09 / 18 · Try to break the inference
When the premises cannot all hold
Recognize consistency, tautology, contradiction, and vacuous consequence.
Builds on Build the case that breaks it
Go to practice ↓A question to keep in mind
Can an argument be valid precisely because its premises are impossible?
A set of premises is satisfiable, or semantically consistent here, if at least one assignment makes them all true. A tautology is true under every assignment; a contradiction is false under every assignment. These are model-wide properties, not observations of one row.
P and ¬P have no shared true assignment. Therefore there can be no assignment making both premises true and any conclusion false. Under classical entailment, inconsistent premises entail every conclusion. This is sometimes called explosion.
This does not give us knowledge of arbitrary facts. An inconsistent set cannot be a set of true premises in this model, so an argument relying on it cannot be sound. Diagnose and repair the inconsistency rather than celebrate a formally valid output.
Work through an example
- A specification demands both 'the file must be public' and 'the file must not be public' under the same conditions and meaning.
- No configuration satisfies both. A checker reporting implication success from these requirements may be detecting an empty set of valid configurations.
- Check satisfiability as well as implication. Different times or meanings might remove the apparent contradiction, so fix the interpretation first.
Your turn
0 / 3Compare a tautology, a contradiction, and a conditional with an impossible antecedent.
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- P=T, Q=T →
P ∨ ¬P: T;P ∧ ¬P: F;(P ∧ ¬P) → Q: T - P=T, Q=F →
P ∨ ¬P: T;P ∧ ¬P: F;(P ∧ ¬P) → Q: T - P=F, Q=T →
P ∨ ¬P: T;P ∧ ¬P: F;(P ∧ ¬P) → Q: T - P=F, Q=F →
P ∨ ¬P: T;P ∧ ¬P: F;(P ∧ ¬P) → Q: T
The columns are always T, always F, and always T respectively. The last column does not establish that Q is actually true.
Which premise set is satisfiable?
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P ∨ Q, ¬P
A single satisfying assignment is enough to establish satisfiability.
A rule checker says every output is allowed, but its requirements contradict each other. What is the useful next step?
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Check and repair the inconsistent requirements.
The lesson is to inspect assumptions and the model's limits, not to abandon rigorous reasoning.
Bring it back to your own work
Write two apparently conflicting project requirements. Specify time and meaning; decide whether they really conflict or merely sound similar.