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For example: Prove that (something)⊨(another thing)

Is it the same as "Prove that (something)⊢(another thing)"?

The single turnstile symbol always appears during sample proofs in my lecture notes. Yet my homework question suddenly got the double turnstile symbol, am I supposed to take it as a single turnstile symbol and do syntactical proving using natural deduction? Thanks!

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$\vDash$ usually refers to semantic entailment. In the presence of a soundness theorem, $P \vdash Q$ implies $P \vDash Q$. – Zhen Lin Sep 2 '12 at 9:47
The person in the best place to answer your question is the person who assigned the homework. – Gerry Myerson Sep 2 '12 at 11:26
Possibly useful: What is the difference between ⊢ and ⊨? – MJD Sep 2 '12 at 16:17

$\vDash$ stands for semantic truth rather than provabilty. It has two common uses:

  • $M\vDash\phi$ where $M$ is a structure, means that formula $\phi$ is always true in $M$. (For ordinary first-order logic, $M$ would consist of a non-empty universe plus concrete realizations for all functions and predicates in the language of $\phi$. For other logics it may be a stranger beast, such as a Kripke structure).

  • $T\vDash\phi$ where $T$ is a theory, means that $M\vDash\phi$ for every structure $M$ that satisfies the axioms of $T$.

The soundness and completeness properties of a formal system state that $T\vDash\phi$ if and only if $T\vdash\phi$ -- but if you're being asked specifically to argue for $T\vDash\phi$, you're probably supposed to do it by arguing more explicitly at the semantic level.

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