If / Then — Conditional Logic Quiz
Master the logic of if/then conditionals (P → Q). 45 problems across three levels: Foundation covers the truth table for P→Q (vacuous truth, the one false case), Modus Ponens, Modus Tollens, and why Affirming the Consequent and Denying the Antecedent are invalid. Intermediate adds the contrapositive, converse, inverse, biconditional (P↔Q), the disjunction equivalence (¬P∨Q), and hypothetical syllogism. Advanced covers chained conditionals, negation of conditionals (¬(P→Q) = P∧¬Q), tautologies, complex truth value analysis, and edge cases.
How to Play If / Then
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Learn the Truth Table
The truth table in the lobby is the foundation of everything. P→Q is FALSE in exactly ONE case: when P is TRUE and Q is FALSE. In all other cases — including when P is FALSE — the conditional is TRUE. This 'vacuous truth' (a false antecedent makes the conditional true) is the most counterintuitive concept for beginners.
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Recognize the Inference Rules
Modus Ponens (P→Q, P, ∴Q) and Modus Tollens (P→Q, ¬Q, ∴¬P) are valid. Denying the Antecedent (P→Q, ¬P, ∴¬Q) and Affirming the Consequent (P→Q, Q, ∴P) are INVALID fallacies. The Foundation tier tests all four.
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Master the Equivalent Forms
The Intermediate tier tests four related forms: the CONTRAPOSITIVE (¬Q→¬P, equivalent to original), the CONVERSE (Q→P, NOT equivalent), the INVERSE (¬P→¬Q, NOT equivalent), and the BICONDITIONAL (P↔Q, true only when P and Q match). Also: P→Q ≡ ¬P∨Q is the disjunction equivalence.
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Handle Advanced Cases
Advanced questions explore: chained hypothetical syllogism (A→B→C→D gives A→D); what we can determine if P→Q is known to be FALSE (P=T, Q=F); tautologies like (P→Q)∨(Q→P); and the negation ¬(P→Q) = P∧¬Q. These cover content from discrete mathematics and formal logic courses.
Key Features
45 Problems Across 3 Levels
Foundation (15): The full truth table for P→Q with all four rows; identifying the one false case (P=T, Q=F); Modus Ponens (P→Q, P, ∴Q); Modus Tollens (P→Q, ¬Q, ∴¬P); why Denying the Antecedent and Affirming the Consequent are invalid; and vacuous truth (false antecedent = always true). Intermediate (15): The contrapositive (¬Q→¬P), converse (Q→P), and inverse (¬P→¬Q); which pairs are logically equivalent; the biconditional (P↔Q); the disjunction equivalence (P→Q ≡ ¬P∨Q); the negation of a conditional; and hypothetical syllogism. Advanced (15): Chained conditionals; tautologies; knowing P→Q is false forces P=T and Q=F; combining conditionals to prove biconditionals; determining Q from both P→Q and ¬P→Q; and the logical meaning of P→P.
Interactive Truth Table Lobby
The lobby displays the complete truth table for P→Q with color-coded truth values — green chips for TRUE, red for FALSE — and highlights the one row that produces FALSE. This serves as a persistent reference and teaches the central mechanic before the quiz begins.
Semantic Choice Coloring
Answer tiles that say TRUE are subtly tinted green and tiles that say FALSE are tinted red even before selection, matching their semantic meaning. This builds the association between truth value labels and their meaning, rather than treating all choices as visually identical.
Monospace Logic Notation
Questions and explanations use standard logical notation: P→Q, ¬P, P∧Q, P∨Q, P↔Q, ∴. The monospace font makes logical structure scannable and mirrors how conditionals appear in discrete math, computer science, and SAT/LSAT formal reasoning sections.
Frequently Asked Questions
Why is 'If false, then anything' TRUE?
This is called vacuous truth. A conditional 'If P, then Q' is a promise: 'Whenever P happens, Q will happen.' If P never happens (P is false), the promise is never tested — and an untested promise can't be broken. So the conditional is technically true. This feels odd but is logically consistent: a false antecedent can never violate the conditional.
What's the difference between the contrapositive, converse, and inverse?
Original: If P, then Q (P→Q). Contrapositive: If not Q, then not P (¬Q→¬P) — EQUIVALENT to original. Converse: If Q, then P (Q→P) — NOT equivalent. Inverse: If not P, then not Q (¬P→¬Q) — NOT equivalent (but equivalent to the converse). Memory trick: Original and Contrapositive are equivalent; Converse and Inverse are equivalent to each other.
What is Modus Ponens?
Modus Ponens ('affirming the antecedent'): P→Q, P, ∴Q. If the conditional holds and the antecedent is true, the consequent must be true. Example: 'If it rains, the ground gets wet. It is raining. Therefore, the ground is wet.' This is the most basic and universally valid form of deductive inference.
What is Modus Tollens?
Modus Tollens ('denying the consequent'): P→Q, ¬Q, ∴¬P. If the conditional holds and the consequent is FALSE, the antecedent must be false. Example: 'If it rains, the ground gets wet. The ground is NOT wet. Therefore, it did NOT rain.' This is valid because the conditional would be violated (P=T, Q=F) if P were true.
Why is 'Affirming the Consequent' invalid?
Affirming the Consequent: P→Q, Q, ∴P is invalid. Even if the conditional holds and Q is true, P might be false — Q could have been caused by something else. Example: 'If it rains, the ground gets wet. The ground is wet. Therefore, it rained.' Invalid — the ground could be wet from a sprinkler. The conditional doesn't say rain is the ONLY cause of wetness.
How does scoring work?
Correct answers earn 10 pts (Foundation), 15 pts (Intermediate), or 20 pts (Advanced). Consecutive correct answers add a 5-point streak bonus per answer after the first. A wrong answer resets the streak to zero.