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I know that the symbol $⊨$ means entails, and that $A⊨B$ means that if A is true then B must be true. However, I'm confused about the $\large⊭$ symbol. Which of the following is it?

  1. $A⊭B$ means if A is true then B is false. That is, $A⊨¬B$
  2. $A⊭B$ means the trueness of A is not any guarantee of B.
  3. $A⊭B$ means if A is false then B must be true. That is, $¬A⊨B.$
ryang
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Peyman
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1 Answers1

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$A \vDash B$ means

For all assignments $v$, if $A$ is true under $v$, then $B$ is true under $v$.

$A \not \vDash B$ simply is the (meta-logical) negation of this statement, that is

Not for all assignments $v$ it is the case that if $A$ is true under $v$, then $B$ is true under $v$

which is equivalent to

There is at least one assignment $v$ such that $A$ is true under $v$ but $B$ is not

which means your second option is the right one.

The other two options (1 and 3) are notated in the way you already figured out by yourself.

Natalie Clarius
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  • well, so if $A⊭B$ then A must be satisfiable I think. nice, thank you. – Peyman Jun 13 '19 at 21:11
  • @Peymanmohsenikiasari Correct. Because if $A$ is not satisfiable (= is a contradiction), then it entails any formula. – Natalie Clarius Jun 13 '19 at 21:12
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    To emphasize, $A\nvDash B$ literally means "it is not the case that $A\vDash B$". That is, it is simply the meta-logical negation of $A\vDash B$. That is why there is not special discussion of it. We can, as lemontree does, work out a more pleasant, equivalent formulation. – Derek Elkins left SE Jun 13 '19 at 22:16