What does the semantic entailment relation mean (M |= A)?

Question

I read many acticles about it. They described it as :

In logics, meaning is often described by a satisfaction relation

M |= A

that describes when a situation M satisfies a formula A.

So, I also searched some examples. I found the examples following :

True |= False = false

False |= True = true

I don't understand at all. What does it mean in these cases?

Answer 1

(assuming that you talk about propositional logic (it is similar for other logics such as pred. logic))

for two formulas A and B:

A |= B 
"B evaluates to true under all evaluations that evaluate A to true"

for a set of formulas M and a formula B:

M |= B
"for every evaluation: B evaluates to true if only all elements of M
 evaluate to true"

coming to your examples:

true |= false 
is incorrect since evaluations exist


false |= A
is correct for any formula A, since 'false' is never evaluated to 'true'
under any evaluation

rgrds.