Which of the following correctly describes what Γ ⊨ φ means?
AThere exists a proof of φ from Γ using inference rules
BEvery interpretation that satisfies all formulas in Γ also satisfies φ
Cφ logically implies every formula in Γ
Dφ is a tautology whenever Γ contains at least one tautology
Γ ⊨ φ is a semantic (model-theoretic) definition: φ holds in every interpretation where all of Γ holds. It makes no reference to proofs or derivation rules. Option (a) describes syntactic derivability (Γ ⊢ φ), which is related but distinct — they coincide only because of the soundness and completeness theorems. Options (c) and (d) get the direction and conditions wrong.
Question 2 True / False
If A ⊨ B (A semantically entails B), then the material conditional A → B is a tautology.
TTrue
FFalse
Answer: True
If every interpretation satisfying A also satisfies B, then no interpretation makes A true and B false. But making A true and B false is the only way to make A → B false. So A → B is true in every interpretation — it is a tautology. This equivalence is the deduction theorem for semantic consequence: {A} ⊨ B if and only if ⊨ (A → B).
Question 3 Short Answer
What is the difference between semantic entailment (Γ ⊨ φ) and syntactic derivability (Γ ⊢ φ), and what theorem relates them?
Think about your answer, then reveal below.
Model answer: Semantic entailment holds when every model of Γ satisfies φ (truth-based). Syntactic derivability holds when there is a formal proof of φ from Γ using inference rules (proof-based). The soundness and completeness theorems together establish that Γ ⊨ φ if and only if Γ ⊢ φ for standard proof systems.
Soundness says every derivable formula is semantically valid (proofs preserve truth), and completeness says every semantic entailment is derivable (truth implies provability). Together they show the proof system captures exactly the semantic content of the logic — a deep result meaning you can study truth by studying proofs and vice versa.