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We know what A ⊢ ⊥ means: it means that there's a proof from premises A of ⊥.

A ⊢ ⊥

What does putting the line through mean? Just that there's no proof of ⊥ from premises A.

A ⊬ ⊥

This is also easy to understand ...

A ⊨ B

A ⊭ B

An argument is **logically valid **just if there’s no possible situation in which the premises are true and the conclusion false.

But what does A ⊨ ⊥ mean?

The argument is logically valid ... no possible situation where ... but ⊥ is always false, so no possible situation where premises A are all true

A ⊨ ⊥

A ⊭ ⊥

A ⊨_{TT}⊥

A ⊭_{TT}⊥