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
Go to practice ↓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
- Assume: a correct implementation passes this test. Passing the test does not exclude an incorrect implementation that happens to pass.
- Let P = correct, Q = passes. Set P = F and Q = T. Both P → Q and Q are true, but P is false.
- That same assignment also refutes denying the antecedent: P → Q and ¬P hold, but ¬Q fails.
Your turn
0 / 3Refute affirming the consequent. Set P and Q so every premise is true and the conclusion false.
Read solution · does not award completion
P = F · Q = T
P = F and Q = T preserves both premises. Q can hold without P.
Now refute denying the antecedent.
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.
P → Q and ¬Q are given. What follows deductively?
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.