Which of the following statements is true ?
2017
Which of the following statements is true ?
- A.
The sentence 𝑆 is a logical consequence of 𝑆1,…,𝑆𝑛 if and only if 𝑆1 ∧ 𝑆2 ∧⋯∧ 𝑆𝑛 → 𝑆 is satisfiable.
- B.
The sentence 𝑆 is a logical consequence of 𝑆1,…,𝑆𝑛 if and only if 𝑆1 ∧ 𝑆2 ∧⋯∧ 𝑆𝑛 → 𝑆 is valid.
- C.
The sentence 𝑆 is a logical consequence of 𝑆1,…,𝑆𝑛 if and only if 𝑆1 ∧ 𝑆2 ∧⋯∧ 𝑆𝑛 ∧ ¬𝑆 is consistent.
- D.
The sentence 𝑆 is a logical consequence of 𝑆1,…,𝑆𝑛 if and only if 𝑆1 ∧ 𝑆2 ∧⋯∧ 𝑆𝑛 ∧ 𝑆 is inconsistent.
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Correct answer: B
Key idea: a sentence S is a logical consequence of premises S1,...,Sn exactly when every interpretation that satisfies all premises also satisfies S.
If (S1 ∧ S2 ∧ ... ∧ Sn) → S is valid, then in every interpretation where all premises S1,...,Sn hold the implication yields S, so S is a logical consequence.
Conversely, if S is a logical consequence of S1,...,Sn, then every interpretation that makes all premises true also makes S true; hence (S1 ∧ S2 ∧ ... ∧ Sn) → S is true in every interpretation, i.e., valid.
Alternative (equivalent) characterization:
S is a logical consequence of S1,...,Sn iff (S1 ∧ S2 ∧ ... ∧ Sn ∧ ¬S) is unsatisfiable (has no model).
This follows because a model of S1,...,Sn that falsifies S is exactly a model of S1 ∧ ... ∧ Sn ∧ ¬S; absence of such a model is equivalent to entailment.
Conclusion: the correct characterization of logical consequence is that the implication (S1 ∧ S2 ∧ ... ∧ Sn) → S is valid (true in every interpretation).