what is the symbol $⊭$ in logic?

I know that $$⊨$$ symbol is entails symbol and $$A⊨B$$ means that if A is True then B must be True.

But I'm confused about the $$⊭$$ symbol. which one is it?

• $$A⊭B$$ means if A is true then B is False? = $$A⊨¬B$$
• $$A⊭B$$ means the Trueness of A is not any guarantee for B?
• $$A⊭B$$ means if A is False then B must be True? = $$¬A⊨B$$

thank you in advance.

• Clearly not the third, as that is consistent with $A⊨B$ – Henry Jun 13 at 21:05
• @Henry, one and three are consistent with that. the problem was I couldn't find any resource for it on the web! all is just about $⊨$. – Peyman mohseni kiasari Jun 13 at 21:09
• @Peymanmohsenikiasari No, option 1 is not consistent with $A \vDash B$. If $A \vDash B$, then all assignments which make $A$ true must also make $B$ true, but then $\neg B$ can never be the case. – lemontree Jun 13 at 21:11
• @Henry Oh! my mistake. thank you. – Peyman mohseni kiasari Jun 13 at 21:12

1 Answer

$$A \vDash B$$ 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.

• well, so if $A⊭B$ then A must be satisfiable I think. nice, thank you. – Peyman mohseni kiasari Jun 13 at 21:11
• @Peymanmohsenikiasari Correct. Because if $A$ is not satisfiable (= is a contradiction), then it entails any formula. – lemontree Jun 13 at 21:12
• 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 Jun 13 at 22:16