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Logic

Valid Argument Forms

Lesson

Four classic valid argument forms that always produce a true conclusion from true premises.

Modus ponens (MP)

p→q, p ∴ qp \to q, \ p \ \therefore \ q

Modus tollens (MT)

p→q, ¬q ∴ ¬pp \to q, \ \neg q \ \therefore \ \neg p

Hypothetical syllogism (HS)

p→q, q→r ∴ p→rp \to q, \ q \to r \ \therefore \ p \to r

Disjunctive syllogism (DS)

p∨q, ¬p ∴ qp \vee q, \ \neg p \ \therefore \ q

Common fallacies (INVALID)

  • Affirming the consequent: p→q, q ∴ pp \to q, \ q \ \therefore \ p.
  • Denying the antecedent: p→q, ¬p ∴ ¬qp \to q, \ \neg p \ \therefore \ \neg q.

How to type your answer

1 MP, 2 MT, 3 HS, 4 DS, 5 invalid.

Practice

Work through these. Stuck? Click Get a hint.

Warm-Up

Quick problems to get going.

Problem 1

p→q, p ∴qp \to q, \ p \ \therefore q

Problem 2

p→q, ¬q ∴¬pp \to q, \ \neg q \ \therefore \neg p

Problem 3

p→q, q→r ∴p→rp \to q, \ q \to r \ \therefore p \to r

Problem 4

p∨q, ¬p ∴qp \vee q, \ \neg p \ \therefore q

Practice

Standard problems matching the lesson.

Problem 5

If rains, ground wet. Raining. So ground wet.\text{If rains, ground wet. Raining. So ground wet.}

Problem 6

If sun out, shadow. No shadow. So no sun.\text{If sun out, shadow. No shadow. So no sun.}

Problem 7

If A then B. If B then C. So if A then C.\text{If A then B. If B then C. So if A then C.}

Problem 8

P or Q. Not P. So Q.\text{P or Q. Not P. So Q.}

Problem 9

p→q, q ∴pp \to q, \ q \ \therefore p

Problem 10

p→q, ¬p ∴¬qp \to q, \ \neg p \ \therefore \neg q

Problem 11

If even, then div 2. n even. So n div 2.\text{If even, then div 2. n even. So n div 2.}

Problem 12

If rains, wet. Not wet. So no rain.\text{If rains, wet. Not wet. So no rain.}

Problem 13

If pass, graduate. If graduate, job. So if pass, job.\text{If pass, graduate. If graduate, job. So if pass, job.}

Problem 14

Hot dog or burger. Not hot dog. So burger.\text{Hot dog or burger. Not hot dog. So burger.}

Problem 15

p→q, q ∴pp \to q, \ q \ \therefore p

Problem 16

p→q, ¬p ∴¬qp \to q, \ \neg p \ \therefore \neg q

Problem 17

If x>5, then x>3. x>5. ∴x>3.\text{If } x > 5, \text{ then } x > 3. \ x > 5. \ \therefore x > 3.

Problem 18

If cube, has 6 faces. Not 6 faces. So not cube.\text{If cube, has 6 faces. Not 6 faces. So not cube.}

Challenge

Harder problems — edge cases, trickier numbers, multiple steps.

Problem 19

If snows, school closes. Snowing. So closes.\text{If snows, school closes. Snowing. So closes.}

Problem 20

If snows, school closes. Open. So no snow.\text{If snows, school closes. Open. So no snow.}

Problem 21

A→B,B→C∴A→CA \to B, B \to C \therefore A \to C

Problem 22

a∨b, ¬a ∴ba \vee b, \ \neg a \ \therefore b

Problem 23

A→B, B ∴AA \to B, \ B \ \therefore A

Problem 24

A→B, ¬A ∴¬BA \to B, \ \neg A \ \therefore \neg B

Problem 25

If bird, can fly. Cannot fly. So not bird.\text{If bird, can fly. Cannot fly. So not bird.}

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