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07 / 18 · Try to break the inference

Four similar-looking inferences

Distinguish modus ponens and modus tollens from two invalid lookalikes.

Builds on Same truth in every row

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A question to keep in mind

The test passed. Does that prove the implementation is correct?

From P → Q and P, infer Q: this is modus ponens. From P → Q and ¬Q, infer ¬P: this is modus tollens. Both preserve truth when their premises are true. The conditional premise itself may still be wrong in a real case.

Affirming the consequent infers P from P → Q and Q. It is invalid: Q could hold for another reason. Denying the antecedent infers ¬Q from P → Q and ¬P. It is also invalid: the rule never said P was the only route to Q.

Do not diagnose a fallacy from a phrase alone. If Q → P is also a premise, inferring P from Q can be valid. First write all the premises, then check the form.

Work through an example

  1. Assume: a correct implementation passes this test. Passing the test does not exclude an incorrect implementation that happens to pass.
  2. Let P = correct, Q = passes. Set P = F and Q = T. Both P → Q and Q are true, but P is false.
  3. That same assignment also refutes denying the antecedent: P → Q and ¬P hold, but ¬Q fails.

Your turn

0 / 3
Practice 1Not checked

Refute affirming the consequent. Set P and Q so every premise is true and the conclusion false.

Your answer
Premises · all must be trueP → QQ
Conclusion · must be falseP
Read solution · does not award completion

P = F · Q = T

P = F and Q = T preserves both premises. Q can hold without P.

Practice 2Not checked

Now refute denying the antecedent.

Your answer
Premises · all must be trueP → Q¬P
Conclusion · must be false¬Q
Read solution · does not award completion

P = F · Q = T

P = F, Q = T gives true premises and a false ¬Q. Sufficient does not mean necessary.

Practice 3Not checked

P → Q and ¬Q are given. What follows deductively?

Your answer
Read solution · does not award completion

¬P

This is modus tollens. Its validity does not independently verify the real-world conditional.

Bring it back to your own work

Find a 'test passed, therefore correct' claim. Describe a concrete undetected fault instead of only naming a fallacy.

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Sources, scope, and corrections

Original Fineuralab teaching examples; fictional scenarios are not reported research results. Edition 20260903-v1. Page updated 2026-09-03.

  • forall x: Calgary

    Open textbook for further study of formal logic. Used to cross-check scope and terminology; these lessons and exercises are newly authored, not a translation of its chapters.

  • Organizing Instruction and Study to Improve Student Learning

    IES practice guide informing worked examples, retrieval, and concrete/abstract connections. It has not evaluated this course.

  • Carnap

    A separate open-source platform for more extensive formal-logic exercises and proof work.

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