08 / 18 · Try to break the inference
Build the case that breaks it
Search systematically for true premises and a false conclusion.
Builds on Four similar-looking inferences
Go to practice ↓A question to keep in mind
How do you test a rule without just collecting examples that agree?
To test whether premises entail a conclusion, start by trying to make the conclusion false. Then see whether every premise can stay true. This search direction is useful because many assignments with a true conclusion cannot expose invalidity.
For P ∨ Q, ¬P therefore Q, a false Q forces P true to satisfy the first premise, but then ¬P fails. There is no counterexample. With a small number of letters, checking every assignment is a complete decision procedure.
A counterexample need not be typical or probable. It must be allowed by the stated premises. For empirical claims, whether a proposed case is physically possible or accurately described is an additional substantive question.
Work through an example
- Claimed inference: P → Q, Q → R, therefore R. A chain alone does not assert that the chain starts.
- Set R = F. To preserve Q → R choose Q = F. Then choose P = F to preserve P → Q.
- Both conditionals hold, but R does not. Adding P as a premise would change the argument.
Your turn
0 / 3Break the unsupported chain conclusion.
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P = F · Q = F · R = F
All three letters false is a counterexample. Neither premise asserts P or Q.
Check the whole argument as one conditional: ((P ∨ Q) ∧ ¬P) → Q.
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- P=T, Q=T →
((P ∨ Q) ∧ ¬P) → Q: T - P=T, Q=F →
((P ∨ Q) ∧ ¬P) → Q: T - P=F, Q=T →
((P ∨ Q) ∧ ¬P) → Q: T - P=F, Q=F →
((P ∨ Q) ∧ ¬P) → Q: T
Every row is true, so the conjunction of the premises entails Q. This is disjunctive syllogism.
You tried three of eight assignments and found no counterexample. What can you claim?
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No counterexample was found in the three checked assignments.
State the coverage of a test. This habit also matters when evaluating software or scientific models.
Bring it back to your own work
Choose one inference you believe. Try to construct its strongest counterexample while leaving all stated premises intact.
How do I retain independent judgment when using AI here?
Keep your own answer before consulting an AI candidate. Check sources, calculations, inference and conditions; explain adoption or rejection. Then solve a new-context task independently and apply the checks to this lesson.
Practice the verification workflow and keep a recordMy learning status
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