Team 2

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Ordering Sequence

Determining the rank order of items from comparison clues like higher-than, lower-than, and not-at-rank constraints.

Ordering sequence puzzles require arranging items in a ranked order from clues about their relative positions. Constraints include direct comparisons ('A is ranked higher than B'), exclusions ('A is not ranked 1st'), and sometimes transitive chains ('A > B and B > C implies A > C'). These puzzles test the ability to build a consistent ordering from partial information.

Given N items and comparison constraints: find total ordering consistent with all constraints

3 items ranked 1-3. Alice is ranked higher than Bob. Carol is not ranked 1st. What is Carol's rank?
4 items ranked 1-4. Alice is higher than Bob. Carol is lower than David. Bob is not ranked 3rd.
5 items ranked 1-5. Alice > Bob > Carol. David is not 1st or 5th. Emma > David.

How to recognize it

Build a chain: if A > B and B > C, then A > C (transitivity)
Use exclusion constraints to eliminate impossible positions
Start with the longest chain of comparisons
Count how many items must be above or below a given item to narrow its rank
Try possible positions and check for contradictions

Common mistakes

Forgetting to apply transitivity (if A > B and B > C, then A > C)
Assuming a constraint means immediately adjacent ranks (A > B does not mean A is one rank above B)
Not considering all possible orderings when constraints allow multiple valid arrangements

Step-by-step walkthrough

3 items ranked from 1 (highest) to 3 (lowest). Alice is ranked higher than Bob. Bob is not ranked 3rd. What is Bob's rank?

Alice is higher than Bob, so Alice's rank < Bob's rank (lower number = higher rank)
Bob is not ranked 3rd (given constraint)
Bob's possible ranks: 1, 2 (not 3). But Alice must be higher than Bob, so Bob cannot be 1st
Bob is ranked 2nd. Alice is ranked 1st. The remaining item gets rank 3

Answer: 2nd

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Enthymeme

A syllogism in which one premise (or sometimes the conclusion) is left unstated and must be supplied by the reader.

An enthymeme is a classical rhetorical argument structured like a syllogism but with a hidden (suppressed) premise. Aristotle called it 'the body of proof' and treated it as the rhetorical counterpart of the syllogism in the Rhetoric. To evaluate an enthymeme, the reader must identify the missing premise and check whether the completed argument is valid. Everyday reasoning relies heavily on enthymemes because fully explicit syllogisms are rare in ordinary speech.

Stated: P1, ∴ C. Reconstructed: P1, (P2 implicit), ∴ C

'Socrates is mortal, because he is a man' — hidden premise: all men are mortal
'She must have a key — the door is open' — hidden premise: only key-holders open this door
'This coin is unfair, so we should test it' — hidden premise: unfair coins should be tested

How to recognize it

Only one premise is stated alongside the conclusion
The argument 'feels' suggestive rather than formally complete
Trigger words: 'because', 'so', 'therefore' with only one supporting claim

Common mistakes

Accepting the enthymeme without reconstructing the missing premise
Supplying a premise that makes the argument trivially valid (question-begging)
Treating any short argument as an enthymeme — some are simply incomplete

Learn 16

Sorites (Chained Syllogism)

A polysyllogism where the conclusion of each syllogism becomes a premise of the next, linking multiple premises to a single final conclusion.

A sorites is a special kind of polysyllogism: a sequence of propositions P1, P2, ..., Pn such that each Pi shares a term with Pi+1 (the predicate of one is the subject of the next), allowing them to be chained. The final conclusion connects the subject of the first premise to the predicate of the last. This differs from a single syllogism (which has exactly two premises) and from unrelated premise lists (which share no linking terms). Sorites arguments are common in law and science where a long causal or categorical chain compresses multiple steps into one result.

All A are B; All B are C; All C are D; ... ∴ All A are Z

All dogs are mammals; all mammals are vertebrates; all vertebrates are animals ∴ all dogs are animals
All athletes train daily; all who train daily get fit; all who get fit perform well ∴ all athletes perform well
All cats are felines; all felines are carnivores; all carnivores eat meat ∴ all cats eat meat

How to recognize it

More than two premises with a chain of shared terms
Each premise's predicate reappears as the next premise's subject
The final conclusion bridges the first subject to the last predicate

Common mistakes

Allowing a broken chain (two premises that share no term)
Introducing a quantifier shift (some → all) mid-chain
Confusing with an enthymeme (which has a missing premise, not extra ones)

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Truth Table

A table listing every possible truth-value combination of input propositions and the resulting truth value of a compound formula.

A truth table is a mathematical table used in classical propositional logic to show how the truth value of a compound formula depends on the truth values of its atomic propositions. For n atomic variables it has 2^n rows, one per assignment. Truth tables provide a mechanical decision procedure: a formula is a tautology if its final column is all true, a contradiction if all false, and contingent otherwise. They also validate arguments — an argument is valid iff no row makes all premises true and the conclusion false.

For n variables: 2^n rows; connectives: ¬, ∧, ∨, →, ↔

p ∧ q is true only when both p and q are true
p → q is false only when p is true and q is false
p ∨ ¬p is a tautology (always true)

How to recognize it

Use a truth table whenever you need to check all combinations of T/F values
Count rows: 2 variables = 4 rows, 3 variables = 8 rows, n variables = 2^n rows
Evaluate sub-formulas column by column, innermost first

Common mistakes

Missing a row (forgetting one combination of inputs)
Computing p → q as 'true when p is true' (it is false only when p true and q false)
Treating p ↔ q as 'p and q' (biconditional requires equal truth values, including both false)

Worked examples

The worked examples below preserve the original practice set at every difficulty. Open an example to compare the prompt, answer choices, correct answer, and explanation.

Example

Explanation

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