Take the 2-minute tour ×
Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It's 100% free, no registration required.

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!

share|improve this question
2  
$\vDash$ 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
2  
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
1  
Possibly useful: What is the difference between ⊢ and ⊨? –  MJD Sep 2 '12 at 16:17
add comment

1 Answer 1

$\vDash$ 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.

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.